$((2X_{1}-1)\partial_{2}^{2}+3\partial_{2}+X_{1}^{2})/X_{1},\quad\partial_{1}-(2X_{2}+1)/X_{1}\partial_{2}-1-1/X_{1}.$

This renders the elements in the $\Gamma^{k}\Theta$ are spanned by
$\\{X_{1}^{a}X_{2}^{b}:a+b\leq k\\}$ and
$\\{X_{1}^{a^{\prime}{}}X_{2}^{b^{\prime}{}}\partial_{2}:a+b\leq k-1\\}$.
Hence the $\Theta$ shares the same Hilbert polynomial with that studied in
Example 2.26 so that

$\dim_{\mathbb{C}}\mathrm{Gr}^{k}M=k^{2}.$

It follows that the $\Theta$ is holonomic, hence dimension two and with
multiplicity two. ∎

Eliminating $\partial_{1}$ again from the above two relations of
$\mathcal{A}_{2}$ from the Definition $\Theta$ (4.9) yields the following
element of $\mathcal{A}_{2}$-module.

######  Proposition 4.11 ((Three-term recursion)).

The natural map

$\dfrac{\mathcal{A}_{2}}{\mathcal{A}_{2}[(2X_{1}-1)\partial_{2}^{2}+3\partial_{2}+X_{1}^{2}]}\longrightarrow\Theta$

is a well-defined left $\mathcal{A}_{2}$-linear surjection.

The following records the relationship between the half-Bessel module $\Theta$
defined above and the second order half-Bessel operator $L_{n}$.

######  Proposition 4.12 ((Weyl-algebraic reverse Bessel-polynomial sequence
of $\Theta$)).

The natural map

$\dfrac{\mathcal{A}_{2}}{\mathcal{A}_{2}[X_{1}\partial_{1}^{2}-2(X_{1}+X_{2}\partial_{2})\partial_{1}+2X_{2}\partial_{2}]}\longrightarrow\Theta$

is a well-defined left $\mathcal{A}_{2}$-linear surjection.

The derivation of this result is given in the Appendix A.2.

###### Proof.

See the Appendix. ∎

####  4.2.2. Classical reverse Bessel polynomials

To study Poisson generating functions, i.e., generating functions of the form
$\displaystyle\sum_{\mathbb{N}}a_{n}{t^{n}}/{n!}$, one uses the $D$-module
structure endowed on spaces $\mathbb{C}^{\mathbb{N}_{0}}$ and
$\mathcal{O}^{\mathbb{N}_{0}}$ of (one-sided) sequences as in Example 2.15.
The purpose of the section is to illustrate that many classically known
defining formulae about the reverse Bessel polynomials can be derived and
presented in a coherent manner as consequences of manifestation of the
$D$-module structures discussed in the last section. We shall show that the
same $D$-module structure with a different manifestation would allow us to
reach an analogous theory for difference reverse Bessel polynomials in the
next subsection. Thus except for the transmutation formulae derived from
above, the remaining formulae that we derive here are well-known in the
classical Bessel polynomial literature, e.g., [32]. Reader who are looking for
new formulae about difference Bessel polynomials may proceed directly to the
next section.

We deduce from Theorem 4.6 the expected result:

######  Proposition 4.13.

Let $n\in\mathbb{N}_{0}$. Then the left $\mathcal{A}$-linear map

$\begin{array}[]{rclll}\mathcal{A}/\mathcal{A}\big{(}X\partial^{2}-2(n+X)\partial+2n)\big{)}&\stackrel{{\scriptstyle\times\Theta_{n}(X)}}{{\longrightarrow}}&\mathcal{A}/\mathcal{A}\partial&\stackrel{{\scriptstyle\times
1}}{{\longrightarrow}}&\mathcal{O}\end{array}$

sends $1$ to the classical reversed Bessel polynomial

$\theta_{n}(x)=\sum_{k=0}^{n}\frac{(n+k)!}{2^{k}(n-k)!\,k!}\,x^{n-k}.$

######  Example 4.14.

Consider the sequence of classical Bessel polynomials $(\theta_{n})$. Then the
maps

$\begin{array}[]{rcl}\mathfrak{j}:\Theta&\stackrel{{\scriptstyle\times(\theta_{n})}}{{\longrightarrow}}&\mathcal{O}^{\mathbb{N}_{0}}\end{array}$
(4.14)

is well-defined and left $\mathcal{A}_{2}$-linear.

The following differential-difference equations and three-term recurrence, as
well as Corollaries 4.7 and 4.8 about classical Bessel polynomials below
present solution of the module $\Theta$ in $\mathcal{O}_{d}$. They follow from
Example 4.14, hence giving new proofs to these classical results. We shall see
how solutions of the module $\Theta$ in $\mathcal{O}_{\Delta}$ gives their
difference analogues below.

######  Theorem 4.15 ((differential-difference equations [32], p. 19)).

The reverse classical Bessel polynomials $\\{\theta_{n}(x)\\}$ satisfy the
following (well-known) formulae

$\begin{split}&\theta_{n}^{\prime}(x)-\theta_{n}(x)+x\theta_{n-1}(x)=0,\\\
&x\theta_{n}^{\prime}(x)-(x+2n+1)\theta_{n}(x)+\theta_{n+1}(x)=0\end{split}$
(4.15)

for all $n\geq 1$ and $x$.

###### Proof.

This follows from Corollary 4.7 with the manifestation of analytic function
space $\mathcal{O}_{d}$ as a $\mathcal{A}_{1}$-module given by the Example
2.3. ∎

######  Proposition 4.16 ((Three-term recurrence [32], p. 18)).

The sequence $(\theta_{n})$ satisfies the (well-known) three-term recurrence
formula

$\theta_{n+2}(x)-(2n+3)\theta_{n+1}(x)-x^{2}\theta_{n}(x)=0$ (4.16)

for every $n\in\mathbb{N}_{0}$ and every $x$.

###### Proof.

This follows from the last Corollary or directly from the Corollary 4.8 with
the manifestation of the $\mathcal{A}_{1}$-module $\mathcal{O}_{d}$ as given
by the Example 2.3. ∎

Recently, there are also complex analytic studies of the PDEs satisfied by
generating functions of some classical special functions, including the Bessel
polynomials [34, 33].

####  4.2.3. Difference reverse Bessel polynomials

Now we consider the left $\mathcal{A}_{2}$-module structure of
$\mathcal{O}_{\Delta}^{\mathbb{N}_{0}}$ as in Example 2.16. Recalling that
$\Theta_{n}$ is a solution of $L_{n}:=X\partial^{2}-2(n+X)\partial+2n$, we
immediately obtain the following

######  Theorem 4.17 ((Difference equation)).

For each integer $n\geq 0$, let $\theta^{\Delta}_{n}$ be the difference
reverse Bessel polynomial of degree $n$ defined by

$\theta^{\Delta}_{n}(x):=\sum_{k=0}^{n}\frac{(n+k)!}{2^{k}(n-k)!\,k!}\,(x)_{n-k}.$
(4.17)

Then for each integer $n\geq 0$ and each $x$, we have

$(x-2n)\,\theta_{n}^{\Delta}(x+1)-4(x-n)\,\theta_{n}^{\Delta}(x)+3x\,\theta_{n}^{\Delta}(x-1)=0.$
(4.18)

The following delay-differential relations and three-term recurrence are
direct consequences of the module $\Theta$, as well as Corollaries 4.7 and 4.8
with $N=\mathcal{O}_{\Delta}$.

######  Theorem 4.18 ((Delay-difference relations)).

The sequence of difference reverse Bessel polynomials $(\theta^{\Delta}_{n})$
satisfy the following two delay-difference equations

$\begin{split}\theta^{\Delta}_{n}(x+1)-2\theta^{\Delta}_{n}(x)+x\theta^{\Delta}_{n-1}(x-1)=0,\\\
\theta^{\Delta}_{n+1}(x)+(x-2n-1)\theta^{\Delta}_{n}(x)-2x\,\theta^{\Delta}_{n}(x-1)=0,\end{split}$
(4.19)

for every $n\geq 1$ and every $x$.

###### Proof.

This follows from Corollary 4.7 with the manifestation of the analytic
function space $\mathcal{O}_{\Delta}$ as a $\mathcal{A}_{1}$-module given by
the Example 2.4. ∎

It follows from Proposition 4.11 and the structure of the above
$\mathcal{A}_{2}$-module the new three-term recursion formula for the reverse
difference Bessel polynomials.

######  Theorem 4.19 ((Three-term recurrence)).

The sequence of difference reversed Bessel polynomials $(\theta^{\Delta}_{n})$
satisfies the following recurrence relation

$\theta^{\Delta}_{n+2}(x)-(2n+3)\theta^{\Delta}_{n+1}(x)-x(x-1)\theta^{\Delta}_{n}(x-2)=0$
(4.20)

for every $n\in\mathbb{N}_{0}$ and every $x$.

###### Proof.

This follows from the last Corollary or directly from the Corollary 4.8 with
the manifestation of the analytic function space $\mathcal{O}_{d}$ as a
$\mathcal{A}_{1}$-module given by the Example 2.4. ∎

###  4.3. Characteristics of modified half-Bessel modules

We recall from (4.12) the definition of the half-Bessel module. We may change
one of the two generators from the half-Bessel module (4.12) by
$(2X_{2}-1)\partial_{2}^{2}+3\partial_{2}+X_{1}^{2}$, as suggested by
Corollary 4.8.

######  Theorem 4.20.

The half-Bessel module $\Theta$ is isomorphic to the left
$\mathcal{A}_{2}$-module

$\dfrac{\mathcal{A}_{2}}{\mathcal{A}_{2}[(2X_{2}-1)\partial_{2}^{2}+3\partial_{2}+X_{1}^{2}]+\mathcal{A}_{2}[X_{1}\partial_{1}+(1-2X_{2})\partial_{2}-1-X_{1}]}.$

######  Theorem 4.21.

The map

$\dfrac{\mathcal{A}_{2}}{\mathcal{A}_{2}[(2X_{2}-1)\partial_{2}^{2}+3\partial_{2}+X_{1}^{2}]+\mathcal{A}_{2}[X_{1}\partial_{1}+(1-2X_{2})\partial_{2}-1-X_{1}]}\longrightarrow\Theta$

is a left $\mathcal{A}_{2}$-linear isomorphism.

######  Theorem 4.22.

Let

$\mathcal{A}_{2}(\rho):=\frac{\mathcal{A}_{2}(\rho,{1}/{\rho})}{\langle\rho^{2}-1+2X_{2}\rangle}$

and let $S$ be the element

$S=C_{1}\rho^{-1}{\mathop{\rm E}}[X_{1}(1-\rho)]+C_{2}\rho^{-1}{\mathop{\rm
E}}[X_{1}(1+\rho)],$ (4.21)

for every choice of $C_{1},\,C_{2}\in\mathbb{C}$, where the ${\mathop{\rm E}}$
is the Weyl exponential defined in Example 3.3. Then the map

$\begin{split}\Theta(\rho):=\dfrac{\mathcal{A}_{2}(\rho)}{\mathcal{A}_{2}(\rho)[(2X_{2}-1)\partial_{2}^{2}+3\partial_{2}+X_{1}^{2}]+\mathcal{A}_{2}(\rho)[X_{1}\partial_{1}+(1-2X_{2})\partial_{2}-1-X_{1}]}\\\
\stackrel{{\scriptstyle\times
S}}{{\longrightarrow}}\overline{\dfrac{\mathcal{A}_{2}(\rho)}{\mathcal{A}_{2}(\rho)\partial_{1}+\mathcal{A}_{2}(\rho)\partial_{2}}}\end{split}$
(4.22)

is left $\mathcal{A}_{2}(\rho)$-linear.

###### Proof.

Let

$\Lambda_{1}=\partial_{1}\quad\Lambda_{2}=\rho\partial_{2},\quad
Z_{1}=X_{1},\quad Z_{2}=-\rho,$ (4.23)

where $\rho^{2}=1-2X_{2}$. Since

$\partial_{2}\rho-\rho\partial_{2}=-\frac{1}{\rho},$

we deduce from direct computations that

$[\Lambda_{1},Z_{1}]=1,\quad[\Lambda_{2},Z_{2}]=1,\quad[\Lambda_{1},\Lambda_{2}]=0,\quad[Z_{1},Z_{2}]=0,\quad[\Lambda_{i},\,Z_{j}]=0,\quad
i\not=j$ (4.24)

hold.

It is routine to check that

$(2X_{2}-1)\partial_{2}^{2}+3\partial_{2}+X_{1}^{2}=-\Lambda_{2}^{2}-\frac{2}{Z_{2}}\Lambda_{2}+Z_{1}^{2}.$
(4.25)

Let us make use of the characteristic change of elements by (4.23) in the
second generator $X_{1}\partial_{1}+(1-2X_{2})\partial_{2}-1-X_{1}$ to obtain

$Z_{1}\Lambda_{1}-Z_{2}\Lambda_{2}-1-Z_{1}=Z_{1}(\Lambda_{1}-1)-\Lambda_{2}Z_{2}.$

Let
$\widehat{\mathcal{A}_{2}}=\langle\Lambda_{1},\,\Lambda_{2},\,Z_{1},\,Z_{2}\rangle$
and

$\widehat{\Theta}:=\frac{\widehat{\mathcal{A}_{2}}}{\widehat{\mathcal{A}_{2}}[(Z_{2}\Lambda_{2}^{2}+2\Lambda_{2}-Z_{2}Z_{1}^{2}]+\widehat{\mathcal{A}_{2}}[(Z_{1}(\Lambda_{1}-1)-\Lambda_{2}Z_{2})]}$

Note that the above quotient $\widehat{\mathcal{A}_{2}}$-module
$\widehat{\Theta}$ is holonomic with dimension and multiplicity both equal to
$2$, see e.g., Example 2.26.

In order to find the map

$\widehat{\Theta}\
\stackrel{{\scriptstyle\times\widehat{S}}}{{\longrightarrow}}\overline{\frac{\widehat{\mathcal{A}_{2}}}{{\widehat{\mathcal{A}_{2}}}\Lambda_{1}+{\widehat{\mathcal{A}_{2}}}\Lambda_{2}}},$

we left multiply the expression (4.25) by $Z_{2}$. Further simplification
yield

$\begin{split}(Z_{2}\Lambda_{2})\Lambda_{2}+2\Lambda_{2}-Z_{2}Z_{1}^{2}&=\Lambda_{2}^{2}Z_{2}-Z_{2}Z_{1}^{2}\\\
&=(\Lambda_{2}^{2}-Z_{1}^{2})Z_{2}\\\
&=(\Lambda_{2}+Z_{1})(\Lambda_{2}-Z_{1})Z_{2}.\end{split}$

Since the two first-order operators above commute, so we deduce that

$\frac{1}{Z_{2}}F(Z_{1})E(Z_{1}Z_{2})+\frac{1}{Z_{2}}G(Z_{1})E(-Z_{1}Z_{2})$

is the general solution to the above operator for some $F(Z_{1})$ and
$G(Z_{1})$. This is a sum of two solutions which also conform with our
knowledge that the $\widehat{\Theta}$ has multiplicity $2$ discussed above.

We now make use of the second generator of $\widehat{\Theta}$ to determine the
exact forms of the solutions $F$ and $G$. We first note that

$\displaystyle\Lambda_{1}{\mathop{\rm E}}(Z_{1}Z_{2})=Z_{2}{\mathop{\rm
E}}(Z_{1}Z_{2})+{\mathop{\rm E}}(Z_{1}Z_{2})\Lambda_{1},$ (4.26)
$\displaystyle\Lambda_{2}{\mathop{\rm E}}(Z_{1}Z_{2})=Z_{1}{\mathop{\rm
E}}(Z_{1}Z_{2})+{\mathop{\rm E}}(Z_{1}Z_{2})\Lambda_{2}.$ (4.27)

It follows from (4.26) and (4.27) that

$\begin{split}&\big{(}Z_{1}(\Lambda_{1}-1)-\Lambda_{2}Z_{2}\big{)}\frac{1}{Z_{2}}F(Z_{1}){\mathop{\rm
E}}(Z_{1}Z_{2})\\\ &=Z_{1}(\Lambda_{1}-1)F(Z_{1}){\mathop{\rm
E}}(Z_{1}Z_{2})-\Lambda_{2}F(Z_{1}){\mathop{\rm E}}(Z_{1}Z_{2})\\\
&=(Z_{1}/Z_{2})(\Lambda_{1}-1)F(Z_{1}){\mathop{\rm
E}}(Z_{1}Z_{2})-F(Z_{1})Z_{1}{\mathop{\rm E}}(Z_{1}Z_{2})-F(Z_{1}){\mathop{\rm
E}}(Z_{1}Z_{2})\Lambda_{2}\\\
&=(Z_{1}/Z_{2})(\Lambda_{1}-1)F(Z_{1}){\mathop{\rm
E}}(Z_{1}Z_{2})-F(Z_{1})Z_{1}{\mathop{\rm
E}}(Z_{1}Z_{2})\quad\mod\widehat{\mathcal{A}}_{2}\Lambda_{2}\\\
&=(Z_{1}/Z_{2})\big{(}{\mathop{\rm
E}}(Z_{1}Z_{2})\Lambda_{1}F(Z_{1})-F(Z_{1})\Lambda_{1}{\mathop{\rm
E}}(Z_{1}Z_{2})\big{)}-(Z_{1}/Z_{2})F(Z_{1})E(Z_{1}Z_{2})\\\
&\qquad-F(Z_{1})Z_{1}{\mathop{\rm E}}(Z_{1}Z_{2})\\\
&=(Z_{1}/Z_{2}){\mathop{\rm
E}}(Z_{1}Z_{2})\Lambda_{1}F(Z_{1})-(Z_{1}/Z_{2})F(Z_{1})Z_{2}{\mathop{\rm
E}}(Z_{1}Z_{2})-(Z_{1}/Z_{2})F(Z_{1})E(Z_{1}Z_{2})\\\
&\qquad-F(Z_{1})Z_{1}{\mathop{\rm E}}(Z_{1}Z_{2})\\\
&=(Z_{1}/Z_{2}){\mathop{\rm
E}}(Z_{1}Z_{2})\Lambda_{1}F(Z_{1})-(Z_{1}/Z_{2})F(Z_{1})E(Z_{1}Z_{2})\\\
&=(Z_{1}/Z_{2}){\mathop{\rm E}}(Z_{1}Z_{2})(\Lambda_{1}-1)F(Z_{1})\\\
&=0\quad\mod\widehat{\mathcal{A}}_{2}\Lambda_{1}+\widehat{\mathcal{A}}_{2}\Lambda_{2},\end{split}$

where the last equal sign holds provided that

$(\Lambda_{1}-1)F(Z_{1})=0\quad\mod\widehat{\mathcal{A}}_{2}\Lambda_{1}.$

But this equation could hold when we have chosen $F(Z_{1})=-{\mathop{\rm
E}}(Z_{1})$, see Example 3.3, except perhaps for a constant multiple.

It remains to consider the following and a similar calculation yields

$\begin{split}&\big{(}Z_{1}(\Lambda_{1}-1)-\Lambda_{2}Z_{2}\big{)}\frac{1}{Z_{2}}G(Z_{1}){\mathop{\rm
E}}(-Z_{1}Z_{2})\\\ &={Z_{1}}/{Z_{2}}\Big{(}{\mathop{\rm
E}}(-Z_{1}Z_{2})\Lambda_{1}G(Z_{1})-Z_{2}{\mathop{\rm
E}}(-Z_{1}Z_{2})G(Z_{1})\Big{)}-{Z_{1}}/{Z_{2}}G(Z_{1}){\mathop{\rm
E}}(-Z_{1}Z_{2})\\\ &\qquad-\Big{(}{\mathop{\rm
E}}(-Z_{1}Z_{2})\Lambda_{2}G(Z_{1})-Z_{1}{\mathop{\rm
E}}(-Z_{1}Z_{2})G(Z_{1})\Big{)}\qquad\mod\widehat{\mathcal{A}}_{2}\Lambda_{1}+\widehat{\mathcal{A}}_{2}\Lambda_{2}\\\
&={\mathop{\rm
E}}(-Z_{1}Z_{2})/Z_{2}\Big{(}Z_{1}(\Lambda_{1}-1)+Z_{2}\Lambda_{2}\Big{)}G(Z_{1})\qquad\mod\widehat{\mathcal{A}}_{2}\Lambda_{1}+\widehat{\mathcal{A}}_{2}\Lambda_{2}\\\
&=(1/Z_{2}){\mathop{\rm E}}(-Z_{1}Z_{2})Z_{1}(\Lambda_{1}-1)G(Z_{1})\\\
&=0\qquad\mod\widehat{\mathcal{A}}_{2}\Lambda_{1}+\widehat{\mathcal{A}}_{2}\Lambda_{2},\end{split}$

where the last equal sign would hold if

$G(Z_{1})={\mathop{\rm
E}}(Z_{1})\quad\mod\widehat{\mathcal{A}}_{2}\Lambda_{1},$

which may differ by a constant multiple. This establishes the (4.21) and hence
completes the proof. ∎

###  4.4. Bessel polynomials module

In this subsection, we define the Weyl Bessel polynomial of degree $n\geq 0$
by

$Y_{n}(X)=X^{n}\Theta_{n}(1/X)=\sum_{k=0}^{n}\frac{(n+k)!}{(n-k)!k!2^{k}}X^{k}.$
(4.28)

Right multiplication by $Y_{n}(X)$ gives a left $\mathcal{A}$-linear map

$\mathcal{A}/\mathcal{A}(X^{2}\partial^{2}+(2X+2)\partial-n(n+1)\stackrel{{\scriptstyle\times
Y_{n}(X)\ \ }}{{\longrightarrow}}\overline{\mathcal{A}/\mathcal{A}\partial}.$

We may further define when $n=-1$, then $Y_{-1}(X)=1$.

We deduce from Corollary 4.7 the followings.

######  Corollary 4.23.

For each $n$, let $Y_{n}$ be defined as above, which is a solution of
$X^{2}\partial^{2}+(2X+2)\partial-n(n+1)$ in a certain left
$\mathcal{A}$-module $\mathbb{C}[X]$. Then

$\displaystyle(X^{2}\partial+(n+1)X+1)Y_{n}-Y_{n+1}=0,$ (4.29)
$\displaystyle(X^{2}\partial-nX+1)Y_{n}-Y_{n-1}=0.$ (4.30)

as elements in $N$.

###### Proof.

Replacing $X$ by $1/X$ and $\partial$ by $-X^{2}\partial$ in Corollary 4.7, we
obtain

$\displaystyle(-X\partial-1/X-2n-1)\Theta_{n}(1/X)+\Theta_{n+1}(1/X)=0,$
$\displaystyle(-X^{2}\partial-1)\Theta_{n}(1/X)+(1/X)\Theta_{n-1}(1/X)=0.$

Left-multiplying both sides of the first equation by $X^{n+1}$ and both sides
of the second equation by $X^{n}$, we obtain

$\displaystyle(-X^{2}\partial-1-(n+1)X)X^{n}\Theta_{n}(1/X)+X^{n+1}\Theta_{n+1}(1/X)=0,$
$\displaystyle(-X^{2}\partial-1+nX)X^{n}\Theta_{n}(1/X)+X^{n-1}\Theta_{n-1}(1/X)=0,$

and hence the result follows. ∎

######  Corollary 4.24 ((Three-term recurrence)).

For each $n$, let $Y_{n}$ be defined as above, which is a solution of
$X^{2}\partial^{2}+(2X+2)\partial-n(n+1)$ in a certain left
$\mathcal{A}$-module $C[x]$. Then

$Y_{n+1}-(2n+1)XY_{n}-Y_{n-1}=0$ (4.31)

as an element in $N$.

We introduce $(\partial_{2}f)_{n}=f_{n+1}$ and $(X_{2}f)_{n}=nf_{n-1}$ to
constrruct an $\mathcal{A}_{2}$-module, such that

$[\partial_{1},X_{1}]=1,\quad[\partial_{2},X_{2}]=1,\quad[X_{1},X_{2}]=0,\quad[\partial_{1},\partial_{2}]=0.$

Replacing $n$ by $n-1$ in (4.29) and representing the resulting formula in
terms of $X_{1},\partial_{i},i=1,2$ yield

$X_{1}^{2}\partial_{1}+X_{1}X_{2}\partial_{2}+1-\partial_{2}.$ (4.32)

Similarly, the equation (4.30) can be rewritten as

$X_{1}^{2}\partial_{1}-X_{1}X_{2}\partial_{2}+1-\partial_{2}^{-1}.$

Right multiplying $\partial_{2}$ to the above formula gives

$X_{1}^{2}\partial_{1}\partial_{2}-X_{1}X_{2}\partial^{2}_{2}+\partial_{2}-1.$
(4.33)

Thus the Bessel polynomial module is given by

######  Definition 4.25.

The Bessel polynomial module is the left $\mathcal{A}_{2}$-module defined by

$\mathcal{Y}=\dfrac{\mathcal{A}_{2}}{\mathcal{A}_{2}(X_{1}^{2}\partial_{1}\partial_{2}-X_{1}X_{2}\partial^{2}_{2}+\partial_{2}-1)+\mathcal{A}_{2}(X_{1}^{2}\partial_{1}+X_{1}X_{2}\partial_{2}+1-\partial_{2})}.$
(4.34)

Similarly, we can represent (4.31) in $\mathcal{A}_{2}$ by

$\partial_{2}^{2}-2X_{1}X_{2}\partial_{2}^{2}-X_{1}\partial_{2}-1.$ (4.35)

######  Proposition 4.26.

The map

$\dfrac{\mathcal{A}_{2}}{\mathcal{A}_{2}(X_{1}^{2}\partial_{1}^{2}+(2X_{1}+2)\partial_{1}-X^{2}_{2}\partial_{2}^{2})}\longrightarrow\mathcal{Y}$

is a well-defined left $\mathcal{A}_{2}$-linear surjection.

####  4.4.1. Classical Bessel polynomials

######  Proposition 4.27.

Let $n\in\mathbb{N}_{0}$. Then the classical Bessel polynomial

$y_{n}(x)=\sum_{k=0}^{n}\frac{(n+k)!}{2^{k}(n-k)!\,k!}\,x^{k},$

for $n\geq 0$ and $y_{-1}(x)=1$, is the image of $1$ via the left
$\mathcal{A}$-linear map

$\begin{array}[]{rclll}\mathcal{A}/\mathcal{A}\big{(}X^{2}\partial^{2}+2(X+2)\partial-n(n+1)\big{)}&\stackrel{{\scriptstyle\times
Y_{n}(X)}}{{\longrightarrow}}&\mathcal{A}/\mathcal{A}\partial&\stackrel{{\scriptstyle\times
1}}{{\longrightarrow}}&\mathcal{O}\end{array}$

The following PDEs (delay-differential relations) and three-term recurrence
are direct consequences of the module $\mathcal{Y}$, as well as Corollaries
4.23 and 4.24 with $N=\mathcal{O}_{d}$.

######  Theorem 4.28 ((Delay-differential equations)).

The classical Bessel polynomials $\\{y_{n}(x)\\}$ [32, p. 19] satisfy the
following (well-known) formulae

$\begin{split}&x^{2}y^{\prime}_{n-1}(x)-[(n-1)x-1]y_{n-1}(x)-y_{n-2}(x)=0,\\\
&x^{2}y^{\prime}_{n-1}(x)-y_{n}(x)+(nx+1)y_{n-1}(x)=0.\end{split}$ (4.36)

for all $n$ and all $x$.

###### Proof.

This follows from the two generators (4.32) and (4.33) of the Bessel
polynomial module $\mathcal{Y}$ in the Definition 4.25, follows by the
manifestation of the analytic function space $\mathcal{O}_{d}$ as a
$\mathcal{A}_{1}$-module given by the Example 2.3. ∎

######  Proposition 4.29 ((Three-term recurrence [32], p. 18)).

The $(y_{n})$ satisfies the (well-known) three-term recurrence formula

$y_{n+1}(x)-(2n+1)xy_{\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}n}(x)-y_{n-1}(x)=0.$
(4.37)

or every $n\in\mathbb{N}_{0}$ and every $x$.

###### Proof.

This either follows from the last corollary or directly from the Corollary
4.24 with the manifestation of the analytic function space $\mathcal{O}_{d}$
as a $\mathcal{A}_{1}$-module given by the Example 2.3. ∎

####  4.4.2. Difference Bessel polynomials

######  Proposition 4.30.

Let $n\in\mathbb{N}_{0}$. Then the difference Bessel polynomial

$y^{\Delta}_{n}(x):=\sum_{k=0}^{n}\frac{(n+k)!}{2^{k}(n-k)!\,k!}\,(x)_{k},$

where we define $y^{\Delta}_{-1}(x):=1$, is the image of $1$ via the left
$\mathcal{A}$-linear map

$\begin{array}[]{rclll}\mathcal{A}/\mathcal{A}\big{(}X^{2}\partial^{2}+2(X+2)\partial-n(n+1)\big{)}&\stackrel{{\scriptstyle\times
Y_{n}(X)}}{{\longrightarrow}}&\mathcal{A}/\mathcal{A}\partial&\stackrel{{\scriptstyle\times
1}}{{\longrightarrow}}&\mathcal{O}_{\Delta}.\end{array}$

The following delay-differential relations and three-term recurrence are
direct consequences of the module $\mathcal{Y}$, as well as Corollaries 4.23
and 4.24 with $M=\mathcal{O}_{\Delta}$.

######  Theorem 4.31 ((Delay-difference equations)).

The difference Bessel polynomials $(y_{n})$ satisfy the following two delay-
difference equations

$\begin{split}x(x-1)\Delta
y^{\Delta}_{n-1}(x-2)-(nx-x)y^{\Delta}_{n-1}(x-1)+y^{\Delta}_{n-1}(x)-y^{\Delta}_{n-2}(x)=0,\\\
x(x-1)\Delta
y^{\Delta}_{n-1}(x-2)-y^{\Delta}_{n}(x)+nxy^{\Delta}_{n-1}(x-1)+y^{\Delta}_{n-1}(x)=0\end{split}$
(4.38)

for every $n$ and every $x$.

###### Proof.

∎

######  Proposition 4.32 ((Three-term recurrence)).

The difference Bessel polynomials $(y^{\Delta}_{n})$ satisfy the three-term
recurrence formula

$y^{\Delta}_{n+1}(x)-x(2n+1)y^{\Delta}_{n}(x-1)-y^{\Delta}_{n-1}(x)=0$ (4.39)

for every $n$ and $x$.

###  4.5. Characteristic of Bessel polynomial module

We change one of the two generators from the defining Bessel polynomial module
(4.34) by the element from the three-term recursion (4.35) to obtain following
theorem.

######  Theorem 4.33.

The natural map

$\dfrac{\mathcal{A}_{2}}{\mathcal{A}_{2}((1-2X_{1}X_{2})\partial_{2}^{2}-X_{1}\partial_{2}-1)+\mathcal{A}_{2}(X_{1}^{2}\partial_{1}-\partial_{2}+X_{1}X_{2}\partial_{2}+1)}\longrightarrow\mathcal{Y}$

is a left $\mathcal{A}_{2}$-linear isomorphism.

######  Theorem 4.34.

Let

$\mathcal{A}_{2}(\eta):=\dfrac{\mathcal{A}_{2}\langle\eta\rangle}{\langle\eta^{2}-1+2X_{1}X_{2}\rangle},$
$\mathcal{Y}(\eta)=\dfrac{\mathcal{A}_{2}(\eta)}{\mathcal{A}_{2}(\eta)(\eta^{2}\partial_{2}^{2}-X_{1}\partial_{2}-1)+\mathcal{A}_{2}(\eta)(X_{1}^{2}\partial_{1}-\partial_{2}+X_{1}X_{2}\partial_{2}+1)}$

and let

$S:=C_{1}\,{\mathop{\rm E}}(\frac{1-\eta}{X_{1}})+C_{2}\,{\mathop{\rm
E}}(\frac{1+\eta}{X_{1}})$

for every choice of $C_{1},\,C_{2}\in\mathbb{C}$, where ${\mathop{\rm E}}$
denote the Weyl exponential introduced in Example 3.3. Then
$\mathcal{Y}(\eta)$ is a left $\mathcal{A}_{2}(\eta)$-module and the map

$\mathcal{Y}(\eta)\xrightarrow[]{\times
S}\overline{\mathcal{A}_{2}(\eta)/(\mathcal{A}_{2}(\eta)\partial_{1}+\mathcal{A}_{2}(\eta)\partial_{2})},$
(4.40)

is left $\mathcal{A}_{2}(\eta)$-linear.

Since the proof of Theorem 4.34 is similar to that of Theorem 4.22, so we
place it in the Appendix A.3.

##  5\. Newton transformations

To handle more sophisticated generating functions from Bessel modules in
relation to difference calculus, we would like to introduce a transformation
that allows us to “transform” entities with respect to differential operators
to difference operators under different sittings. An origin of the transform
can be traced to spectral analysis and integral forms of gamma functions, see
e.g., [66, §12.22]:

$\Gamma(x)=-\frac{1}{2i\sin\pi
x}\int_{\infty}^{(0+)}e^{-\lambda}(-\lambda)^{x-1}d\lambda\quad(|\arg(-\lambda)|<\pi)$

which is valid on $\mathbb{C}$ except at negative integers including the
origin, where the upper and lower limits denote the standard Hankel contour
which starts from $+\infty$ above the real-axis, then circles around the
origin in the counter-clockwise direction before returns to $+\infty$ below
the real-axis. If we replace $x$ by $-x$, then the integral becomes

$\Gamma(-x)=\frac{e^{-i\pi x}}{2i\sin\pi
x}\int_{-\infty}^{(0+)}e^{\lambda}(-\lambda)^{-x-1}d\lambda\quad(|\arg(\lambda)|<\pi)$
(5.1)

where the integral contour starts at $-\infty$ before the negative real-axis,
then circle around the origin in the counter-clockwise direction, before
returns to $-\infty$ above negative real-axis.

######  Theorem 5.1 ((Newton transformations)).

Let $\mathcal{O}_{\Delta}$ be the space of entire functions endowed with the
structure of a left $\mathcal{A}$-module as in Example 2.9, and
$\mathcal{O}_{d}$ be the space of entire functions $f$ with growth rate

$|f(x)\,e^{mx}|\to 0\quad\mbox{ as }\quad x\to-\infty$

for some $m<1$, endowed with the structure of a left $\mathcal{A}$-module as
in Example 2.3. Then the map

$\begin{array}[]{rcl}\mathfrak{N}:\mathcal{O}_{d}&\longrightarrow&\mathcal{O}_{\Delta}\\\
f&\mapsto&\dfrac{e^{-i\pi x}}{2i\sin(\pi
x)\Gamma(-x)}\displaystyle\int_{-\infty}^{(0+)}\,f(s)\,e^{s}(-s)^{-x-1}\,ds\end{array}$
(5.2)

is left $\mathcal{A}$-linear. This map is called the Newton transformation.

###### Proof.

We just need to verify that $\mathfrak{N}(\partial f)=\partial(\mathfrak{N}f)$
and $\mathfrak{N}(Xf)=X(\mathfrak{N}f)$ as follows.

$\displaystyle\mathfrak{N}(\partial f)(x)$ $\displaystyle=\dfrac{e^{-i\pi
x}}{2i\sin(\pi
x)\Gamma(-x)}\displaystyle\int_{-\infty}^{(0+)}\,f^{\prime}(s)\,e^{s}(-s)^{-x-1}\,ds$
$\displaystyle=-\dfrac{e^{-i\pi x}}{2i\sin(\pi
x)\Gamma(-x)}\displaystyle\int_{-\infty}^{(0+)}\,f(s)[e^{s}(-s)^{-x-1}+(x+1)e^{s}(-s)^{-x-2}]\,ds$
$\displaystyle=(\mathfrak{N}f)(x+1)-(\mathfrak{N}f)(x)=\partial(\mathfrak{N}f)(x);$
$\displaystyle\mathfrak{N}(Xf)(x)$ $\displaystyle=\dfrac{e^{-i\pi
x}}{2i\sin(\pi
x)\Gamma(-x)}\displaystyle\int_{-\infty}^{(0+)}\,sf(s)\,e^{s}(-s)^{-x-1}\,ds$
$\displaystyle=-\dfrac{e^{-i\pi x}}{2i\sin(\pi
x)\Gamma(-x)}\displaystyle\int_{-\infty}^{(0+)}\,f(s)\,e^{s}(-s)^{-x}\,ds$
$\displaystyle=x(\mathfrak{N}f)(x-1)=X(\mathfrak{N}f)(x).$

∎

######  Theorem 5.2.

The Newton transformation defined above is an injection.

###### Proof.

It follows from the injectivity of the Mellin transformation and the
observation that the integral above is nothing but the Mellin transformation
of $f$ times the exponential function. ∎

######  Example 5.3.

Let us revisit the Weyl exponential ${\mathop{\rm E}}(aX)$ from Example 3.3
derived when finding a $\mathcal{A}$-linear map as a solution $S$ to

$\mathcal{A}/\mathcal{A}(\partial-a)\stackrel{{\scriptstyle\times
S}}{{\longrightarrow}}\mathcal{A}/\mathcal{A}\partial.$

Since it is known that
$\mathcal{A}/\mathcal{A}(\partial-a)\xrightarrow{\times{\mathop{\rm
E}}(aX)}\mathcal{A}/\partial\mathcal{A}\xrightarrow{\times 1}\mathcal{O}_{d}$
can be given by the classical function $e^{ax}$ where $\mathcal{O}_{d}$
endowed with the $\mathcal{A}$-module structure described in Example 2.3. Then
the Newton transformation above completes the following commutative diagram

${\mathcal{D}/\mathcal{D}(\partial-a)}$${\mathcal{O}_{d}}$${\mathcal{O}_{\Delta}}$$\scriptstyle{\times
e^{ax}}$$\scriptstyle{\times\exp_{\Delta}(a;\,X)}$$\scriptstyle{\mathfrak{N}}$

where the space of analytic functions $\mathcal{O}_{\Delta}$ is endowed with a
$\mathcal{A}$-module structure defined in Example 2.4. It is straightforward
to verify that

$\mathfrak{N}:x^{n}\ \ \longmapsto\ \ x(x-1)\cdots(x-n+1),$ (5.3)

for every integer $n$, so that the above commutative diagram gives the
difference exponential function

$\exp_{\Delta}(a;\,x):=(a+1)^{x}=\sum_{k=0}^{\infty}\frac{a^{k}}{k!}\,x(x-1)\cdots(x-k+1).$

converges absolutely for any complex $x$, if $|a|<1$ directly. The
$\exp_{\Delta}$ was already derived in Example 3.30.

Moreover, it is easy to see that

$\mathfrak{N}:x^{\nu}\ \ \longmapsto\ \
\frac{\Gamma(x+1)}{\Gamma(x+1-\nu)}=(x)_{\nu},$ (5.4)

for every $\nu\in\mathbb{C}$.

We shall apply similar rationale as the above example to derive generating
functions for certain half-Bessel modules with respect to difference operator
to be considered in subsequent sections. In fact, similar transformations
already appeared in the literature in different applications spread over many
decades. See for example [3], in integral form [62, p. 5 (25)] for Laguerre
polynomials, and again in [30, p. 401] in ad hoc manners.

######  Remark 5.4.

Variants of the Newton transformation are possible. For example, if
$\mathcal{O}_{d}$ in Theorem 5.1 is replaced by the space of entire functions
$f$ with growth rate such that

$|f(x)\,e^{mx}|\to 0\mbox{ as }x\to\pm i\infty,$

holds for every $m\in\mathbb{C}$, then one replaces the Hankel-type contour in
the above Newton transformation by, for example, Then the conclusion of
Theorem 5.1 remains valid.

##  6\. Poisson-type generating functions for (difference) Bessel polynomials

We are ready to apply the Theorem 4.22 derived from the last section to
recover the well-known (Poisson-type) generating function for classical Bessel
polynomials $\theta_{n}(x)$ and to derive new difference Bessel polynomials
$\theta_{n}^{\Delta}(x)$ in this section.

######  Theorem 6.1 ((Poisson transformation I)).

Let $\mathcal{O}_{d}^{\mathbb{N}_{0}}$ denote the space of sequences of
analytic functions $(f_{n}(x))_{0}^{\infty}$ with appropriately restricted
growth rate be endowed with the $\mathcal{A}_{2}$-module structure defined in
Example 2.15. Let $\mathcal{O}_{dd}$ denote the space of analytic functions be
endowed with the $\mathcal{A}_{2}$-module structure as in the Theorem 3.24.
Then the map

$\begin{array}[]{rcl}\mathfrak{p}:\mathcal{O}^{\mathbb{N}_{0}}&\to&\mathcal{O}_{dd}\\\
(f_{n})&\mapsto&\displaystyle\sum_{n=0}^{\infty}f_{n}\dfrac{t^{n}}{n!},\end{array}$
(6.1)

called the “Poisson transformation” is a left $\mathcal{A}_{2}$-linear map.

###### Proof.

This is a straightforward verification. ∎

For finite difference operator $\Delta$, we have the following analogue.

######  Theorem 6.2 ((Poisson transformation II)).

Let $\mathcal{O}_{\Delta}^{\mathbb{N}_{0}}$ denote the space of bilateral
sequences of analytic functions $(f_{n}(x))$ with appropriately restricted
growth rate be endowed with the $\mathcal{A}_{2}$-module structure defined in
Example 2.16. Let $\mathcal{O}_{\Delta d}$ denote the space of analytic
functions be endowed with the $\mathcal{A}_{2}$-module structure as in the
Theorem 3.24. Then the map

$\begin{array}[]{rcl}\mathfrak{p}_{\Delta}:\mathcal{O}^{\mathbb{N}_{0}}&\to&\mathcal{O}_{\Delta
d}\\\
(f_{n})&\mapsto&\displaystyle\sum_{n=0}^{\infty}f_{n}\dfrac{t^{n}}{n!},\end{array}$
(6.2)

called the “difference Poisson transformation” is a left
$\mathcal{A}_{2}$-linear map.

######  Remark 6.3.

The left $\mathcal{A}$-module structure of $\mathbb{C}^{\mathbb{Z}}$ as in
Example 2.5 is specifically designed so that the z-transform
$\mathfrak{z}:\mathbb{C}^{\mathbb{Z}}\to\mathcal{O}_{d}$ in Theorem 2.6 is
left $\mathcal{A}$-linear. In a similar way, the left $\mathcal{A}$-module
structure of $\mathbb{C}^{\mathbb{N}}$ as in Example 2.15 and Example 2.16 is
specifically designed so that the above Poisson transforms are left
$\mathcal{A}$-linear.

###  6.1. Classical reverse Bessel polynomials $\theta_{n}(x)$

######  Example 6.4.

Let
$\mathcal{A}_{2}(\rho):=\dfrac{\mathcal{A}_{2}\langle\rho,{1}/{\rho}\rangle}{\langle\rho^{2}-1+2X_{2}\rangle}$
be defined as in Theorem 4.22, and

$\Theta(\rho):=\dfrac{\mathcal{A}_{2}(\rho)}{\mathcal{A}_{2}(\rho)(-\rho^{2}\partial_{2}^{2}+3\partial_{2}+X_{1}^{2})+\mathcal{A}_{2}(\rho)(X_{1}\partial_{1}+\rho^{2}\partial_{2}-1-X_{1})}.$

Then $\Theta(\rho)$ is a left $\mathcal{A}_{2}(\rho)$-module. On the other
hand, the space $\mathcal{O}_{2}$ of two-variable analytic functions with
appropriate growth rate endowed with

$\begin{split}(\partial_{1}f)(x,t))&=f_{x}(x,t),\quad(X_{1}f)(x,t)=xf(x,t),\\\
(\partial_{2}f)(x,t)&=f_{t}(x,t),\quad(X_{2}f)(x,t)=tf(x,t)\end{split}$ (6.3)

and by spectra analysis, $\rho^{2}f$ can be defined so that

$(\rho^{2}f)(x,t)=(1-2t)f(x,t),$

$\mathcal{O}_{2}$ is a left $\mathcal{A}_{2}(\rho)$-module again denoted by
$\mathcal{O}_{dd}$, and the same space $\mathcal{O}_{2}$ endowed with (2.4)

$\begin{array}[]{ll}\partial_{1}f(x,\,t)=f(x+1,\,t)-f(x,\,t),&X_{1}f(x,\,t)=xf(x-1,\,t);\\\
\partial_{2}f(x,\,t)=f_{t}(x,\,t),&X_{2}f(x,\,t)=tf(x,\,t),\end{array}$ (6.4)

and

$(\rho^{2}f)(x,t)=(1-2t)f(x,t),$

is a left $\mathcal{A}_{2}(\rho)$-module again denoted by $\mathcal{O}_{\Delta
d}$; and the space $\mathcal{O}^{\mathbb{N}_{0}}$ of all sequences of analytic
functions endowed with (2.8) and $\rho^{2}f$ can be defined so that

$(\rho^{2}f)_{n}=f_{n}-2nf_{n-1}$ (6.5)

is also a left $\mathcal{A}_{2}(\rho)$-module.

We are ready to state our first main result in this subsection for which the
generating function of the classical reverse Bessel polynomials derived by
Burchnall [8] is a special case.

######  Theorem 6.5.

  1. (1)

Let $(\vartheta_{n})$ be a sequence of analytic functions which is a solution
of the reverse Bessel polynomial module $\Theta$ in
$\mathcal{O}_{d}^{\mathbb{N}_{0}}$. Then there exist complex constants
$C_{1},\,C_{2}$ such that

$\frac{1}{\sqrt{1-2t}}\Big{\\{}C_{1}\exp\big{[}x(1-\sqrt{1-2t})\big{]}+C_{2}\exp\big{[}x(1+\sqrt{1-2t})\big{]}\Big{\\}}=\sum_{n=0}^{\infty}\vartheta_{n}(x)\,\frac{t^{n}}{n!}$
(6.6)

which converges uniformly in each compact subset of $\mathbb{C}\times
B(0,\,\frac{1}{2})$.

  2. (2)

Moreover, the generating function in the part $\mathrm{(i)}$ above satisfies
the system of partial differential equations

$\begin{split}&f_{xt}(x,\,t)-f_{t}(x,\,t)+xf(x,\,t)=0,\\\
&xf_{x}(x,\,t)+(1-2t)f_{t}(x,\,t)-(1+x)f(x,\,t)=0\end{split}$ (6.7)

for all $x$ and $t$.242424See also the Appendix C.5..

We skip the proof to this theorem since it is very similar to that of the
following special case.

######  Theorem 6.6 (([32], p. 42)).

  1. (1)

The reverse Bessel polynomials $(\theta_{n})$ have the generating function

$\frac{1}{\sqrt{1-2t}}\exp\big{[}x(1-\sqrt{1-2t})\big{]}=\sum_{n=0}^{\infty}\theta_{n}(x)\,\frac{t^{n}}{n!}$
(6.8)

which converges uniformly in each compact subset of $\mathbb{C}\times
B(0,\,\frac{1}{2})$.

  2. (2)

Moreover, the generating function in the part $\mathrm{(i)}$ above satisfies
the system of partial differential equations (6.7)252525See also the Appendix
C.5..

###### Proof.

Recall the map $\mathfrak{p}$ as defined in (6.1) and the Example 6.4 just
discussed above that $\Theta(\rho)$ is a $\mathcal{A}_{2}(\rho)$-modules
above. Then we have the diagram

${\Theta(\rho)}$${\mathcal{O}^{\mathbb{N}_{0}}}$${\widetilde{A}_{2}(\rho)}$${\mathcal{O}_{dd}}$$\scriptstyle{\times(\theta_{n}(x))}$$\scriptstyle{\times\\{C_{1}\rho^{-1}{\mathop{\rm
E}}[X_{1}(1-\rho)]+C_{2}\rho^{-1}{\mathop{\rm
E}}[X_{1}(1+\rho)]\\}}$$\scriptstyle{\mathfrak{p}}$$\scriptstyle{\times 1}$
(6.9)

in which
$\widetilde{A}_{2}(\rho):=\overline{\mathcal{A}_{2}(\rho)/\big{[}\mathcal{A}_{2}(\rho)\partial_{1}+\mathcal{A}_{2}(\rho)\partial_{2}\big{]}}$.
Since the above diagram commutes, the sum
$\sum_{n=0}^{\infty}\theta_{n}(x)\,t^{n}/n!$ being the image of $1$ in
$\mathcal{O}_{dd}$ via the top-right path, is also a solution to the system of
differential-difference equations

$\begin{split}&\theta_{n}^{\prime}(x)-\theta_{n}(x)+x\theta_{n-1}(x)=0,\\\
&x\theta_{n}^{\prime}(x)-(x+2n+1)\theta_{n}(x)+\theta_{n-1}(x)=0.\end{split}$
(6.10)

mentioned in the Theorem 4.15. Now the function
$\frac{1}{\sqrt{1-2t}}\exp\big{[}x(1-\sqrt{1-2t})\big{]}$ is a solution to the
system of PDEs (6.7)

$\displaystyle f_{xt}(x,\,t)-f_{t}(x,\,t)+xf(x,\,t)=0,$ (6.11) $\displaystyle
xf_{x}(x,\,t)+(1-2t)f_{t}(x,\,t)-(1+x)f(x,\,t)=0.$ (6.12)

Since the PDEs (6.11) and (6.12) are images of the two generators of the
Bessel module $\Theta$ in Definition 4.9, so the dimension of the local
solution space of the PDEs coincides with the multiplicity two as stated in
Theorem 4.10.The Theorem 4.22 directly verifies that the solution space of the
PDEs has $\mathbb{C}$-dimension two by solving the explicit solution as a
linear combination of $\rho^{-1}{\mathop{\rm E}}[X_{1}(1-\rho)]$ and
$\rho^{-1}{\mathop{\rm E}}[X_{1}(1+\rho)]$ of $\Theta(\rho)$ in
$\tilde{A}_{2}(\rho)$ that effects the bottom-left path in the commutative
diagram (6.9). So

$\frac{1}{\sqrt{1-2t}}\big{\\{}C_{1}\exp\big{[}x(1-\sqrt{1-2t})\big{]}+C_{2}\exp\big{[}x(1+\sqrt{1-2t})\big{]}\big{\\}}=\sum_{n=0}^{\infty}\theta_{n}(x)\,\frac{t^{n}}{n!}$

for complex scalars $C_{1},\,C_{2}$. Substituting $t=0$ into the above
equation yields $C_{1}=1$ and $C_{2}=0$. One way to verify the convergence of
(6.8) is to combine (6.11) and (6.12) into a single ODE. Subtracting the
equation (6.11) from (6.12) after it being differentiated partially with
respect to $t$ yields the ODE

$(1-2t)f_{tt}(x,\,t)-3f_{t}(x,\,t)-x^{2}f(x,\,t)=0.$

For each fixed, but otherwise arbitrary, $x$ the ODE has a regular singularity
at $t=\frac{1}{2}$. Hence the infinite sum in (6.8) converges uniformly in
each compact subset of $\mathbb{C}\times B(0,\,\frac{1}{2})$. This completes
the proof. ∎

###  6.2. Difference reverse Bessel polynomials $\theta^{\Delta}_{n}(x)$

######  Example 6.7.

Let $\Theta(\rho)$ and $(\theta^{\Delta}_{n}(x))$ to denote the Bessel
polynomial module defined in Example 6.4 and infinite sequence of difference
Bessel polynomials respectively. Suppose that
$\mathcal{O}^{\mathbb{N}_{0}}=\mathcal{O}^{\mathbb{N}_{0}}_{\Delta}$ is
endowed with the $\mathcal{A}_{2}$-module structure as defined in Example
2.16. Then the map

$\begin{array}[]{rcl}\mathfrak{j}_{\Delta}:\Theta(\rho)&\stackrel{{\scriptstyle\times(\theta^{\Delta}_{n})}}{{\longrightarrow}}&\mathcal{O}^{\mathbb{N}_{0}}_{\Delta}\end{array}$
(6.13)

is left $\mathcal{A}_{2}$-linear.

We are now ready to make use of the Example 6.4 and to apply Newton’s
transform $\mathfrak{N}$ to “map” the classical generating for the reverse
Bessel polynomials derived in the last subsection to the difference reverse
Bessel polynomials.

######  Theorem 6.8.

  1. (1)

The difference reverse Bessel polynomials have the generating function

$\frac{e^{-i\pi x}}{2i\sin\pi
x\Gamma(-x)}\int_{-\infty}^{(0+)}e^{\lambda}(-\lambda)^{-x-1}\frac{1}{\sqrt{1-2t}}\exp\big{[}\lambda(1-\sqrt{1-2t})\big{]}\,d\lambda=\sum_{n=0}^{\infty}\theta_{n}^{\Delta}(x)\,\frac{t^{n}}{n!},$
(6.14)

where the infinite series converges uniformly in each compact subset of
$\mathbb{C}\times B(0,\,\frac{1}{2})$.

  2. (2)

Moreover, the generating function in part $\mathrm{(i)}$ above satisfies the
delay-differential equations262626See also the Appendix C.5.

$\begin{split}&f_{t}(x+1,\,t)-2f_{t}(x,\,t)+xf(x-1,\,t)=0,\\\
&(1-2t)f_{t}(x,\,t)+(x-1)f(x,\,t)-2xf(x-1,\,t)=0.\end{split}$ (6.15)

###### Proof.

We first draw on the simple fact from the Example 6.4 that $\Theta(\rho)$ is a
$\mathcal{A}_{2}(\rho)$-module and from the Example 6.7 that the map
$\mathfrak{j}_{\Delta}:\Theta(\rho)\stackrel{{\scriptstyle\times(\theta^{\Delta}_{n})}}{{\longrightarrow}}\mathcal{O}^{\mathbb{N}_{0}}_{\Delta}$
is $\mathcal{A}_{2}(\rho)$-linear. Then the map
$\mathfrak{p}_{\Delta}:\mathcal{O}^{\mathbb{N}_{0}}\longrightarrow\mathcal{O}_{\Delta
d}$ as defined in (6.2) and their composition is also an
$\mathcal{A}_{2}(\rho)$-linear. Thus
$\mathfrak{p}_{\Delta}\circ\mathfrak{j}_{\Delta}$ acting on the identity
yields the right-side of (6.14). Moreover, we have the commutative diagram

${\Theta(\rho)}$${\widetilde{\mathcal{A}}_{2}(\rho)}$${\mathcal{O}_{d}^{\mathbb{N}_{0}}}$${\mathcal{O}_{\Delta}^{\mathbb{N}_{0}}}$${\mathcal{O}_{dd}}$${\mathcal{O}_{\Delta
d}}$$\scriptstyle{\times\rho^{-1}{\mathop{\rm
E}}[X_{1}(1-\rho)]}$$\scriptstyle{\mathfrak{j}}$$\scriptstyle{\mathfrak{j}_{\Delta}}$$\scriptstyle{\times
1}$$\scriptstyle{\mathfrak{p}}$$\scriptstyle{\mathfrak{p}_{\Delta}}$$\scriptstyle{\mathfrak{N}}$
(6.16)

in which
$\widetilde{\mathcal{A}}_{2}(\rho):=\overline{\mathcal{A}_{2}(\rho)/\big{[}\mathcal{A}_{2}(\rho)\partial_{1}+\mathcal{A}_{2}(\rho)\partial_{2}\big{]}}$,
and the growth of the analytic functions in $\mathcal{O}_{dd}$ and
$\mathcal{O}_{\Delta d}$ are suitably restricted. Let

$y(x,t)=\int_{-\infty}^{(0+)}e^{\lambda}(-\lambda)^{-x-1}\frac{1}{\sqrt{1-2t}}\exp\big{[}\lambda(1-\sqrt{1-2t})\big{]}\,d\lambda.$

Since $\frac{1}{\sqrt{1-2t}}\exp\big{[}x(1-\sqrt{1-2t})\big{]}$ is a solution
to the system of PDEs (6.7) according to the leftmost path of the above
diagram, this $y(x,\,t)$ satisfies

$\begin{split}&[\partial_{2}\partial_{1}-\partial_{2}+X_{1}]y(x,\,t)=0,\\\
&\big{[}X_{1}\partial_{2}+(1-2X_{2})\partial_{2}-1-X_{1}\big{]}y(x,\,t)=0,\end{split}$

by Theorem 5.1 in the analytic function space $\mathcal{O}_{\Delta d}$ endowed
with a $\mathcal{A}_{2}$-module structure given by the Example 2.9. Hence the
$y(x,\,t)$ satisfies the system of delay-differential equations (6.15). We
instead offer an alternative approach by direct verification that the
$y(x,\,t)$ to satisfies the equivalent system272727This would make the
verification shorter.

$\begin{split}&[(2X_{1}-1)\partial_{2}^{2}+3\partial_{2}+X_{1}^{2}]y(x,\,t)=0,\\\
&\big{[}X_{1}\partial_{1}+(1-2X_{2})\partial_{2}-1-X_{1}\big{]}y(x,\,t)=0,\end{split}$

where the first expression above can be found in Proposition 4.11. It is
straightforward to note that

$\partial_{2}y=\frac{1}{\Gamma(-x)}\int_{-\infty}^{(+0)}e^{\lambda}(-\lambda)^{-x-1}\big{[}(1-2t)^{-1}+\lambda(1-2t)^{-\frac{1}{2}}\big{]}\frac{1}{\sqrt{1-2t}}\exp\big{[}\lambda(1-\sqrt{1-2t})\big{]}\,d\lambda$

and

$\begin{split}\partial^{2}_{2}y&=\frac{1}{\Gamma(-x)}\int_{-\infty}^{(+0)}e^{\lambda}(-\lambda)^{-x-1}\big{[}3(1-2t)^{-2}+3\lambda(1-2t)^{-\frac{3}{2}}+\lambda^{2}(1-2t)^{-1}\big{]}\\\
&\quad\times\frac{1}{\sqrt{1-2t}}\exp\big{[}\lambda(1-\sqrt{1-2t})\big{]}\,d\lambda\\\
\end{split}$

so that

$\begin{split}(2X_{2}-1)\partial_{2}^{2}y=&\frac{1}{\Gamma(-x)}\int_{-\infty}^{(+0)}e^{\lambda}(-\lambda)^{-x-1}\big{[}-3(1-2t)^{-1}-3\lambda(1-2t)^{-\frac{1}{2}}-\lambda^{2}\big{]}\\\
&\quad\times\frac{1}{\sqrt{1-2t}}\exp\big{[}\lambda(1-\sqrt{1-2t})\big{]}\,d\lambda\\\
&=(-3\partial_{2}-X_{1}^{2})y.\end{split}$

Hence

$\big{[}(2X_{2}-1)\partial_{2}^{2}+3\partial_{2}+X_{1}^{2}\big{]}y=0.$

On the other hand, we have

$X_{1}\partial_{1}y=\frac{1}{\Gamma(-x)}\int_{-\infty}^{(+0)}e^{\lambda}(-\lambda)^{-x-1}(x-\lambda)\frac{1}{\sqrt{1-2t}}\exp\big{[}\lambda(1-\sqrt{1-2t})\big{]}\,d\lambda\\\
$

and

$\begin{split}\big{[}(1-2X_{2})\partial_{2}-1-X_{1}\big{]}y&=\frac{1}{\Gamma(-x)}\int_{-\infty}^{(+0)}e^{\lambda}(-\lambda)^{-x}(1-\sqrt{1-2t})\frac{1}{\sqrt{1-2t}}\exp\big{[}\lambda(1-\sqrt{1-2t})\big{]}\,d\lambda\\\
=&\frac{1}{\Gamma(-x)}\int_{-\infty}^{(+0)}e^{\lambda}(-\lambda)^{-x}\frac{1}{\sqrt{1-2t}}d\exp\big{[}\lambda(1-\sqrt{1-2t})\big{]}.\end{split}$

Integration-by-parts of the above improper integral over an Hankel-type
contour yields

$\frac{1}{\Gamma(-x)}e^{\lambda}(-\lambda)^{-x}\frac{1}{\sqrt{1-2t}}\exp\big{[}\lambda(1-\sqrt{1-2t})\big{]}\Big{|}_{-\infty}^{(0+)}=0$

so that

$\begin{split}\big{[}(1-2X_{2})\partial_{2}-1-X_{1}\big{]}y=&-\frac{1}{\Gamma(-x)}\int_{-\infty}^{(+0)}\frac{1}{\sqrt{1-2t}}\exp\big{[}\lambda(1-\sqrt{1-2t})\big{]}\,d(e^{\lambda}(-\lambda)^{-x})\\\
=&-X_{1}\partial_{1}\frac{1}{\Gamma(-x)}\int_{-\infty}^{(+0)}e^{\lambda}(-\lambda)^{-x-1}\frac{1}{\sqrt{1-2t}}\exp\big{[}\lambda(1-\sqrt{1-2t})\big{]}\,d\lambda.\end{split}$

Hence

$\big{[}X_{1}\partial_{1}+(1-2X_{2})\partial_{2}-1-X_{1}\big{]}y=0$

as desired. That is, $y(x,\,t)$ satisfies

$\displaystyle(2t-1)f_{tt}(x,\,t)+3f_{t}(x,t)+x(x-1)f(x-2,t)=0,$ (6.17)
$\displaystyle(1-2t)f_{t}(x,t)+(x-1)f(x,t)-2xf(x-1,t)=0.$ (6.18)

Hence $y(x,\,t)$ satisfies the original system of delay-differential equations
(6.15).

Let us return to the right-half of the commutative diagram above. We notice
that the rightmost path of the commutative diagram (6.16), the sum
$\sum_{n=0}^{\infty}\theta_{n}^{\Delta}(x)\,t^{n}/n!$ is a (formal sum)
solution to the system of delay-difference equations

$\displaystyle\theta^{\Delta}_{n}(x+1)-2\theta^{\Delta}_{n}(x)+x\theta^{\Delta}_{n-1}(x-1)$
$\displaystyle=0,$
$\displaystyle\theta^{\Delta}_{n+1}(x)+(x-2n-3)\theta^{\Delta}_{n}(x)-2x\,\theta^{\Delta}_{n}(x-1)$
$\displaystyle=0$

for each $n$, which were derived in the Theorem 4.17.

Let us determine the region of convergence of (6.14). First we differentiate
the equation (6.18) with respect to $t$. This yields

$(1-2t)f_{tt}(x,\,t)+(x-3)f_{t}(x,\,t)-2xf_{t}(x-1,\,t)=0.$ (6.19)

Replace $x$ by $x-1$ in equation (6.18) and multiply the resulting equation by
$x$ throughout yield

$x(1-2t)f_{t}(x-1,t)+x(x-2)f(x-1,t)-2x(x-1)f(x-2,\,t)=0.$ (6.20)

Multiply the equation (6.17) throughout by $2$ and added to the (6.20) yield

$2(2t-1)f_{tt}(x,t)+6f_{t}(x,t)+x(1-2t)f_{t}(x-1,t)+x(x-2)f(x-1,t)=0.$ (6.21)

Multiply the equation (6.19) throughout by $1-2t$ yields

$(1-2t)^{2}f_{tt}(x,t)+(1-2t)(x-3)f_{t}(x,t)-2(1-2t)xf_{t}(x-1,t)=0.$ (6.22)

Multiply the equation (6.21) by $2$ and add the resulting equation to (6.22)
yield

$[(1-2t)^{2}+4(2t-1)]f_{tt}(x,t)+[(1-2t)(x-3)+12]f_{t}(x,t)+2x(x-2)f(x-1,t)=0.$
(6.23)

Now multiply the equation (6.18) throughout by the factor $x-2$ yields

$(x-2)(1-2t)f_{t}(x,t)+(x-2)(x-1)f(x,t)-2x(x-2)f(x-1,t)=0.$ (6.24)

It is now clear that the result of adding (6.23) and (6.24) yields, after
simplification, the equation

$(2t-1)(2t+3)f_{tt}(x,t)+[(1-2t)(2x-5)+12]f_{t}(x,t)+(x-1)(x-2)f(x,t)=0.$
(6.25)

For a fixed $x$, the above equation (6.25) is a second order linear ODE with
two finite regular singularities and $t=\frac{1}{2}$ is the one closest to the
origin $t=0$ which is an ordinary point. Hence we conclude that the expansion
in (6.14) converges uniformly in each compact subset of $\mathbb{C}\times
B(0,\,\frac{1}{2})$ as asserted. This completes the proof of the theorem. ∎

###  6.3. Classical Bessel polynomials $y_{n}(x)$

######  Example 6.9.

Let
$\mathcal{A}_{2}(\eta):=\dfrac{\mathcal{A}_{2}\langle\eta\rangle}{\langle\eta^{2}-1+2X_{1}X_{2}\rangle}$
be as defined in Theorem 4.34. Then $\mathcal{Y}(\eta)$ as defined in Theorem
4.34 is a left $\mathcal{A}_{2}(\eta)$-module. Other examples of
$\mathcal{A}_{2}(\eta)$-modules include $\mathcal{O}_{dd}$,
$\mathcal{O}_{\Delta d}$, $\mathcal{O}^{\mathbb{N}_{0}}$ and so on.

In much the same spirit of the derivation of the generating function for the
classical reverse Bessel polynomials in Theorem 6.6, we have the corresponding
theorems for the generating function of the classical Bessel polynomials
$(y_{n}(x))$ [41, 32].

######  Theorem 6.10.

  1. (1)

Let $(\mathscr{Y}_{n})$ be a sequence of analytic functions which is a
solution of the Bessel polynomial module $\mathcal{Y}$ in
$\mathcal{O}_{d}^{\mathbb{N}_{0}}$. Then there exist complex constants
$C_{1},\,C_{2}$ such that

$C_{1}\exp\left(\frac{1-(1-2xt)^{1/2}}{x}\right)+C_{2}\exp\left(\frac{1+(1-2xt)^{1/2}}{x}\right)=\sum_{n=0}^{\infty}\mathscr{Y}_{n-1}(x)\,\frac{t^{n}}{n!},$
(6.26)

which holds for $|t|<\left|\frac{1}{2x}\right|$, with an appropriate choice of
$\mathscr{Y}_{-1}$.

  2. (2)

Moreover, the holonomic system of PDEs of $\mathcal{Y}$ from the Definition
4.25 when manifested in $\mathcal{O}_{dd}$ is given by

$\begin{split}&x^{2}f_{xt}(x,\,t)-xtf_{tt}(x,\,t)+f_{t}(x,\,t)-f(x,\,t)=0,\\\
&x^{2}f_{x}(x,\,t)-f_{t}(x,\,t)+xtf_{t}(x,\,t)+f(x,\,t)=0,\end{split}$ (6.27)

for which both sides of (6.26) satisfy.

######  Theorem 6.11.

  1. (1)

The classical Bessel polynomials have the generating function

$\exp\left(\frac{1-(1-2xt)^{1/2}}{x}\right)=\sum_{n=0}^{\infty}y_{n-1}(x)\,\frac{t^{n}}{n!}$
(6.28)

which holds for $|t|<\left|\frac{1}{2x}\right|$.

  2. (2)

Moreover, the generating function from part (i) satisfies the PDE system
(6.27)282828See also the Appendix C.5..

###### Proof.

Recall the Poisson transform $\mathfrak{p}$ as defined in (6.1). Then we have
the commutative diagram

${\mathcal{Y}(\eta)}$${\mathcal{O}^{\mathbb{N}_{0}}_{d}}$${\widetilde{A}_{2}(\eta)}$${\mathcal{O}_{dd}}$$\scriptstyle{\times(y_{n-1}(x))}$$\scriptstyle{{\mathop{\rm
E}}(\frac{1-\eta}{X_{1}})}$$\scriptstyle{\mathfrak{p}}$$\scriptstyle{\times
1}$ (6.29)

in which
$\widetilde{\mathcal{A}_{2}}(\eta):=\overline{\mathcal{A}_{2}(\eta)/\big{[}\mathcal{A}_{2}(\eta)\partial_{1}+\mathcal{A}_{2}(\eta)\partial_{2}\big{]}}$
and the analytic functions in $\mathcal{O}_{dd}$ are suitably restricted. Due
to the similarity of the proof of Theorem 6.6, the remaining of the proof will
be completed in the Appendix §A.4. ∎

###  6.4. Difference Bessel polynomials $y^{\Delta}_{n}(x)$

######  Theorem 6.12.

  1. (1)

The difference Bessel polynomials $(y^{\Delta}_{n}(x))$ have the Poisson-type
generating function

$\displaystyle\frac{e^{-i\pi x}}{2i\sin\pi
x\Gamma(-x)}\int_{-\infty}^{(0+)}e^{\lambda+\frac{1-\sqrt{1-2\lambda
t}}{\lambda}}(-\lambda)^{-x-1}d\lambda=\sum_{n=0}^{\infty}y^{\Delta}_{n-1}(x)\,\frac{t^{n}}{n!},$
(6.30)

which converges uniformly in each compact subset of $\mathbb{C}\times
B(0,\frac{1}{2})$.

  2. (2)

Moreover, the generating function from part $\mathrm{(i)}$ above satisfies the
PDEs (delay-differential equations)292929See also the Appendix C.5.

$\begin{split}&x(x-1)[f_{t}(x-1,\,t)-f_{t}(x-2,\,t)]-xtf_{tt}(x-1,\,t)+f_{t}(x,\,t)-f(x,\,t)=0,\\\
&x(x-1)[f(x-1,\,t)-f(x-2,\,t)]-f_{t}(x,\,t)+xtf_{t}(x-1,\,t)+f(x,\,t)=0.\end{split}$
(6.31)

###### Proof.

Recall the maps $\mathfrak{p}$ and $\mathfrak{p}_{\Delta}$ as defined in (6.1)
and (6.2) respectively. Then we have the diagram

${\mathcal{Y}(\eta)}$${\widetilde{\mathcal{A}}_{2}(\eta)}$${\mathcal{O}^{\mathbb{N}_{0}}}$${\mathcal{O}^{\mathbb{N}_{0}}}$${\mathcal{O}_{dd}}$${\mathcal{O}_{\Delta
d}}$$\scriptstyle{{\mathop{\rm
E}}(\frac{1-\eta}{X_{1}})}$$\scriptstyle{\times(y_{n-1}(x))}$$\scriptstyle{\times(y^{\Delta}_{n-1}(x))}$$\scriptstyle{\times
1}$$\scriptstyle{\mathfrak{p}}$$\scriptstyle{\mathfrak{p}_{\Delta}}$$\scriptstyle{\times\mathfrak{N}}$
(6.32)

in which
$\widetilde{\mathcal{A}}_{2}(\eta):=\overline{\mathcal{A}_{2}/\big{[}\mathcal{A}_{2}(\eta)\partial_{1}+\mathcal{A}_{2}(\eta)\partial_{2}\big{]}}$
and the analytic functions in $\mathcal{O}_{\Delta d}$ have suitably
restricted growth. The remaining of the proof, being similar to that of the
Theorem 6.8, will be completed in the Appendix §A.5. ∎

##  7\. Half-Bessel modules II: Glaisher’s generating functions

In a number of publications between the years 1873-1881 (see e.g., [31]), the
derivations of trigonometric-type generating functions for Bessel functions

$\begin{split}&\sqrt{\frac{2}{\pi
x}}\cos\sqrt{x^{2}-2xt}=\sum_{n=0}^{\infty}J_{n-\frac{1}{2}}(x)\,\frac{t^{n}}{n!},\\\
&\sqrt{\frac{2}{\pi
x}}\sin\sqrt{x^{2}+2xt}=\sum_{n=0}^{\infty}J_{-n+\frac{1}{2}}(x)\,\frac{t^{n}}{n!}\end{split}$

obtained by Glaisher are known to be “formidable” to compute as noted by
Watson [64, p. 144]. We shall show in this section that how a suitable
“characteristic change of variables” from the generators of the Bessel module
of order $n+\frac{1}{2}$ in Definition 3.12 can simplify Glaisher’s
computation [31, pp. 774-781]. Moreover, in addition to recovering and
extending Glaisher’s results for classical Bessel functions, we also derive
analogous generating functions for the difference Bessel functions in this
section. Analogous generating functions for $q$-Bessel functions will be
investigated in a separate publication. On the other hand, Glaisher’s results
have been superseded by Lommel’s results from a different perspective, see
also [64, p. 144]. Weisner [65] re-derived Lommel’s results with Lie-algebraic
method, whose idea can also be found in [47]. We comment that no explicit Lie
algebraic method has been employed in our approach.

###  7.1. Characteristics of Glaisher modules

####  7.1.1. Negative Glaisher module

The characteristic change of variables we shall apply are

$\begin{array}[]{ll}\Xi_{1}=\partial_{1},&W_{1}=X_{1}\\\
\Xi_{2}={1}/{X_{2}},&W_{2}=X_{2}\partial_{2}X_{2}.\end{array}$ (7.1)

Then one can easily verify that

$[\pm\Xi_{1},\pm
W_{1}]=1,\quad[\Xi_{2},W_{2}]=1,\quad[\Xi_{1},\Xi_{2}]=0,\quad[W_{1},W_{2}]=0,\quad[\Xi_{i},\,W_{j}]=0,\
i\not=j$

hold.

Substitute the above characteristic change of variables into (3.14) and choose
$\nu=-\frac{1}{2}$ yield

$\displaystyle W_{1}\Xi_{1}+(W_{2}\Xi_{2}-\frac{1}{2})-W_{1}\Xi_{2}^{-1},$
(7.2) $\displaystyle W_{1}\Xi_{1}-(W_{2}\Xi_{2}-\frac{1}{2})+W_{1}\Xi_{2}.$
(7.3)

######  Definition 7.1.

Let $W_{i}$ and $\Xi_{i}$ be defined as in (7.1). The

$\mathcal{G}_{-}=\dfrac{\mathcal{A}_{2}}{\mathcal{A}_{2}\big{(}W_{1}\Xi_{1}+(W_{2}\Xi_{2}-\frac{1}{2})-W_{1}\Xi_{2}^{-1}\big{)}+\mathcal{A}_{2}\big{(}W_{1}\Xi_{1}-(W_{2}\Xi_{2}-\frac{1}{2})+W_{1}\Xi_{2}\big{)}}$
(7.4)

is called the negative Glaisher module.

######  Theorem 7.2 ((Integrability)).

The negative Glaisher module $\mathcal{G}_{-}$ is a left
$\mathcal{A}_{2}$-module that has dimension two and multiplicity two. In
particular, it is holonomic.

###### Proof.

Let us effect the following “change of variables”

$\widetilde{W}_{2}=\Xi_{2},\quad\widetilde{\Xi}_{2}=-W_{2}$

in the two generators of the negative Glaisher module (7.4). Note that
$[\widetilde{\Xi}_{2},\,\widetilde{W}_{2}]=1$. This yields

$\begin{split}&W_{1}\Xi_{1}+(\widetilde{W}_{2}\widetilde{\Xi}_{2}-\frac{1}{2})-W_{1}\widetilde{W}_{2}^{-1},\\\
&W_{1}\Xi_{1}-(\widetilde{W}_{2}\widetilde{\Xi}_{2}-\frac{1}{2})+W_{1}\widetilde{W}_{2}.\end{split}$

Subtracting and adding the above two relations yield

$\begin{split}&\widetilde{\Xi}_{2}+\frac{1}{2}\Big{(}\frac{1}{\widetilde{W}_{2}}+W_{1}+\frac{W_{1}}{\widetilde{W}_{2}^{2}}\Big{)}\\\
&\Xi_{1}+\frac{1}{2}\Big{(}\widetilde{W}_{2}-\frac{1}{\widetilde{W}_{2}}\Big{)}.\end{split}$

One immediately verifies that the two relations satisfy the integrability
condition discussed from Example 2.30.

A disadvantage of the above argument is although the “change of variables”
method preserves holonomicity, there is no guarantee that the multiplicity of
the negative Glaisher’s module remains unchanged. Indeed, we could follow the
argument used in the proof of Theorem 4.10 to conclude directly that the
dimension and multiplicity of the module are both two. We skip the details. ∎

We immediately obtain from the negative Glaisher module (7.4) the following
surjection.

######  Proposition 7.3.

The natural map

$\dfrac{\mathcal{A}_{2}}{\mathcal{A}_{2}[(W_{1}\Xi_{1})^{2}+W_{1}^{2}-(W_{2}\Xi_{2}-\frac{1}{2})^{2}]}\rightarrow\mathcal{B}_{-\frac{1}{2}}$

is a well-defined $\mathcal{A}_{2}$-linear surjection.

We omit its routine verification.

######  Theorem 7.4.

Let $\mathcal{G}_{-}$ be the negative Glaisher module. Set
$\omega_{-}^{2}=W_{1}^{2}-2W_{1}W_{2}$ and

$\mathcal{A}_{2}(\omega_{-})=\langle\Xi_{1},\Xi_{2},W_{1},W_{2},\omega_{-}^{\pm
1}\rangle.$

Then the

$S=C_{1}\mathop{\rm Cos}\omega_{-}+C_{2}\mathop{\rm Sin}\omega_{-}$

for each choice of complex constants $C_{1},C_{2}$, where $\mathop{\rm Sin}$
and $\mathop{\rm Cos}$ are the Weyl sine and the Weyl cosine defined in (3.3)
and (3.4), is an element of $\mathcal{A}_{2}(\omega_{-})$, such that the map

$\mathcal{G}_{-}\xrightarrow{\times
S}\overline{\mathcal{A}_{2}(\omega_{-})/[\mathcal{A}_{2}(\omega_{-})(W_{1}\Xi_{1}+\frac{1}{2})+\mathcal{A}_{2}(\omega_{-})\Xi_{2}]}$

is well-defined and left $\mathcal{A}_{2}$-linear.

###### Proof.

Suppose

$\delta_{1}=\Xi_{1}\,\omega_{-}-\omega_{-}\,\Xi_{1},\qquad\delta_{2}=\Xi_{2}\,\omega_{-}-\omega_{-}\,\Xi_{2}$

and

$\delta_{1}\,\omega=\omega_{-}\,\delta_{1},\quad\omega_{-}\delta_{2}=\delta_{2}\omega_{-},\quad[\omega_{-},\,W_{1}]=0,\quad[\omega_{-},\,W_{2}]=0.$
(7.5)

Then it is easy to compute

$\Xi_{1}\,\omega_{-}^{2}-\omega_{-}^{2}\,\Xi_{1}=2(W_{1}-W_{2}),\quad\Xi_{2}\,\omega_{-}^{2}-\omega_{-}^{2}\,\Xi_{2}=-2W_{1}.$

Hence

$\delta_{1}=\Xi_{1}\,\omega_{-}-\omega_{-}\,\Xi_{1}=\frac{W_{1}-W_{2}}{\omega_{-}},\quad\delta_{2}=\Xi_{2}\,\omega_{-}-\omega_{-}\,\Xi_{2}=-\frac{W_{1}}{\omega_{-}}.$
(7.6)

Subtracting the two elements (7.2) and (7.3) yields the expression

$-2(W_{2}\Xi_{2}-\frac{1}{2}\big{)}+W_{1}\Xi_{2}+W_{2}/\Xi_{2}.$

Left multiplication to the above expression throughout by $\Xi_{2}$ on the
left yields

$\Xi_{2}[(W_{1}-2W_{2})\Xi_{2}+1+W_{1}/\Xi_{2}]=(W_{1}-2W_{2})\Xi_{2}^{2}-\Xi_{2}+W_{1}.$

Right multiplication of the above expression by $W_{1}$ throughout yields

$W_{1}(W_{1}-2W_{2})\Xi_{2}^{2}-W_{1}\Xi_{2}+W_{1}^{2}=\omega^{2}\,\Xi_{2}^{2}-W_{1}\Xi_{2}+W_{1}^{2}.$

Substituting the second expression from (7.6) into the above expression and
utilising the (7.5) lead to

$(\omega_{-}\,\Xi_{2})^{2}+W_{1}^{2}=W_{1}^{2}\big{[}\big{(}\frac{\omega_{-}\,\Xi_{2}}{W_{1}}\big{)}^{2}+1\big{]}=W_{1}^{2}(\Xi^{2}+1).$

Let us denote

$\Xi=-\frac{\omega_{-}\,\Xi_{2}}{W_{1}}.$ (7.7)

It follows from the second expression of (7.6) that

$[\Xi,\,\omega_{-}]=1.$

Therefore, it follows from the above commutator and (7.3) that it is
appropriate to set

$\displaystyle\aleph:$
$\displaystyle=\Xi_{1}+\Big{(}1-\frac{W_{2}}{W_{1}}\Big{)}\Xi_{2},$ (7.8)
$\displaystyle W:$ $\displaystyle=W_{1}.$ (7.9)

Then it is easy to check that

$[\aleph,W]=1,\quad[\Xi,\,\omega_{-}]=1,\quad[\aleph,\Xi]=0,\quad[\aleph,\omega_{-}]=0,\quad[\Xi,W]=0,\quad[\omega_{-},W]=0$

hold. Hence we may consider

$\widehat{\mathcal{A}}_{2}=\mathbb{C}\langle\aleph,\Xi,W,\omega_{-}\rangle,$

and to rewrite the negative Glaisher module (7.4) in the form with the above
“change of variables”:

$\widehat{\mathcal{G}}_{-}:=\frac{\widehat{\mathcal{A}}_{2}}{\widehat{\mathcal{A}}_{2}(W\aleph+\frac{1}{2})+\widehat{\mathcal{A}}_{2}(\Xi^{2}+1)}.$

We observe from Example 2.26 that the $\widehat{\mathcal{G}}_{-}$ is a
holonomic module. In particular, its dimension and multiplicity are both equal
to $2$. On the other hand, it follows from Example 3.4 that there exists a
left $\widehat{\mathcal{A}_{2}}$-linear map

$\widehat{\mathcal{A}}_{2}/\widehat{\mathcal{A}}_{2}(\Xi^{2}+1)\
\xrightarrow{\times S}\
\overline{\widehat{\mathcal{A}}_{2}/\widehat{\mathcal{A}}_{2}\Xi}$

of the form

$S=C_{1}\mathop{\rm Cos}(\omega_{-})+C_{2}\mathop{\rm Sin}(\omega_{-})$

for some complex numbers $C_{1},\,C_{2}$.

Hence a solution map $S$ for

$\widehat{\mathcal{G}}_{-}\ \xrightarrow{\times S}\
\overline{\widehat{\mathcal{A}}_{2}/\big{[}\widehat{\mathcal{A}}_{2}(W\aleph+\frac{1}{2})+\widehat{\mathcal{A}}_{2}\Xi\big{]}}$

must assume the form

$S=f_{1}(W)\mathop{\rm Cos}(\omega_{-})+f_{2}(W)\mathop{\rm Sin}(\omega_{-}),$

where the maps $f_{1},f_{2}$ depends on $W$ only.

Without loss of generality, we may consider, $f_{1}$ only. As a result,

$\begin{split}0=&\big{(}W\aleph+\dfrac{1}{2})f_{1}(W)\,\mathrm{Cos}\,\omega_{-}\
\mod\widehat{\mathcal{A}_{2}}(W\aleph+\dfrac{1}{2})\\\
=&\mathrm{Cos}\,\omega_{-}(W\aleph+\dfrac{1}{2}\big{)}f_{1}(W)\
\mod\widehat{\mathcal{A}_{2}}(W\aleph+\dfrac{1}{2})\\\ =&W\aleph
f_{1}(W),\end{split}$

which implies $\aleph f_{1}(W)=0$, Hence $f_{1}(W)$ reduces to a constant.
Similarly $f_{2}(W)$ also reduces to a constant.

Hence it follows from (7.7), (7.8) and (7.9) that we have derived the asserted
map $S$

$\mathcal{G}_{-}(\omega_{-})\xrightarrow{\times(C_{1}\,\mathrm{Cos}\,\omega_{-}+C_{2}\,\mathrm{Sin}\,\omega_{-})}\overline{\mathcal{A}_{2}(\omega_{-})/[\mathcal{A}_{2}(\omega_{-})(W_{1}\Xi_{1}+1/2)+\mathcal{A}_{2}(\omega_{-})\Xi_{2}}].$

This completes the proof. ∎

####  7.1.2. Positive Glaisher module

The characteristic change of variables we shall apply are specific sign
changes of those of (7.1)

$\begin{array}[]{ll}\Xi_{1}=-\partial_{1},&W_{1}=-X_{1},\\\
\Xi_{2}={1}/{X_{2}},&W_{2}=X_{2}\partial_{2}X_{2}\end{array}$ (7.10)

with $\nu=\frac{1}{2}$ in (3.14) and choose

$\begin{split}&W_{1}\Xi_{1}+(W_{2}\Xi_{2}-\frac{1}{2})+W_{1}\Xi_{2}^{-1},\\\
&W_{1}\Xi_{1}-(W_{2}\Xi_{2}-\frac{1}{2})-W_{1}\Xi_{2}.\end{split}$ (7.11)

######  Definition 7.5.

Let $W_{i}$ and $\Xi_{i}$ be defined as in (7.10). Then

$\mathcal{G}_{+}=\dfrac{\mathcal{A}_{2}}{\mathcal{A}_{2}\big{(}W_{1}\Xi_{1}+(W_{2}\Xi_{2}-\frac{1}{2})+W_{1}\Xi_{2}^{-1}\big{)}+\mathcal{A}_{2}\big{(}W_{1}\Xi_{1}-(W_{2}\Xi_{2}-\frac{1}{2})-W_{1}\Xi_{2}\big{)}}$
(7.12)

a positive Glaisher module.

We obviously have the following companion theorem for the negative Glaisher
module.

######  Theorem 7.6.

The positive Glaisher module $\mathcal{G}_{+}$ is a left
$\mathcal{A}_{2}$-module that has dimension two and multiplicity two. In
particular, it is holonomic.

We state without giving details the theorem

######  Theorem 7.7.

Let $\mathcal{G}_{+}$ be defined in Definition 7.5. Set
$\omega_{+}^{2}=W_{1}^{2}+2W_{1}W_{2}$ and

$\mathcal{A}_{2}(\omega_{+})=\langle-\Xi_{1},\Xi_{2},-W_{1},W_{2},\omega_{+}^{\pm
1}\rangle.$

Then the expression

$S=C_{1}\mathop{\rm Cos}\omega_{+}+C_{2}\mathop{\rm Sin}\omega_{+},$

where $C_{1},C_{2}$ are constants, $\mathop{\rm Sin}$ and $\mathop{\rm Cos}$
are the Weyl sine and the Weyl cosine defined in (3.3) and (3.4) respectively,
is an element of $\mathcal{A}_{2}(\omega_{+})$, such that the map

$\overline{\mathcal{G}_{+}}\xrightarrow{\times
S}\overline{\mathcal{A}_{2}(\omega_{+})/[\mathcal{A}_{2}(\omega_{+})(W_{1}\Xi_{1}+\frac{1}{2})+\mathcal{A}_{2}(\omega_{+})\Xi_{2}]}$

is left $\mathcal{A}_{2}(\omega_{+})$-linear.

###  7.2. Glaisher’s generating functions

Combining the discussion from the previous subsections concerning the positive
and negative Glaisher-modules, which are modified half-Bessel modules, yields
the followings.

######  Theorem 7.8.

  1. (1)

Let $(\mathscr{C}_{n+\nu})_{n}$ be a bilateral sequence of analytic functions
which is a solution of the Glaisher modules $\mathcal{G}_{\mp}$ in
$\mathcal{O}^{\mathbb{Z}}$. Then there exist complex numbers $C_{1},\,C_{2}$
such that

$x^{-\frac{1}{2}}\big{[}C_{1}\cos\sqrt{x^{2}\pm 2xt}+C_{2}\sin\sqrt{x^{2}\pm
2xt}\big{]}=\sum_{n=0}^{\infty}\mathscr{C}_{\mp
n\pm\frac{1}{2}}(x)\,\frac{t^{n}}{n!},$ (7.13)

where the convergence of the infinite sum in (7.13) is uniform in each compact
subset of
$\mathbb{G}:=\\{(x,\,t)\in\mathbb{C}^{\dagger}\times\mathbb{C}:2|t|<|x|\\}$,
$\mathbb{C}^{\dagger}:=\mathbb{C}\backslash\\{x:x\leq 0\\}$.

  2. (2)

Moreover, the holonomic systems of PDEs that generates the Glaisher’s modules
$\mathcal{G}_{-}$ (7.4) and $\mathcal{G}_{+}$ (7.12) with manifestation in
$\mathcal{O}_{dd}$ as defined in (2.1) from Example 2.3 are given,
respectively, by303030See also the Appendix C.5 for the list of holonomic PDE
systems.

$\displaystyle
xf_{xt}(x,\,t)+tf_{tt}(x,\,t)+\frac{1}{2}f_{t}(x,\,t)-xf(x,t)=0,$ (7.14)
$\displaystyle xf_{x}(x,\,t)+(x-t)f_{t}(x,\,t)+\frac{1}{2}f(x,\,t)=0$ (7.15)

and

$\displaystyle
xf_{xt}(x,\,t)+tf_{tt}(x,\,t)+\frac{1}{2}f_{t}(x,\,t)+xf(x,t)=0,$ (7.16)
$\displaystyle xf_{x}(x,\,t)-(x+t)f_{t}(x,\,t)+\frac{1}{2}f(x,\,t)=0.$ (7.17)

###### Proof.

Recall the map $\mathfrak{p}$ as defined in (6.1). Then we have the diagram

${\mathcal{G}_{\pm}}$${\mathcal{O}^{\mathbb{N}_{0}}}$${\widetilde{\mathcal{A}_{2}}(\omega_{\pm})}$${\mathcal{O}_{dd}}$$\scriptstyle{\times(\mathscr{C}_{\pm
n\mp\frac{1}{2}})}$$\scriptstyle{\mathop{\rm Cos}\omega_{\pm}/\mathop{\rm
Sin}\omega_{\pm}}$$\scriptstyle{\mathfrak{p}}$$\scriptstyle{\times
x^{-\frac{1}{2}}}$

in which

$\widetilde{\mathcal{A}_{2}}(\omega_{\pm}):=\overline{\displaystyle{\mathcal{A}_{2}(\omega_{\pm})}/{\big{[}\mathcal{A}_{2}(\omega_{\pm})(W_{1}\Xi_{1}+\frac{1}{2})+\mathcal{A}_{2}(\omega_{\pm})\Xi_{2}\big{]}}},$

where the left-vertical map given in terms of Weyl cosine and sine
$\mathop{\rm Cos}\omega_{\pm}/\mathop{\rm Sin}\omega_{\pm}$ which are defined
as in Example 3.4, is guaranteed by the Theorem 7.4. We also note that the
analytic functions in $\mathcal{O}_{dd}$ are suitably restricted.

We first consider the case of negative Glaisher’s module $\mathcal{G}_{-}$.
Since $\omega_{-}^{2}=W_{1}^{2}-2W_{1}W_{2}$, the lower left path of the above
commutative diagram shows that

$x^{-\frac{1}{2}}\big{(}C_{1}\cos\sqrt{x^{2}-2xt}+C_{2}\sin\sqrt{x^{2}-2xt}\big{)}$
(7.18)

solves the PDEs (7.2) and (7.3) when manifested in the
$\mathcal{A}_{2}$-module $\mathcal{O}_{dd}$ endowed with the interpretation in
the Example 2.3, i.e.,

$\displaystyle
xf_{xt}(x,\,t)+tf_{tt}(x,\,t)+\frac{1}{2}f_{t}(x,\,t)-xf(x,t)=0,$
$\displaystyle xf_{x}(x,\,t)+(x-t)f_{t}(x,\,t)+\frac{1}{2}f(x,\,t)=0$

which are precisely the PDEs (7.14) and (7.15). On the other hand, via the
top-right path of the diagram, the sum

$\sum_{n=0}^{\infty}J_{n-\frac{1}{2}}(x)\,\frac{t^{n}}{n!}$

is a solution to the system of PDEs (7.2) and (7.3). Since we know from
Theorem 7.2 that the $\mathcal{G}_{-}$ is holonomic with both dimension and
multiplicity equal to two, so the dimension of the local solution space about
$t=0$ for the PDEs (7.2) and (7.3) is therefore two. We conclude that the
formula (7.13) holds. We next investigate the convergence region of the series
(7.13).

Differentiating the equation (7.15) with respect to $t$ yields

$xf_{xt}(x,\,t)+(x-t)f_{tt}(x,\,t)-\frac{1}{2}f_{t}(x,\,t)=0.$ (7.19)

Substituting the equation (7.14) into equation (7.19) yields the new equation

$(x-2t)f_{tt}(x,\,t)-f_{t}(x,t)+xf(x,\,t)=0.$ (7.20)

The equation is a second order linear ODE for a fixed $x$ and it has an
ordinary point at $t=0$ and a regular singularity at $t={x}/{2}$. Hence the
expansion (7.13) converges uniformly in each compact subset of
$\mathbb{C}\times\mathbb{C}$ where $2|t|<|x|$.

The handling of the second case when $\omega_{+}=W_{1}^{2}+2W_{1}W_{2}$ when
$\nu=\frac{1}{2}$ was dealt with in the last subsection which is similar to
that of the case $\mathcal{G}_{-}$ and is therefore omitted. We only note that
the PDEs (7.14) and (7.15) are replaced by

$\displaystyle
xf_{xt}(x,\,t)+tf_{tt}(x,\,t)+\frac{1}{2}f_{t}(x,\,t)+xf(x,t)=0,$
$\displaystyle xf_{x}(x,\,t)-(x+t)f_{t}(x,\,t)+\frac{1}{2}f(x,\,t)=0$

which are precisely the (7.16) and (7.17) A similar procedure that has led to
the equation (7.20) applied to the above equations (7.16) and (7.17) yields

$(x+2t)f_{tt}(x,\,t)+f_{t}(x,\,t)+xf(x,\,t)=0.$ (7.21)

This equation is similar to the equation (7.20) which also has an ordinary
point at $t=0$ and a regular singular point at $t=-x/2$. So we omit the
details. 313131It is unclear to the authors of the reason why Watson indicated
in [64, p. 140] that the restriction $2|t|<|x|$ in the above theorem only
applies to the case $\nu=\frac{1}{2}$, that is to $\mathcal{G}_{+}$ instead to
both since $t=x/2$ is a common regular singularity for both of the
differential equations (7.20) and (7.21).

The above analysis shows that the commutative diagram (7.43) holds. This
completes the proof. ∎

One can deduce the following Glaisher-type generating functions for other
classical Bessel functions from the above Theorem.

######  Corollary 7.9.

Let $\mathbb{G}:=\\{2|t|<|x|:\
(x,\,t)\in\mathbb{C}^{\dagger}\times\mathbb{C}\\}$ where
$\mathbb{C}^{\dagger}:=\mathbb{C}\backslash\\{x:x\leq 0\\}$. Then the
following series converge uniformly in any compact subset of
$\mathbb{G}$.323232Modified forms of the (i) and (ii) can be found from [1, p.
439] for $j_{n}(x)=\sqrt{\pi/2x}J_{n+\frac{1}{2}}(x)$ and
$y_{n}(x)=\sqrt{\pi/2x}Y_{n+\frac{1}{2}}(x)$.

  1. (1)

$\begin{split}&\sqrt{\frac{2}{\pi
x}}\cos(x^{2}-2xt)^{\frac{1}{2}}=\sum_{n=0}^{\infty}J_{n-\frac{1}{2}}(x)\,\frac{t^{n}}{n!},\\\
&\sqrt{\frac{2}{\pi
x}}\sin(x^{2}+2xt)^{\frac{1}{2}}=\sum_{n=0}^{\infty}J_{-n+\frac{1}{2}}(x)\,\frac{t^{n}}{n!};\end{split}$
(7.22)

  2. (2)

$\begin{split}&\sqrt{\frac{2}{\pi
x}}\sin(x^{2}-2xt)^{\frac{1}{2}}=\sum_{n=0}^{\infty}Y_{n-\frac{1}{2}}(x)\,\frac{t^{n}}{n!},\\\
-&\sqrt{\frac{2}{\pi
x}}\cos(x^{2}+2xt)^{\frac{1}{2}}=\sum_{n=0}^{\infty}Y_{-n+\frac{1}{2}}(x)\,\frac{t^{n}}{n!};\end{split}$
(7.23)

  3. (3)

$\begin{split}&\sqrt{\frac{2}{\pi
x}}\cos(-x^{2}-2xt)^{\frac{1}{2}}=\sum_{n=0}^{\infty}I_{n-\frac{1}{2}}(x)\,\frac{t^{n}}{n!},\\\
-&i\sqrt{\frac{2}{\pi
x}}\sin(-x^{2}-2xt)^{\frac{1}{2}}=\sum_{n=0}^{\infty}I_{-n+\frac{1}{2}}(x)\,\frac{t^{n}}{n!};\end{split}$
(7.24)

  4. (4)

$\sqrt{\frac{\pi}{2x}}e^{i(-x^{2}+2xt)^{1/2}}=\sum_{n=0}^{\infty}K_{n-\frac{1}{2}}(x)\,\frac{t^{n}}{n!}=\sum_{n=0}^{\infty}K_{-n+\frac{1}{2}}(x)\,\frac{t^{n}}{n!}.$
(7.25)

###### Proof.

  1. (1)

By choosing $t=0$ in equation (7.13), we immediately have

$x^{-\frac{1}{2}}\big{[}C_{1}\cos x+C_{2}\sin
x\big{]}=\mathscr{C}_{-\frac{1}{2}}(x)$

and

$x^{-\frac{1}{2}}\big{[}C_{1}\sin x+C_{2}\cos
x\big{]}=\mathscr{C}_{\frac{1}{2}}(x).$

Since

$J_{-\frac{1}{2}}(x)=\sqrt{2/\pi}x^{-\frac{1}{2}}\cos x,\qquad
J_{\frac{1}{2}}(x)=\sqrt{2/\pi}x^{-\frac{1}{2}}\sin x.$

Hence $C_{1}=C_{2}=\sqrt{2/\pi}$ and we obtain (7.22).

  2. (2)

Applying the well-known formulae

$Y_{-\frac{1}{2}}(x)=\sqrt{2/\pi}x^{-\frac{1}{2}}\sin x,\qquad
Y_{\frac{1}{2}}(x)=-\sqrt{2/\pi}x^{-\frac{1}{2}}\cos x$

to $\mathscr{C}_{-\frac{1}{2}}(x)$ and $\mathscr{C}_{\frac{1}{2}}(x)$, we
obtain (7.23) immediately.

  3. (3)

We have

$\displaystyle\sum_{n=0}^{\infty}I_{n-\frac{1}{2}}(x)\,\frac{t^{n}}{n!}$
$\displaystyle=\begin{cases}\sum_{n=0}^{\infty}(e^{-\frac{i\pi}{2}(n-\frac{1}{2})}J_{n-\frac{1}{2}}(xe^{\frac{i\pi}{2}}))\,\frac{t^{n}}{n!},&-\pi<\arg
x\leqslant\frac{1}{2}\pi\\\
\sum_{n=0}^{\infty}(e^{\frac{3}{2}i\pi(n-\frac{1}{2})}J_{n-\frac{1}{2}}(xe^{\frac{-3i\pi}{2}}))\,\frac{t^{n}}{n!},&\frac{\pi}{2}<\arg
x\leqslant\pi,\end{cases}$
$\displaystyle=\begin{cases}e^{\frac{i\pi}{4}}\sum_{n=0}^{\infty}J_{n-\frac{1}{2}}(xe^{\frac{i\pi}{2}})\,\frac{(te^{-\frac{i\pi}{2}})^{n}}{n!},&-\pi<\arg
x\leqslant\frac{1}{2}\pi,\\\
e^{-\frac{3}{4}i\pi}\sum_{n=0}^{\infty}J_{n-\frac{1}{2}}(xe^{\frac{-3i\pi}{2}})\,\frac{(te^{\frac{3i\pi}{2}})^{n}}{n!},&\frac{\pi}{2}<\arg
x\leqslant\pi.\end{cases}$
$\displaystyle=\begin{cases}\sqrt{2/\pi}x^{-\frac{1}{2}}\cos(-x^{2}-2xt)^{\frac{1}{2}},&-\pi<\arg
x\leqslant\frac{1}{2}\pi,\\\
\sqrt{2/\pi}x^{-\frac{1}{2}}\cos(-x^{2}-2xt)^{\frac{1}{2}}&\frac{\pi}{2}<\arg
x\leqslant\pi.\end{cases}$

from (i). Similarly, we have

$\sum_{n=0}^{\infty}I_{-n+\frac{1}{2}}(x)\,\frac{t^{n}}{n!}=\begin{cases}e^{-\frac{i\pi}{4}}\sum_{n=0}^{\infty}J_{-n+\frac{1}{2}}(xe^{\frac{i\pi}{2}})\,\frac{(te^{\frac{i\pi}{2}})^{n}}{n!},&-\pi<\arg
x\leqslant\frac{1}{2}\pi,\\\
e^{\frac{3}{4}i\pi}\sum_{n=0}^{\infty}J_{-n+\frac{1}{2}}(xe^{\frac{-3i\pi}{2}})\,\frac{(te^{\frac{-3i\pi}{2}})^{n}}{n!},&\frac{\pi}{2}<\arg
x\leqslant\pi.\end{cases}$

We can now substitute the generating function for $J_{-n+\frac{1}{2}}(x)$
derived from (i) above to obtain the desired result.

  4. (4)

Recall the relationships

$K_{n-\frac{1}{2}}(x)=\frac{\pi}{2}(-1)^{n}\big{[}I_{n-\frac{1}{2}}(x)-I_{-n+\frac{1}{2}}(x)\big{]},$

and

$K_{-n+\frac{1}{2}}(x)=\frac{\pi}{2}(-1)^{n}\big{[}I_{n-\frac{1}{2}}(x)-I_{-n+\frac{1}{2}}(x)\big{]}$

and we obtain the corresponding generating functions.

∎

###  7.3. Difference Glaisher’s generating functions

######  Theorem 7.10.

  1. (1)

The difference Bessel functions have the Glaisher’s generating function

$\begin{split}\dfrac{e^{-i\pi x}}{2i\sin(\pi
x)}\int_{-\infty}^{(0+)}&\lambda^{-\frac{1}{2}}\frac{\big{[}C_{1}(x)\sin\sqrt{\lambda^{2}\mp
2\lambda t}+C_{2}(x)\cos\sqrt{\lambda^{2}\mp 2\lambda
t}\big{]}}{\Gamma(-x)}e^{\lambda}(-\lambda)^{-x-1}\,d\lambda\\\
&=\sum_{n=0}^{\infty}\mathscr{C}^{\Delta}_{\pm
n\mp\frac{1}{2}}(x)\,\frac{t^{n}}{n!},\end{split}$ (7.26)

where the infinite sum converges uniformly in each compact subset of
$\mathbb{C}\times\mathbb{C}$.

  2. (2)

The holonomic systems of PDEs (delay-differential equations) that generates
Glaisher’s modules $\mathcal{G}_{-}$ (7.4) and $\mathcal{G}_{+}$ (7.12) with
manifestation in $\mathcal{O}_{\Delta d}$ as defined by (2.4) from Example 2.9
are given, respectively, by333333See also the Appendix C.5.

$\displaystyle
tf_{tt}(x,\,t)+\big{(}x+\frac{1}{2}\big{)}f_{t}(x,\,t)-xf_{t}(x-1,\,t)-xf(x-1,\,t)=0,$
(7.27) $\displaystyle
tf_{t}(x,\,t)-xf_{t}(x-1,\,t)+xf(x-1,\,t)-\big{(}x+\frac{1}{2}\big{)}f(x,\,t)=0$
(7.28)

and

$\displaystyle
tf_{tt}(x,\,t)+\big{(}x+\frac{1}{2}\big{)}f_{t}(x,\,t)-xf_{t}(x-1,\,t)+xf(x-1,\,t)=0,$
(7.29) $\displaystyle
tf_{t}(x,\,t)+xf_{t}(x-1,\,t)+xf(x-1,\,t)-\big{(}x+\frac{1}{2}\big{)}f(x,\,t)=0.$
(7.30)

###### Proof.

Recall the maps $\mathfrak{p}$, $\mathfrak{p}_{\Delta}$ and $\mathfrak{N}$ as
defined in (6.1), (6.2) and (5.2) respectively. Then we have the diagram

${\mathcal{G}_{\pm}(\omega_{\pm})}$${\widetilde{\mathcal{A}_{2}}(\omega_{\pm})}$${\mathcal{O}^{\mathbb{N}_{0}}}$${\mathcal{O}^{\mathbb{N}_{0}}}$${\mathcal{O}_{dd}}$${\mathcal{O}_{\Delta
d}}$$\scriptstyle{\mathop{\rm Cos}\omega/\mathop{\rm
Sin}\omega}$$\scriptstyle{\times(\mathscr{C}_{\pm
n\mp\frac{1}{2}})}$$\scriptstyle{\times(\mathscr{C}^{\Delta}_{\pm
n\mp\frac{1}{2}})}$$\scriptstyle{\times
x^{-\frac{1}{2}}}$$\scriptstyle{\mathfrak{p}}$$\scriptstyle{\mathfrak{p}_{\Delta}}$$\scriptstyle{\times\mathfrak{N}}$
(7.31)

in which
$\widetilde{\mathcal{A}_{2}}(\omega_{\pm}):=\overline{\displaystyle\frac{\mathcal{A}_{2}(\omega_{\pm})}{\big{[}\mathcal{A}_{2}(\omega_{\pm})(W_{1}\Xi_{1}+\frac{1}{2})+\mathcal{A}_{2}(\omega_{\pm})\Xi_{2}\big{]}}}$,
the Weyl cosine and sine $\mathop{\rm Cos}\omega_{\pm}/\mathop{\rm
Sin}\omega_{\pm}$ are as defined in Example 3.4, and the analytic functions in
$\mathcal{O}_{\Delta d}$ are suitably restricted.

Again, it suffices to consider the case for the negative Glaisher module
$\mathcal{G}_{-}$ for the proof of $\mathcal{G}_{+}$ being similar. Since
$\omega_{-}^{2}=W_{1}^{2}-2W_{1}W_{2}$, the leftmost path in the above
commutative diagram asserts that the analytic function from (7.18)

$x^{-\frac{1}{2}}\big{(}C_{1}\cos\sqrt{x^{2}-2xt}+C_{2}\sin\sqrt{x^{2}-2xt}\big{)}$
(7.32)

satisfies the system of PDEs (7.14) and (7.15). A further composition of this
analytic function under the Newton transformation (5.2) gives

$f_{-}(x,\,t)=\int_{-\infty}^{(0+)}\lambda^{-\frac{1}{2}}\frac{\big{[}C_{1}(x)\sin\sqrt{\lambda^{2}-2\lambda
t}+C_{2}(x)\cos\sqrt{\lambda^{2}-2\lambda
t}\big{]}}{\Gamma(-x)}e^{\lambda}(-\lambda)^{-x-1}\,d\lambda,$ (7.33)

which satisfies the system of delay-differential equations (7.2) and (7.3) by
Theorem 5.1, i.e.,

$\displaystyle
tf_{tt}(x,\,t)+\big{(}x+\frac{1}{2}\big{)}f_{t}(x,\,t)-xf_{t}(x-1,\,t)-xf(x-1,\,t)=0,$
(7.34) $\displaystyle
tf_{t}(x,\,t)-xf_{t}(x-1,\,t)+xf(x-1,\,t)-\big{(}x+\frac{1}{2}\big{)}f(x,\,t)=0.$
(7.35)

Without loss of generality, we may assume that the $1$-periodic factor
${e^{-i\pi x}}/{2i\sin(\pi x)}$ is included in the unknown $1$-periodic
functions $C_{1}(x),\,C_{2}(x)$ in the following computation. The handling of
the second case when $\omega_{+}=W_{1}^{2}+2W_{1}W_{2}$ when $\nu=\frac{1}{2}$
is similar and is therefore omitted.

On the other hand, the rightmost path in the above commutative diagram shows
that the sum
$\sum_{n=0}^{\infty}\mathscr{C}^{\Delta}_{n-\frac{1}{2}}(x)\,t^{n}/{n!}$ is a
solution to the system of PDEs (7.34) and (7.35). It is clear that the
$\mathcal{G}_{\pm}(\omega_{\pm})$ defined above is holonomic with multiplicity
two. Hence (7.26) holds apart from two $1$-periodic functions in $x$ which can
easily be incorporated into the constants $C_{1},\,C_{2}$.

We now derive a third order differential equation from the equations (7.34)
and (7.35) in order to determine the range of convergence of the (7.26).
Adding and subtracting the equations (7.34) and (7.35) yields the equations

$tf_{tt}(x,\,t)+\big{(}x+t+\frac{1}{2}\big{)}f_{t}(x,\,t)-2xf_{t}(x-1,\,t)-\big{(}x+\frac{1}{2}\big{)}f(x,\,t)=0$
(7.36)

and

$tf_{tt}(x,\,t)+\big{(}x-t+\frac{1}{2}\big{)}f_{t}(x,\,t)-2xf(x-1,\,t)+\big{(}x+\frac{1}{2}\big{)}f(x,\,t)=0$
(7.37)

respectively. Differentiating (7.37) with respect to $t$ yields

$tf_{ttt}(x,\,t)+\big{(}x-t+\frac{3}{2}\big{)}f_{tt}(x,\,t)+\big{(}x-\frac{1}{2}\big{)}f_{t}(x,\,t)-2xf_{t}(x-1,\,t)=0.$
(7.38)

Subtracting (7.36) from (7.38) yields the equation

$tf_{ttt}(x,\,t)+\big{(}x-2t+\frac{3}{2}\big{)}f_{tt}(x,\,t)-(t+1)f_{t}(x,\,t)+\big{(}x+\frac{1}{2}\big{)}f(x,\,t)=0,$
(7.39)

which is a third order linear ordinary differential equation in $t$ for each
fixed $x$. Clearly, the equation (7.39) has only one finite regular
singularity at $t=0$, so that any power series solution of it must converge
unformly in each compact subset of $\mathbb{C}\times\mathbb{C}$. Substituting
a Frobenius solution $f(x,\cdot)=\sum_{k=0}^{\infty}a_{k}x^{\sigma+k}$ into
the third-order equation (7.39) further confirming that $\sigma=0,\,1$ to be
the only integer indicial roots. Hence the (7.26) follows.

The handling of the $\mathcal{G}_{\frac{1}{2}}$ is similar to that for
$\mathcal{G}_{-\frac{1}{2}}$ and is therefore omitted. This completes the
proof. ∎

######  Remark 7.11.

We offer a direct verification that the (7.33) solves the delay-differential
equations (7.34) and (7.35). Since the computation of the sine and cosine
functions are similar, so it suffice to consider only the special case with
cosine kernel. Moreover, the $1$-periodic function $C_{1}(x)$ can be factored
out in the computation below since it is independent of $t$ and so it is also
omitted. Let

$y(x,t)=\displaystyle\int_{-\infty}^{(0+)}e^{\lambda}\dfrac{\cos(\lambda^{2}-2\lambda
t)^{\frac{1}{2}}}{\Gamma(-x)(-\lambda)^{x+1}\lambda^{\frac{1}{2}}}d\lambda.$

We now show that the $y(x,\,t)$ is a solution to the delay-differential
equation (7.35). We split the computation of this equation into two parts. We
substitute this $y$ into the third and the fourth terms from (7.35). This
yields

$xy(x,t)-xy(x-1,t)+\frac{1}{2}y(x,t)\\\
=\displaystyle\frac{1}{\Gamma(-x)}\int_{-\infty}^{(0+)}(x-\lambda+\frac{1}{2})e^{\lambda}(-\lambda)^{-x-1}\cos(\lambda^{2}-2\lambda
t)^{\frac{1}{2}}\lambda^{-\frac{1}{2}}d\lambda.$ (7.40)

We now substitute the $y(x,\,t)$ into the first two terms in (7.35) before
performing an integration-by-parts once yield

$\begin{split}&x\frac{dy(x-1,t)}{dt}-t\frac{dy(x,t)}{dt}=\displaystyle\frac{1}{\Gamma(-x)}\int_{-\infty}^{(0+)}e^{\lambda}(-\lambda)^{-x-1}\lambda^{\frac{1}{2}}\,d[-\cos(\lambda^{2}-2\lambda
t)^{\frac{1}{2}}]\\\
&=-e^{\lambda}\frac{(-\lambda)^{-x-1}}{\Gamma(-x)}\lambda^{\frac{1}{2}}\cos(\lambda^{2}-2\lambda
t)^{\frac{1}{2}}\Big{|}_{-\infty}^{(0+)}+\displaystyle\frac{1}{\Gamma(-x)}\int_{-\infty}^{(0+)}\cos(\lambda^{2}-2\lambda
t)^{\frac{1}{2}}d(e^{\lambda}(-\lambda)^{-x-1}\lambda^{\frac{1}{2}}).\end{split}$

Since $\mathop{\rm Re}(x)<-\frac{1}{2}$, then the limit of the first term
above vanishes. That is,

$-e^{\lambda}\frac{(-\lambda)^{-x-1}}{\Gamma(-x)}\lambda^{\frac{1}{2}}\cos(\lambda^{2}-2\lambda
t)^{\frac{1}{2}}\Big{|}_{-\infty}^{(0+)}=0.$

Thus the above expression becomes, after this integration-by-parts,

$\begin{split}&x\frac{dy(x-1,t)}{dt}-t\frac{dy(x,t)}{dt}\\\
&=\displaystyle\frac{1}{\Gamma(-x)}\int_{-\infty}^{(0+)}\cos(\lambda^{2}-2\lambda
t)^{\frac{1}{2}}e^{\lambda}(-\lambda)^{-x-1}\lambda^{-\frac{1}{2}}(\lambda-x-\frac{1}{2})d\lambda\\\
\end{split}$

which is precisely the negative of the expression above (7.40). This verifies
that $y(x,\,t)$ solves the first equation from (7.35). The verification of
$y(x,t)$ to the (7.34) is similar and is therefore omitted.

The following difference analogues for the Glaisher-type generating functions
(7.22) for difference Bessel functions $J_{n+\frac{1}{2}}^{\Delta}(x)$ are
amongst the special cases described in the above theorem.

######  Theorem 7.12.

The series

  1. (1)

$\sqrt{\frac{2}{\pi}}\dfrac{e^{-i\pi x}}{2i\sin(\pi
x)}\int_{-\infty}^{(0+)}\frac{e^{\lambda}(-\lambda)^{-x-1}\lambda^{-\frac{1}{2}}\cos\sqrt{\lambda^{2}-2\lambda
t}}{\Gamma(-x)}\,d\lambda=\sum_{n=0}^{\infty}J^{\Delta}_{n-\frac{1}{2}}(x)\frac{t^{n}}{n!},$
(7.41)

  2. (2)

and

$\sqrt{\frac{2}{\pi}}\dfrac{e^{-i\pi x}}{2i\sin(\pi
x)}\int_{-\infty}^{(0+)}\frac{e^{\lambda}(-\lambda)^{-x-1}\lambda^{-\frac{1}{2}}\sin\sqrt{\lambda^{2}+2\lambda
t}}{\Gamma(-x)}\,d\lambda=\sum_{n=0}^{\infty}J^{\Delta}_{-n+\frac{1}{2}}(x)\frac{t^{n}}{n!}$
(7.42)

converge uniformly in each compact subset of $\mathbb{C}\times\mathbb{C}$.

It may be possible to derive the two generating functions of the above theorem
from the Theorem 7.10, but we choose a more direct method by applying the
Newton transform Theorem 5.1 to Glaisher’s formulae directly.

###### Proof.

Due to the similarity of the two relations (7.41) and (7.42), so it suffices
to only establish the (7.41).

Let

$L=(X\partial)^{2}+(X^{2}-\nu^{2}),$

and to recall that a $\mathcal{D}-$linear map (3.26)

$S_{-\nu}=\sum_{k=0}^{\infty}\frac{(-1)^{k}X^{2k}}{2^{\nu+2k}k!\Gamma(-\nu+k+1)}$

in the following diagram

${\displaystyle\frac{\mathbb{C}[[X\partial,X]]}{\mathbb{C}[[X\partial,X]]L}}$${\displaystyle\frac{\mathbb{C}[[X\partial,X]]}{\mathbb{C}[[X\partial,X]](X\partial+\nu)}}$${\mathcal{O}_{d}}$${\mathcal{O}_{\Delta}}$$\scriptstyle{\times
S_{-\nu}}$$\scriptstyle{\times 1}$$\scriptstyle{\times
1}$$\scriptstyle{\times\mathfrak{N}}$ (7.43)

where the $\mathfrak{N}$ again denote the Newton transformation. Since (5.4).
Let us apply a truncated Newton transformation $\mathfrak{N}_{R}$ of (5.2)
which has the truncated Hankel-type contour $\Gamma_{R}\ (R>0)$ starts from
$-R$ below the negative real-axis, goes around the origin in the counter-
clockwise direction before returning to $x=-R$

$\begin{split}\mathfrak{N}_{R}(J_{-\nu}(x))&=\dfrac{e^{-i\pi x}}{2i\sin(\pi
x)\Gamma(-x)}\int_{Ne^{-i\pi}}^{Ne^{i\pi}}\sum_{k=0}^{\infty}\frac{(-1)^{k}\lambda^{-\nu+2k}}{2^{\nu+2k}k!\Gamma(-\nu+k+1)}e^{\lambda}(-\lambda)^{-x-1}\,d\lambda\\\
&=\sum_{k=0}^{\infty}\dfrac{e^{-i\pi x}}{2i\sin(\pi
x)\Gamma(-x)}\int_{Ne^{-i\pi}}^{Ne^{i\pi}}\frac{(-1)^{k}\lambda^{-\nu+2k}}{2^{\nu+2k}k!\Gamma(-\nu+k+1)}e^{\lambda}(-\lambda)^{-x-1}\,d\lambda\\\
&\to\sum_{k=0}^{\infty}\dfrac{e^{-i\pi x}}{2i\sin(\pi
x)\Gamma(-x)}\int_{-\infty}^{(0+)}\frac{(-1)^{k}\lambda^{-\nu+2k}}{2^{\nu+2k}k!\Gamma(-\nu+k+1)}e^{\lambda}(-\lambda)^{-x-1}\,d\lambda\\\
&=\sum_{k=0}^{\infty}\frac{(-1)^{k}}{2^{-\nu+2k}k!\Gamma(-\nu+k+1)}(x)_{-\nu+2k}=J^{\Delta}_{-\nu}(x),\end{split}$

as $R\to\infty$, where the interchange of the integral and summation signs
above is guaranteed by the uniform convergence of the classical Bessel
function $J_{\nu}(x)$ in any compact subset of
$\mathbb{C}^{\dagger}=\mathbb{C}\backslash\\{x:\,x\leq 0\\}$.

Let $\nu=n-\frac{1}{2}$. Apply a truncated Newton transformation (5.2) over
the truncated Hankel-type contour $\Gamma_{R}$ as defined above to both sides
of (7.22) with respect to $\lambda$. This yields

$\begin{split}\sqrt{\frac{2}{\pi}}\dfrac{e^{-i\pi x}}{2i\sin(\pi
x)}&\int_{Ne^{-i\pi}}^{Ne^{i\pi}}\frac{e^{\lambda}(-\lambda)^{-x-1}\lambda^{-\frac{1}{2}}\cos\sqrt{\lambda^{2}-2\lambda
t}}{\Gamma(-x)}\,d\lambda\\\ &=\sum_{n=0}^{\infty}\Big{(}\dfrac{e^{-i\pi
x}}{2i\sin(\pi
x)}\int_{Ne^{-i\pi}}^{Ne^{i\pi}}\frac{e^{\lambda}(-\lambda)^{-x-1}J_{n-\frac{1}{2}}(\lambda)}{\Gamma(-x)}\,d\lambda\Big{)}\frac{t^{n}}{n!},\\\
\end{split}$ (7.44)

where the interchange of the integral and summation signs above is allowed
because of the uniform convergence of the (7.22) in each compact subset of
$\mathbb{G}:=\\{2|t|<|x|:\ (x,\,t)\in\mathbb{C}^{\dagger}\times\mathbb{C}\\}$
where $\mathbb{C}^{\dagger}:=\mathbb{C}\backslash\\{x:x\leq 0\\}$ as shown
there. We finally obtain the formula (7.41) after letting $R\to\infty$ in the
truncated contour $\Gamma_{R}$. We may analytic continue the $x$ to the whole
of $\mathbb{C}$ by deforming the Hankel contour when necessary. The proof of
the (7.42) is similar. This completes the proof. ∎

We give a direct derivation of the integral representations of
$J^{\Delta}_{\pm\frac{1}{2}}$ from the (7.41) which are of different type from
those given in (3.73) and (3.74).

######  Corollary 7.13.

Let $x\in\mathbb{C}$. Then

  1. (1)

$J^{\Delta}_{-\frac{1}{2}}(x)=\sqrt{\frac{2}{\pi}}\frac{e^{-i\pi x}}{2i\sin\pi
x}\int_{-\infty}^{(0+)}\dfrac{e^{\lambda}}{\Gamma(-x)(-\lambda)^{x+1}\lambda^{\frac{1}{2}}}\cos\lambda\,d\lambda$

and

  2. (2)

$J^{\Delta}_{\frac{1}{2}}(x)=\sqrt{\frac{2}{\pi}}\frac{e^{-i\pi x}}{2i\sin\pi
x}\int_{-\infty}^{(0+)}\dfrac{e^{\lambda}}{\Gamma(-x)(-\lambda)^{x+1}\lambda^{\frac{1}{2}}}\sin\lambda\,d\lambda.$

###### Proof.

Expanding the $\cos\lambda$ into series in

$\begin{split}&\frac{e^{-i\pi x}}{2i\sin\pi
x}\int_{-\infty}^{(0+)}\dfrac{e^{\lambda}}{\Gamma(-x)(-\lambda)^{x+1}\lambda^{\frac{1}{2}}}\cos\lambda\,d\lambda\\\
&=\frac{e^{-i\pi x}}{2i\sin\pi
x}\int_{-\infty}^{(0+)}\dfrac{e^{\lambda}}{\Gamma(-x)(-\lambda)^{x+1}\lambda^{\frac{1}{2}}}\sum_{k=0}^{\infty}\frac{(-1)^{k}\lambda^{2k}}{(2k)!}\,d\lambda\\\
&=\sum_{k=0}^{\infty}\frac{(-1)^{k}}{(2k)!}\,G_{k}(x)\end{split}$

where, with applications of the generalised Gamma function and Euler’s
reflection formula,

$\begin{split}G_{k}(x)&:=\frac{e^{-i\pi x}}{2i\sin\pi
x}\frac{e^{-i\frac{\pi}{2}}}{\Gamma(-x)}\int_{-\infty}^{(0+)}e^{\lambda}(-\lambda)^{-x-\frac{1}{2}+2k-1}d\lambda\\\
&=\frac{e^{-i\pi x}}{2i\sin\pi
x}\frac{e^{-i\frac{\pi}{2}}}{\Gamma(-x)}\frac{e^{-i\pi(x+\frac{1}{2}-2k)}}{e^{-i\pi(x+\frac{1}{2}-2k)}}\frac{\sin\pi(-x-\frac{1}{2}+2k)}{\sin\pi(-x-\frac{1}{2}+2k)}\frac{\Gamma(-x-\frac{1}{2}+2k)}{\Gamma(-x-\frac{1}{2}+2k)}\int_{-\infty}^{(0+)}e^{\lambda}(-\lambda)^{-x-\frac{1}{2}+2k-1}\,d\lambda\\\
&=\frac{e^{-i\pi x}}{\sin\pi
x}\frac{e^{-i\frac{\pi}{2}}}{\Gamma(-x)}\frac{\sin\pi(-x-\frac{1}{2}+2k)\Gamma(-x-\frac{1}{2}+2k)}{e^{-i\pi(x+\frac{1}{2}-2k)}}\,\\\
&=\frac{\Gamma(-x-\frac{1}{2}+2k)}{\Gamma(-x)}\frac{\sin(x+\frac{1}{2}-2k)\pi}{\sin(-\pi
x)}\\\ &=\frac{\Gamma(x+1)}{\Gamma(x+\frac{3}{2}-2k)}.\end{split}$

Hence

$\frac{e^{-i\pi x}}{2i\sin\pi
x}\int_{-\infty}^{(0+)}\dfrac{e^{\lambda}}{\Gamma(-x)(-\lambda)^{x+1}\lambda^{\frac{1}{2}}}\cos\lambda\,d\lambda\
=\sum_{k=0}^{\infty}\frac{(-1)^{k}}{(2k)!}\,\frac{\Gamma(x+1)}{\Gamma(x+\frac{3}{2}-2k)}=\sqrt{\frac{\pi}{2}}\,J^{\Delta}_{-\frac{1}{2}}(x)$
(7.45)

since, with applications of the well-known identity
$\Gamma(k+\frac{1}{2})k!\,2^{2k}=\sqrt{\pi}(2k)!$, see for example [16, p.
211], that

$\begin{split}J^{\Delta}_{-\frac{1}{2}}(x)&:=\sum_{k=0}^{\infty}\frac{(-1)^{k}}{2^{-\frac{1}{2}+2k}k!\,\Gamma(\frac{1}{2}+k)}(x)_{-\frac{1}{2}+2k}=\sum_{k=0}^{\infty}\frac{(-1)^{k}}{2^{-\frac{1}{2}}\sqrt{\pi}(2k)!}(x)_{-\frac{1}{2}+2k}\\\
&=\sqrt{\frac{2}{\pi}}\sum_{k=0}^{\infty}\frac{(-1)^{k}}{(2k)!}\frac{\Gamma(x+1)}{\Gamma(x+\frac{3}{2}-2k)}.\end{split}$
(7.46)

Similarly, we have

$\frac{e^{-i\pi x}}{2i\sin\pi
x}\int_{-\infty}^{(0+)}\dfrac{e^{\lambda}}{\Gamma(-x)(-\lambda)^{x+1}\lambda^{\frac{1}{2}}}\sin\lambda\,d\lambda\
=\sum_{k=0}^{\infty}\frac{(-1)^{k}}{(2k+1)!}\,\frac{\Gamma(x+1)}{\Gamma(x+\frac{1}{2}-2k)}=\sqrt{\frac{\pi}{2}}\,J^{\Delta}_{\frac{1}{2}}(x).$
(7.47)

∎

##  8\. Discussion

### Truesdell’s $F$-equation theory

It had been observed by Lommel [44], Nielson [51] and Sonine [60], all from
the nineteenth century, to characterize Bessel functions from the pair of
recurrence formulae, which we call PDEs in this article (Proposition 3.20),

$\begin{split}&x\mathscr{C}_{\nu}^{\prime}(x)+\nu\mathscr{C}_{\nu}(x)-x\mathscr{C}_{\nu-1}(x)=0,\\\
&x\mathscr{C}^{\prime}_{\nu}(x)-\nu\mathscr{C}_{\nu+n}(x)+x\mathscr{C}_{\nu+1}(x)=0,\end{split}$
(8.1)

as analytic functions of two complex variables. They called the functions that
satisfy both of the equations cylinder functions, see also [64, pp. 82-84]. In
fact, according to Watson [64], Sonine refrained from using Bessel’s
differential equation (1.19) in his study. Truesdell published the book [62]
in 1948 perfecting the viewpoint started by the aforementioned nineteenth
century scholars as an analytic theory of differential-difference equation
that can be written in the forms

$\frac{\partial}{\partial
x}f(x,\alpha)=A(x,\alpha)f(x,\alpha)+B(x,\alpha)f(x,\alpha+1).$ (8.2)

If the $f(x,\alpha)$, assumes a specific form that after a suitable change of
variables, can be written in an equivalent form

$\frac{\partial}{\partial x}F(x,\alpha)=F(x,\alpha+1),$ (8.3)

then he demonstrated that derived many known classical special function
identities343434Thirty five formulae have been listed in [61]. as their
generating functions and to discover new ones for different $f$ for all sorts
of classical special functions. Truesdell called the equation (8.3) an
$F$-equation. We quote from [62, p. 7]

> “The aim of this essay is to provide a general theory which motivates,
> discovers, and coordinates such seemingly unconnected relations among
> familiar special functions as the formulas (1) through (35).”

Truesdell’s aimed to find “rational methods of discovery” 353535Truesdell, p.
7. of the numerous classical special functions formulae found in the
literature that often left beginners, and even experienced researchers,
wondering about their origins and the methods of their derivation. For
example, Truesdell commented that if a textbook has the following question:

> “Find a formula which gives Laguerre polynomials in terms of Bessel
> functions.”363636Truesdell, p. 7.

then, it would not be straightforward to any student unless he/she had a
priori knowledge of such formulae. However, once the student sees such a
formula, then it would be a typical textbook exercise for a student to give a
rigorous proof. Thus Truesdell would like to “coordinate”, “to bring order
into the part of collection of known relations concerning special functions by
showing that they are simple special cases of about a dozen general formulas
and by adding to their number some of the missing analogues which do not seem
to have been discovered thus far.” 373737[62].

Truesdell further commented that it is not with the particular special
function formulae (and some of them were claimed to be new) that was his
focus, since

> “it is of no great task to construct ad hoc rigorous proofs and … to any of
> the formulas we have just listed,”

but rather a general property, although it had been observed by earlier
researchers in various literature over a period of several decades in the
second half of the nineteenth century, that was largely neglected 383838[62,
p. 8]. is the result of applying his $F$-equation theory that he wanted to
bring about. Indeed Truesdell was not alone that Ehrenpreis, without explicit
mention, essentially also took the $F$-equation viewpoint in his study of
classical analogues of the Rogers-Ramanujan identities in [24, 25] in which a
generalised system of (1.13) that is related to multi-variable Bessel
functions has been proposed. From the $D$-modules viewpoint employed in this
article, although Truesdell and his predecessors certainly had the insight to
study the classical Bessel functions $J_{\nu}$ via the recursions (8.1), the
“general and neglected property” is replaced by the algebraic property that
the corresponding modules, such as the Bessel module studied in this paper,
are holonomic in the sense of Bernstein-Kashiwara-Sato holonomic modules
theory. Moreover, the essence of the $F$-equation theory which is about the
existence and uniqueness of the (8.3) having analytic solutions is being
replaced by the specific algebraic properties of the $D$-modules determined by
their generators. In the case of the Bessel module $\mathcal{B}_{\nu}$
considered in this paper, the generators are given by (3.14). What is
definitely new are the difference analogues of (8.1) given in (3.35) and other
formulae obtained in this article.

In addition to the well-known works of [54, 15, 14, 70, 68] of applying
$D$-modules to compute for ($q$-)hypergeometric type identities and the
standard works on computations with $D$-modules [56, 36], in fact, rudiments
of $D$-modules ideas had been employed by Boole in his seminar work
[5]393939Boole was awarded the first gold medal in mathematics by the Royal
Society in 1844 for this paper. in which a very general commutation relation
on operational method has been applied to solve a wide variety of linear
differential equations including a variant of confluent hypergeometric
equation. 404040See [10] for a more detailed discussion. Of course, in Boole’s
time, he could not possibly have the knowledge of the modern $D$-modules
theory. However, the use of his generalised commutator in solving for series
solutions of several linear differential equations is a vindication of his
enormous insights into the nature.414141Indeed, the paper together with other
earlier works on differential and difference equations of Boole could be
considered as preludes to his celebrated work on mathematical logic [6], see
[42, §3]. In addition to the difference in notations used and problems
tackled, a major differences between our work and those of Wilf and Zeilberger
and his successors [70, 68, 54, 15, 14] is that the way that the generators
used to define our $D$-modules which is based on an algebraization of
Truesdell’s $F$-equation theory. For example, the two generators (1.4) of the
Bessel module $\mathcal{B}_{\nu}$ are abstractions of the consequences of the
gauge transformations on the Bessel operators parametrised by integers derived
from the the corresponding transmutation formulae in Proposition 3.6. The
$\partial_{1},\partial_{2},X_{1},X_{2}$ in the $\mathcal{B}_{\nu}$ carry
different interpretations (manifestations) after appropriate
$\mathcal{A}_{2}$-linear maps in the construction of Bessel’s generating
functions. Lemma 4.1 in [70], although not needed in this paper, provides a
theoretical basis for the existence of characteristic change of variables made
in the proof of Theorem 3.16, Theorem 4.22, Theorem 4.34, Theorem 7.4 and
Theorem 7.7 in order to find (well-defined) $\mathcal{A}_{2}$-linear maps. The
well-defined $\mathcal{A}_{2}$-linear maps found from these Theorems represent
the generating functions derived (in $\mathcal{O}_{dd}$ and
$\mathcal{O}_{d\Delta}$) in the main theorems of this papers.

### Umbral calculus

We also note that the Weyl-exponential ${\mathop{\rm E}}(X)$, Weyl-sine
$\mathop{\rm Sin}(X)$ and Weyl-cosine $\mathop{\rm Cos}(X)$ derived as
“solutions” to the holomorphic $D$-modules
$\mathcal{A}_{1}/\mathcal{A}_{1}(\partial-a)$ and
$\mathcal{A}_{1}/\mathcal{A}_{1}(\partial^{2}+1)$ relative to
$\mathcal{A}_{1}/\mathcal{A}_{1}\partial$ in Example 3.3 and Example 3.4. The
manifestation of these “solutions” in $\mathcal{O}_{\Delta}$ result in the
difference-exponential, difference-sine and cosine introduced in Example 3.30
and Example 3.31 respectively, can be found in various literature under
“umbral calculus”, such as [18, 30, 45, 55]. In this spirit, it is interesting
to ask if the

${\mathop{\rm E}}[X_{1}(X_{2}-1/X_{2})/2]$

constructed in the Theorem 3.16 for the generating function of Bessel module
$\mathcal{B}_{0}$, the

$\rho^{-1}{\mathop{\rm E}}[X_{1}(1-\rho)],\qquad\rho^{2}=1-2X_{2}$

constructed in Theorem 4.22 for the Bessel polynomials module $\Theta$, and
the

$C_{1}\mathop{\rm Cos}\omega_{\mp}+C_{2}\mathop{\rm
Sin}\omega_{\mp},\qquad\omega_{\mp}^{2}=W_{1}^{2}\mp 2W_{1}W_{2}$

constructed in Theorem 7.4 for the Glaisher modules $\mathcal{G}_{\mp}$ can be
interpreted from umbral calculus’s viewpoint.

##  9\. Conclusion

This paper revisits system of PDEs satisfied by the generating function of the
classical Bessel functions of the first kind $J_{n}(x)$ from an holonomic
$D$-modules viewpoint. The approach unifies the generating functions of the
classical Bessel functions and the recently found difference Bessel functions
$J^{\Delta}_{n}(x)$. One can view this as an algebraization of the
$F$-equation theory, which is analytic in nature, proposed by Truesdell in
1948. The key ingredients of our arguments involve new transmutation formulae
for the Bessel functions and Bessel polynomials in certain
$\mathcal{A}_{1}$-modules, the derivation of well-defined
$\mathcal{A}_{2}$-linear maps of the $\mathcal{A}_{2}$-modules, which includes
the Bessel module $\mathcal{B}_{\nu}$, reverse Bessel polynomial module
$\Theta$, Bessel polynomial module $\mathcal{Y}$, and the Glaisher modules
$\mathcal{G}_{\pm}$. Although the transmutation formulae can be viewed as
abstractions of the relatedf factorization method proposed by Infeld and Hull
in [35], the scope of the study here and details involved are very different.

We have studied “solutions” in each $\mathcal{A}_{2}$-module above (see also
Appendix C.3) in the form of well-defined $\mathcal{A}_{2}$-linear maps that
amount to solving systems of linear PDEs defining the holonomic $D$-modules in
“closed-forms”. We have thus extended the classical Bessel’s generating
function to include $J_{\nu}(x)$ for arbitrary $\nu\in\mathbb{C}$ and its
difference analogue by constructing $\mathcal{A}_{2}$-linear maps from the
Bessel module $\mathcal{B}_{\nu}$ to analytic function spaces of two variables
$\mathcal{O}_{dd}$ and $\mathcal{O}_{\Delta d}$ which are also $D$-modules
with different “manifestations” as indicated by their respective subscripts.
The difference Bessel functions $J^{\Delta}_{\nu}(x)$ that appear in the
aforementioned difference manifestation of Bessel’s generating function
coincide with those found by Bohner and Cuthta in 2017. These new generating
functions are divergent (1-Gevrey) series unless $\nu=0$ which are Borel-
resummable. These new Bessel’s generating functions for general $\nu\not=0$ of
asymptotic type explains the classical Schläfli-Sonine integral representation
(3.47) of the Bessel function $J_{\nu}(x)$ that can be obtained via the usual
procedure of “residue extraction” of $J_{\nu+n}$ for any $n$ from the
asymptotic formula (3.41) instead of series manipulation technique usually
used, see [64, §6.2]. In fact, no knowledge of the series expansion of the
Bessel function $J_{\nu}(x)$ is needed with the “residue extraction” from the
(3.41). This viewpoint conforms with the way of studying the sequence of
Bessel functions $(J_{\nu+n})$ as coefficients of generating functions
discussed in [64, Chapter 2]. The “residue extraction” method has been applied
to the generating function formula (3.70) to obtain a difference analogue of
Schläfli-Sonine integral representation for the difference Bessel functions
$J^{\Delta}_{\nu}(x)$ in (3.73). The results obtained thus far (§3) in this
article suggests that one can approach the classical Bessel functions as
(Weyl-)algebraic entities without necessarily recourse to Bessel’s series
expansions as has been done in most modern literature.

Before we discuss about modifications of the half Bessel module
$\mathcal{B}_{1/2}$, we would like to emphasize that although we have devoted
this entire paper to Bessel functions, the choice of topics studied and
results presented in this paper, are by no means exhaustive, only serves to
illustrate our objectives and methodology of studying special functions, with
the Bessel function as an example, by solving the holonomic system PDEs in the
generalised sense represented by the respective $D$-modules. In fact, it is
clear to us that the $D$-modules approach could apply to many other topics
about Bessel functions found, for example, in Watson’s book [64], and to other
type of special functions (see below). Thus the choice of topics studied in
this paper is not limited by the range of applicability on the methodology,
but is limited rather by confining the paper to within a reasonable scope.

We next turn to the modifications of the Bessel module $\mathcal{B}_{\nu}$
when $\nu=\frac{1}{2}$, that would lead to a new generating functions to the
new difference Bessel polynomials and new difference Glaisher trigonometric
type generating functions in addition to their classical counterparts, which
can be found from §4 and §7 respectively. Each modification of
$\mathcal{B}_{1/2}$ requires certain “quadratic extensions” in addition to
their respective “change of variables”. One important feature of
$\mathcal{B}_{1/2}$ comes from the fact that the half Bessel operators can be
factorised (because it is solvable [50]).

In the case about the Weyl-Bessel polynomials studied in §4, the focus on
$\mathcal{B}_{1/2}$ is that after a suitable “change of variables” the reverse
polynomial operator possesses the transmutation formula

$X\partial^{2}-(2+X)\partial+2n=(X\partial-2n)(\partial-1)=(\partial-1)(X\partial-2n-1)$

that can be further derived from the transmutation formula (4.4). This
factorization of the reverse polynomial operator shows the reason that one can
find closed-form solutions which can always be transformed to a terminated
Frobenius solution or a polynomial solution if needed. This is the case only
when $\nu=1/2$ in $\mathcal{B}_{\nu}$. Indeed, most of the algebraic results
derived for the Weyl-Bessel polynomials in §4.2.1, to some extent, can be
found in [8, 9, 32] without the benefit of $D$-modules language. However, the
different manifestations of the reverse Bessel polynomial operator §4.2.1 in
the analytic function spaces $\mathcal{O}_{d}$ and $\mathcal{O}_{\Delta}$ give
rise to the classical reverse Bessel polynomials $\theta_{n}(x)$ and the new
difference reverse Bessel polynomials $\theta^{\Delta}_{n}(x)$ and their
respectively recursive formulae. The introduction of the reverse Bessel
polynomial module $\Theta$ in the same spirit as the Bessel module
$\mathcal{B}_{\nu}$ allows us to solve the corresponding PDEs that leads to
generating functions of the reverse classical Bessel polynomials and their
difference analogue respectively. The same has been done for the Bessel
polynomial module $\mathcal{Y}$ where the new difference Bessel polynomials
which are denoted by ($y^{\Delta}_{n}(x)$).

The consideration of Glaisher modules $\mathcal{G}_{\pm}$ in §7 is achieved
after the “change of variables”

$\begin{array}[]{ll}\Xi_{1}=\partial_{1},&W_{1}=X_{1}\\\
\Xi_{2}={1}/{X_{2}},&W_{2}=X_{2}\partial_{2}X_{2}.\end{array}$

in (7.1) to $\mathcal{B}_{1/2}$ can be considered a kind of “ Borel transform”
on $\mathcal{B}_{1/2}$, which is instrumental in solving the PDEs in the
Glaisher modules $\mathcal{G}_{\pm}$ leading to the trigonometric type
generating functions in Corollary 7.9, Theorem 7.10 and Theorem 7.12.

To summarise, it is clear from within the limited scope of study made in this
article that there is a close relationships amongst the systems of PDEs (in
the generalised sense) for which the generating functions of various type of
Bessel functions or polynomials satisfy are holonomic, and related integral
representations of Bessel functions. This relationship enables us to derive
the first integral representation for the recently discovered difference
Bessel functions. It also follows from our study that these difference Bessel
functions are on “equal footing” with the classical Bessel functions. Similar
relationship also applies to the newly discovered difference Bessel
polynomials in this article and their classical counterparts. This
investigation show that the space of differential operators annihilating some
generating functions of Bessel functions have rich algebraic structures and
that both the classical results and the new ones for the difference Bessel
functions and polynomials can be derived from the same Bessel module
$\mathcal{B}_{\nu}$, Bessel polynomial modules $\Theta,\mathcal{Y}$ and the
Glaisher modules $\mathcal{G}_{\pm}$.

##  10\. Outlook

The findings from this article have brought up more questions than the
problems solved. For example, it is very likely that our method could also
derive the more sophisicated generating function

$(x+h)^{-\nu/2}J_{\nu}(2\sqrt{x+h})=\sum_{k=0}^{\infty}\frac{(-h)^{k}}{k!}x^{-(\nu+k)/2}J_{\nu+k}(2\sqrt{x}),\quad|h|<|x|,$

found by Lommel [64, §5.22]424242Bessel actually discovered the formula when
$\nu=0$ in 1826., its difference analogue and the likes. What are mathematical
basis behind those systems of PDEs that allows for such generating functions?
Since generating functions can be viewed as a natural way to possess certain
combinatorial structures that re-organise the various symbols
$\partial_{1},\partial_{2},X_{1},X_{2}$, and the generating functions are
solutions to the specific systems of PDEs studied in this article, so a
question is how these combinatorial structures are described and understood
against the algebraic structures hidden in these systems of holonomic PDEs.
Another question is about the “orthogonality” of the new difference Bessel
polynomials found in this paper. The orthogonality issue of the new difference
Bessel polynomials will be dealt with in a separate article [12]. It turns out
that the classical orthogonal polynomial theory [37] does not seem to apply to
these new polynomials immediately which are naturally organised as Newton
polynomials. Thus a new approach that treats “orthogonality” as residue
pairings has been developed in the forthcoming article [12]. An advantage of
such an algebraic approach is that it could treat the orthogonality for both
the classical Bessel polynomials, i.e., the classical theory, and the
difference Bessel polynomials simultaneously. Finally, in spite of the lack of
understanding of deeper algebraic structures encoded in these systems of
holonomic PDEs ($D$-modules) that lead to the Bessel’s generating functions
studied in this article, similar strategy has been applied to other types of
$D$-modules of special functions in another forthcoming article [11].

##  Appendix A Proofs of some theorems/propositions

###  A.1. Proof of Proposition 3.33

###### Proof.

We rewrite the series of $J^{\Delta}_{\nu}(x)$ (3.53) as

$J^{\Delta}_{\nu}(x)=\sum_{k=0}^{\infty}\dfrac{(-1)^{k}\Gamma(x+1)}{k!2^{\nu+2k}\Gamma(\nu+k+1)\Gamma(x+1-\nu-2k)}.$
(A.1)

An asymptotic formula of Gamma function [2, Theorem 1.4.2], [66] for a complex
number $x$ not equal to zero or a negative real number, can be derived from
the formula

$\log\Gamma(x)=\frac{1}{2}\log 2\pi+(x-\frac{1}{2})\log
x-x+\sum_{j=1}^{m}\frac{B_{2j}(0)}{(2j-1)2j}\frac{1}{x^{2j-1}}-\frac{1}{2m}\int_{0}^{\infty}\frac{B_{2m}(t-[t])}{(x+t)^{2m}}dt,$

where the $B_{2m}(t)$ is the Bernoulli polynomial of degree $2m$. The value of
$\log\Gamma(x)$ is the branch with $\log\Gamma(x)$ being real when $x$ is real
and positive.

Set

$g(x)=\sum_{j=1}^{m}\frac{B_{2j}(0)}{(2j-1)2j}\frac{1}{x^{2j-1}}-\frac{1}{2m}\int_{0}^{\infty}\frac{B_{2m}(t-[t])}{(x+t)^{2m}}dt.$

Then

$\Gamma(x)=\sqrt{2\pi}x^{x-1/2}e^{-x+g(x)}.$

We rewrite those Gamma functions that appear in the denominator of
$J^{\Delta}_{\nu}(x)$ as

$\begin{split}&\Gamma(k+1)=\sqrt{2\pi}\,k^{k+1/2}(1/k+1)^{k+1/2}e^{-k-1+g(k+1)},\\\
&\Gamma(\nu+k+1)=\sqrt{2\pi}\,k^{\nu+k+1/2}(\nu/k+1/k+1)^{\nu+k+1/2}e^{-\nu-k-1+g(\nu+k+1)},\\\
&\Gamma(x-\nu-2k+1)=\sqrt{2\pi}\dfrac{(x/2k-\nu/2k+1/2k-1)^{x-\nu-2k+1/2}e^{g(x-\nu-2k+1)}}{(2k)^{-x+\nu+2k-1/2}e^{x-\nu-2k+1}}.\end{split}$

Set

$u_{k}=\dfrac{2^{-\nu-2k}}{k!\Gamma(\nu+k+1)\Gamma(x-\nu-2k+1)}.$

Since $g(x)\to 0$ as $x\to\infty$ with $\mathop{\rm Re}(x)>0$, so that it is
routine to verify that

$u_{k}\approx\frac{(-1)^{2k+\nu-\frac{1}{2}-x}2^{-\nu-2k}}{\sqrt{2}k^{\frac{3}{2}}(2k)^{x}2^{-\nu-2k}}=\frac{(\textrm{Const.})(-1)^{x}}{k^{x+\frac{3}{2}}}=\frac{(\textrm{Const.})e^{\pi\mathop{\rm
Im}x}}{k^{x+\frac{3}{2}}}$

holds as $k\to\infty$. Hence

$\sum_{k=1}^{\infty}u_{k}$

converges uniformly in each compact subset of $\mathbb{C}$ where $\mathop{\rm
Re}(x)>-\frac{1}{2}$. Therefore, the same holds for the (A.1). ∎

###  A.2. Proof of Proposition 4.12

###### Proof.

Multiplying the expression

$\partial_{1}-1+\dfrac{X_{1}}{\partial_{2}}$

by $\partial_{1}\partial_{2}$ yields

$\partial_{2}\partial_{1}^{2}-\partial_{1}\partial_{2}+\partial_{1}X_{1}.$

Left multiplying the expression
$X_{1}\partial_{1}-2X_{2}\partial_{2}-1-X_{1}+\partial_{2}$ by $\partial_{1}$
and taking into account of the last two relations yield

$\begin{split}\partial_{1}X_{1}\partial_{1}-2X_{2}\partial_{1}\partial_{2}-\partial_{1}-\partial_{1}X_{1}+\partial_{1}\partial_{2}&=X_{1}\partial_{1}^{2}-2X_{2}\partial_{1}\partial_{2}+\partial_{2}\partial_{1}^{2}\\\
&=X_{1}\partial_{1}^{2}-2X_{2}\partial_{1}\partial_{2}+\partial_{1}\partial_{2}-\partial_{1}X_{1}\\\
&=X_{1}\partial_{1}^{2}-(X_{1}+2X_{2}\partial_{2})\partial_{1}+\partial_{1}\partial_{2}-1\\\
&=X_{1}\partial_{1}^{2}-2(X_{1}+X_{2}\partial_{2})\partial_{1}+X_{1}\partial_{1}+\partial_{1}\partial_{2}-1\\\
&=X_{1}\partial_{1}^{2}-2(X_{1}+X_{2}\partial_{2})\partial_{1}+X_{1}\partial_{1}+\partial_{2}-X_{1}-1\\\
&=X_{1}\partial_{1}^{2}-2(X_{1}+X_{2}\partial_{2})\partial_{1}+1+2X_{2}\partial_{2}-1\\\
\end{split}$

as desired.

∎

###  A.3. Proof of Theorem 4.34

###### Proof.

It is easy to verify that

$\partial_{1}\eta^{2}-\eta^{2}\partial_{1}=-2X_{2},\qquad\partial_{2}\eta^{2}-\eta^{2}\partial_{2}=-2X_{1}$

hold. Suppose that

$\delta_{1}=\partial_{1}\eta-\eta\partial_{1},\qquad\delta_{2}=\partial_{2}\eta-\eta\partial_{2},$

and $\eta$ commutes with $\delta_{i},i=1,2$, i.e.,

$\delta_{1}\eta=\eta\delta_{1},\qquad\delta_{2}\eta=\eta\delta_{2}.$

Then we obtain

$\partial_{1}\eta-\eta\partial_{1}=-\frac{X_{2}}{\eta},\qquad\partial_{2}\eta-\eta\partial_{2}=-\frac{X_{1}}{\eta}.$

Let

$\Theta_{1}=\partial_{1}-\dfrac{1-X_{1}X_{2}}{X_{1}^{2}}\partial_{2},\qquad
Y_{1}=X_{1},\qquad\Theta_{2}=\eta\partial_{2},\qquad
Y_{2}=-\dfrac{\eta}{X_{1}}.$

We deduce from direct computation that

$[\Theta_{1},Y_{1}]=1,\quad[\Theta_{2},Y_{2}]=1,\quad[Y_{1},Y_{2}]=0,\quad[\Theta_{1},Y_{2}]=0,\quad[\Theta_{2},Y_{1}]=0,\quad[\Theta_{1},\Theta_{2}]=0.$

Then $\Theta_{i},Y_{i},i=1,2$ generate the following module

$\widehat{\mathcal{A}_{2}}=\mathbb{C}\langle\Theta_{1},\,\Theta_{2},\,Y_{1},\,Y_{2}\rangle.$

The two generators of $\mathcal{Y}(\eta)$ in Theorem 4.33 become, when written
in terms of $\Theta_{i},Y_{i},i=1,2$,

$(\Theta_{2}-1)(\Theta_{2}+1),\qquad\Theta_{1}+\dfrac{1}{Y^{2}_{1}}.$

Thus we define a modified $\widehat{\mathcal{A}}_{2}$-module

$\widehat{\mathcal{Y}}=\dfrac{\widehat{\mathcal{A}_{2}}}{\widehat{\mathcal{A}_{2}}((\Theta_{2}-1)(\Theta_{2}+1))+\widehat{\mathcal{A}_{2}}(\Theta_{1}+\dfrac{1}{Y^{2}_{1}})}.$

It follows from Example 2.26 that $\widehat{\mathcal{Y}}$ is a holonomic
module with dimension and multiplicity both equal to two. We apply Example 3.3
to derive that the map

$\widehat{\mathcal{A}}_{2}/\widehat{\mathcal{A}}_{2}(\Theta^{2}_{2}-1)\xrightarrow[]{\times[{\mathop{\rm
E}}(\pm
Y_{2})f(Y_{1})]}\overline{\widehat{\mathcal{A}}_{2}/\widehat{\mathcal{A}}_{2}\Theta_{2}}$

is well-defined left $\widehat{\mathcal{A}}_{2}$-linear if the unknown
relations $f(Y_{1}),g(Y_{1})$ depending only on $Y_{1}$ can be worked out. So
we require

$\begin{split}(\Theta_{1}+\dfrac{1}{Y^{2}_{1}}){\mathop{\rm
E}}(Y_{2})f(Y_{1})=&{\mathop{\rm
E}}(Y_{2})\Theta_{1}f(Y_{1})+\dfrac{1}{Y^{2}_{1}}{\mathop{\rm
E}}(Y_{2})f(Y_{1})\\\ =&{\mathop{\rm
E}}(Y_{2})(\Theta_{1}+\dfrac{1}{Y^{2}_{1}})f(Y_{1})\mod\widehat{\mathcal{A}_{2}}\Theta_{1}\\\
=&0.\end{split}$

This would hold if $f(X_{1})={\mathop{\rm E}}({1}/{Y_{1}})$. Similarly, we can
obtain $g(X_{1})={\mathop{\rm E}}({1}/{Y_{1}})$.

Hence, for each choice of constants $C_{1},C_{2}$, we have

$\widehat{\mathcal{Y}}\xrightarrow[]{\times[C_{1}{\mathop{\rm
E}}(Y_{2}+\frac{1}{Y_{1}})+C_{2}{\mathop{\rm
E}}(-Y_{2}+\frac{1}{Y_{1}})]}\overline{\widehat{\mathcal{A}}_{2}/(\widehat{\mathcal{A}}_{2}\Theta_{1}+\widehat{\mathcal{A}}_{2}\Theta_{2})},$

which gives rise to the $\mathcal{A}_{2}(\eta)$-linear map

$\mathcal{Y}(\eta)\xrightarrow[]{\times[C_{1}{\mathop{\rm
E}}(\frac{1-\eta}{X_{1}})+C_{2}{\mathop{\rm
E}}(\frac{1+\eta}{X_{1}})]}\overline{\mathcal{A}_{2}(\eta)/(\mathcal{A}_{2}(\eta)\partial_{1}+\mathcal{A}_{2}(\eta)\partial_{2})}.$

∎

###  A.4. Completion of proof of Theorem 6.11

###### Proof.

We continue the proof that is left unfinished below the Theorem 6.11.

  1. (1)

The $\mathcal{O}^{\mathbb{N}_{0}}$ being an $\mathcal{A}_{2}(\eta)$-module is
essentially verified from Example 2.15. The composition of the horizontal map
and the Poisson transform $\mathfrak{p}$ of the top-right path from the
commutative diagram (6.29)

$\begin{array}[]{rlcll}\mathcal{Y}(\eta)&\xrightarrow[]{\times(y_{n-1})}&\mathcal{O}^{\mathbb{N}_{0}}&\xrightarrow[]{\times
1}&\mathcal{O}_{dd},\\\ &&&\\\ \end{array}$

is a left $\mathcal{A}_{2}(\eta)$-linear map. Hence the composition map yields

$1\longmapsto\displaystyle\sum_{n=0}^{\infty}y_{n-1}(x)\,\dfrac{t^{n}}{n!},$

which implies that the sum $\sum_{n}y_{n-1}(x)\,{t^{n}}/{n!}$ is a solution to
the following system of PDEs (A.3) and (A.4) below.

  2. (2)

The first map from the bottom-left path of the commutative diagram (6.29) has
already been constructed in Theorem 4.34. On the other hand, the following
maps

$\begin{array}[]{rlcll}\mathfrak{g}_{p}:\mathcal{Y}(\eta)&\xrightarrow{C_{1}{\mathop{\rm
E}}(\frac{1-\eta}{X_{1}})+C_{2}{\mathop{\rm
E}}(\frac{1+\eta}{X_{1}})}&\overline{\mathcal{A}_{2}(\eta)/\big{[}\mathcal{A}_{2}(\eta)\partial_{1}+\mathcal{A}_{2}(\eta)\partial_{2}\big{]}}&\xrightarrow{\times
1}\mathcal{O}_{dd}\end{array}$

and (6.29) show that the

$C_{1}\exp\left(\frac{1-(1-2xt)^{1/2}}{x}\right)+C_{2}\exp\left(\frac{1+(1-2xt)^{1/2}}{x}\right)=\sum_{k=0}^{\infty}\frac{y_{k}(x)}{k!}t^{k}$
(A.2)

holds for some $C_{1},\,C_{2}\in\mathbb{C}$ and that both sides satisfy the
system of PDEs

$(1-2xt)f_{tt}(x,\,t)-f(x,\,t)-xf_{t}(x,\,t)=0,$ (A.3)
$x^{2}f_{x}(x,\,t)-f_{t}(x,\,t)+xtf_{t}(x,\,t)+f(x,\,t)=0.$ (A.4)

Putting $t=0$ in both sides of (A.2) implies that $C_{1}=1$ and $C_{2}=0$.

∎

We recall the well-known theorem that describes which half-plane does a Newton
series and its associated series, called faculty series by Nörlund, would
converge.

###  A.5. Proof of Theorem 6.12

Here we give full details of the proof to Theorem 6.12 in addition to the
commutative diagram (6.32).

###### Proof.

  1. (1)

The composition of the following maps

$\mathcal{Y}(\eta)\xrightarrow[]{\times(y^{\Delta}_{n-1})}\mathcal{O}_{\Delta}^{\mathbb{N}_{0}}\xrightarrow[]{\times
1}\mathcal{O}_{\Delta d}$

is left $\mathcal{A}(\eta)$-linear. Hence the infinite sum from

$1\longmapsto\displaystyle\sum_{n=0}^{\infty}y_{n-1}^{\Delta}(x)\,\frac{t^{n}}{n!}$

is a solution to the following delay-differential equations

$\displaystyle x(x-1)[f(x-1,t)-f(x-2,t)]-f_{t}(x,t)+xtf_{t}(x-1,t)+f(x,t)=0,$
(A.5) $\displaystyle f_{tt}(x,t)-2xtf_{tt}(x-1,t)-f(x,t)-xf_{t}(x-1,t)=0.$
(A.6)

  2. (2)

To deal with the left-side of the equation (6.30), we make use of the
commutative diagram (6.29) from the proof of Theorem 6.11 from the last
subsection. Thus to complete the left-half the commutative diagram (6.32), the
final step of the composition map

$\mathcal{Y}(\eta)\xrightarrow[]{\times{\mathop{\rm
E}}[\frac{1-\eta}{X_{1}}]}{\tilde{\mathcal{A}}_{2}(\eta)}\xrightarrow[]{\times
1}\mathcal{O}_{dd}\xrightarrow[]{\times\mathfrak{N}}\mathcal{O}_{\Delta d}$

that is given by the Newton transformation (5.2) yields

$\mathfrak{N}\circ{\mathop{\rm E}}(\frac{1-\eta}{X_{1}})\cdot
1=\displaystyle\frac{e^{-i\pi x}}{2i\sin\pi
x\Gamma(-x)}\int_{-\infty}^{(0+)}e^{\lambda+\frac{1-\sqrt{1-2\lambda
t}}{\lambda}}(-\lambda)^{-x-1}d\lambda$

as asserted.

Although the $\mathcal{A}_{2}(\eta)$-linearity consideration of the above
composition map is sufficient to complete the proof, we offer a direct
verification that the above $\mathfrak{N}\circ{\mathop{\rm
E}}(\frac{1-\eta}{X_{1}})\cdot 1$ is indeed a solution to the system of PDEs
(A.5) and (A.6) here. Let

$f(x,t)=\displaystyle\frac{e^{-i\pi x}}{2i\sin\pi
x\Gamma(-x)}\int_{-\infty}^{(0+)}e^{\lambda+\frac{1-\sqrt{1-2\lambda
t}}{\lambda}}.(-\lambda)^{-x-1}d\lambda.$

Then we verify

$\begin{split}&(X_{1}^{2}\partial_{1}-\partial_{2}+X_{1}X_{2}\partial_{2}+1)f(x,t)=0,\\\
&(1-2X_{1}X_{2})\partial_{2}^{2}-X_{1}\partial_{2}-1)f(x,t)=0.\end{split}$

We first compute

$\begin{split}&\partial_{2}^{2}\displaystyle\frac{e^{-i\pi x}}{2i\sin\pi
x\Gamma(-x)}\int_{-\infty}^{(0+)}e^{\lambda+\frac{1-\sqrt{1-2\lambda
t}}{\lambda}}(-\lambda)^{-x-1}d\lambda\\\ &=\displaystyle\frac{e^{-i\pi
x}}{2i\sin\pi
x\Gamma(-x)}\int_{-\infty}^{(0+)}e^{\lambda+\frac{1-\sqrt{1-2\lambda
t}}{\lambda}}(-\lambda)^{-x-1}\big{[}(1-2\lambda t)^{-1}+\lambda(1-2\lambda
t)^{-3/2}\big{]}d\lambda,\end{split}$

and

$\begin{split}&(1-2X_{1}X_{2})\partial_{2}^{2}\displaystyle\frac{e^{-i\pi
x}}{2i\sin\pi
x\Gamma(-x)}\int_{-\infty}^{(0+)}e^{\lambda+\frac{1-\sqrt{1-2\lambda
t}}{\lambda}}(-\lambda)^{-x-1}d\lambda\\\ &=\displaystyle\frac{e^{-i\pi
x}}{2i\sin\pi
x\Gamma(-x)}\int_{-\infty}^{(0+)}e^{\lambda+\frac{1-\sqrt{1-2\lambda
t}}{\lambda}}(-\lambda)^{-x-1}\big{[}1+\lambda(1-2\lambda
t)^{-1/2}\big{]}d\lambda.\end{split}$

Combine the above computations yields

$((1-2X_{1}X_{2})\partial_{2}^{2}-X_{1}\partial_{2}-1)\displaystyle\frac{e^{-i\pi
x}}{2i\sin\pi
x\Gamma(-x)}\int_{-\infty}^{(0+)}e^{\lambda+\frac{1-\sqrt{1-2\lambda
t}}{\lambda}}(-\lambda)^{-x-1}d\lambda=0.$

Similarly, we can compute

$\begin{split}&X_{1}^{2}\partial_{1}\displaystyle\frac{1}{\Gamma(-x)}\int_{-\infty}^{0}e^{\lambda+\frac{1-\sqrt{1-2\lambda
t}}{\lambda}}(-\lambda)^{-x-1}d\lambda\\\
&=\displaystyle\frac{1}{\Gamma(-x)}\int_{-\infty}^{0}e^{\lambda+\frac{1-\sqrt{1-2\lambda
t}}{\lambda}}(-\lambda)^{-x+1}\big{[}\dfrac{x-1}{\lambda}-1\big{]}d\lambda\\\
&=-\displaystyle\frac{1}{\Gamma(-x)}\int_{-\infty}^{0}e^{\frac{1-\sqrt{1-2\lambda
t}}{\lambda}}d(e^{\lambda}(-\lambda)^{-x+1}).\end{split}$

Integration-by-part yields

$\begin{split}&(X_{1}^{2}\partial_{1}+1)\displaystyle\frac{1}{\Gamma(-x)}\int_{-\infty}^{0}e^{\lambda+\frac{1-\sqrt{1-2\lambda
t}}{\lambda}}(-\lambda)^{-x-1}d\lambda\\\
&=(1-X_{1}X_{2})\partial_{2}\displaystyle\frac{1}{\Gamma(-x)}\int_{-\infty}^{0}e^{\lambda+\frac{1-\sqrt{1-2\lambda
t}}{\lambda}}(-\lambda)^{-x-1}d\lambda,\end{split}$

which implies

$(X_{1}^{2}\partial_{1}-(1-X_{1}X_{2})\partial_{2}+1)\displaystyle\frac{1}{\Gamma(-x)}\int_{-\infty}^{0}e^{\lambda+\frac{1-\sqrt{1-2\lambda
t}}{\lambda}}(-\lambda)^{-x-1}d\lambda=0.$

Hence $f(x,t)$ satisfies the system of differential-difference equations (A.5)
and (A.6). Similar to the argument used in the proof of Theorem 6.11 that the
solution space of the PDEs (A.5) and (A.6) is two-dimensional. Hence (6.26)
from Theorem 6.10 holds for some constants $C_{1},\,C_{2}$ with
$\mathscr{Y}_{n-1}$ replaced by $y_{n-1}$. Substitute $t=0$ into both sides of
(6.30) yields $C_{1}=1,\,C_{2}=0$. Then we determine the region of convergence
of (6.30). The sum of the two generators yields

$X_{1}\partial_{1}\partial_{2}+X_{1}\partial_{1}-X_{2}\partial^{2}_{2}+X_{2}\partial_{2}$

which gives rise to the equation

$x[f_{t}(x,\,t)-f_{t}(x-1,\,t)]+x[f(x,\,t)-f(x-1,\,t)]-tf_{tt}(x,\,t)+tf_{t}(x,\,t)=0.$

We differentiate above equation with respect to $t$ to obtain

$x[f_{tt}(x,\,t)-f_{tt}(x-1,\,t)]+x[f_{t}(x,\,t)-f_{t}(x-1,\,t)]-tf_{ttt}(x,\,t)+(t-1)f_{tt}(x,\,t)+f_{t}(x,\,t)=0.$
(A.7)

Since the manifestation of equation (4.35) in $\mathcal{O}_{\Delta d}$ is
given by

$f_{tt}(x,\,t)-2xtf_{tt}(x-1,\,t)-xf_{t}(x-1,\,t)-f(x,\,t)=0.$ (A.8)

Multiplying the equation (A.7) by $2t$ and subtract the resulting equation
from (A.8) yield the first equation in (A.9). Subtracting the (A.7) and (A.8)
yields the second equation in (A.9):

$\begin{split}&xf_{t}(x-1,\,t)=\dfrac{1}{1-2t}[(1-2tx+2t-2t^{2})f_{tt}(x,t)+2t^{2}f_{ttt}-2t(x+1)f_{t}(x,t)-f(x,t)],\\\
&xf_{tt}(x-1,\,t)=\dfrac{1}{1-2t}[(x-2+t)f_{tt}(x,t)-tf_{ttt}(x,t)+(x+1)f_{t}(x,t)+f(x,t)].\end{split}$
(A.9)

We replace $x$ by $x-1$ in the equation (A.8) and differentiate the resulting
equation with respect to $t$ yield

$f_{ttt}(x-1,\,t)-2(x-1)tf_{ttt}(x-2,\,t)-3(x-1)f_{tt}(x-2,\,t)-f_{t}(x-1,\,t)=0.$
(A.10)

We multiply $x$ to the equation (A.10) and subtract the resulting equation to
the equation (A.5) after it being differentiated three times and multiply by
the $2t$ throughout the whole equation would eliminate the term
$f_{ttt}(x-2,\,t)$ from the equation (A.10). Multiply the equation (A.10) by
$x$ and subtract 3 times the (A.5) after it being differentiated twice would
eliminate the term $f_{tt}(x-2,\,t)$ from the equation (A.10). This results in
the following fourth order equation

$\begin{split}&(x-2tx^{2}-7xt)f_{ttt}(x-1,\,t)-2xt^{2}f_{tttt}(x-1,\,t)+(-3x^{2}-3x)f_{tt}(x-1,\,t)-xf_{t}(x-1,\,t)\\\
&+2tf_{tttt}(x,\,t)+(3-2t)f_{ttt}(x,\,t)-3f_{tt}(x,\,t)=0.\end{split}$ (A.11)

We then eliminate the terms $f_{tttt}(x-1,\,t)$ and $f_{ttt}(x-1,\,t)$ from
the (A.11) utilizing the equation (A.8) after it being differentiated several
times. The substitutions of the terms $f_{tttt}(x-1,\,t)$ and
$f_{ttt}(x-1,\,t)$ from the equation (A.11) yield

$\begin{split}\dfrac{-3x}{2t}f_{tt}(x-1,t)-xf_{t}(x-1,t)+&tf_{tttt}(x,t)+(\dfrac{4t-4t^{2}-2xt+1}{2t})f_{ttt}(x,t)\\\
&+(t-3)f_{tt}-\dfrac{1-2tx-2t}{2t}f_{t}(x,t)=0.\end{split}$

Substituting (A.9) into above equation yields

$\begin{split}&tf_{tttt}(x,\,t)+\big{(}\dfrac{-4t^{3}+12t^{2}-5t-4xt^{2}+2xt-1}{2t(2t-1)}\big{)}f_{ttt}(x,\,t)\\\
&+\big{(}\dfrac{-10t^{2}-4t^{2}x+11t-6+3x}{2t(2t-1)}\big{)}f_{tt}(x,\,t)+\big{(}\dfrac{3x+4-4t-2xt}{2t(2t-1)}\big{)}f_{t}(x,\,t)\\\
&+\dfrac{3-2t}{2t(2t-1)}f(x,\,t)=0,\end{split}$ (A.12)

which is a fourth order linear ordinary differential equation in $t$ for each
fixed $x$. Since the equation (A.9) has a regular points at $t=0,\,1/2$, so
that every analytic solution (A.9) has a radius of convergence $1/2$ in the
neighbourhood of $x=0$. Hence the (6.30) converges uniformly in each compact
subset of $\mathbb{C}\times B(0;\frac{1}{2})$. Substituting a Frobenius
solution $f(x,\cdot)=\sum_{k=0}^{\infty}a_{k}x^{\sigma+k}$ into the equation
(A.9) further confirming that $\sigma=0,\,1,\,2$ are the only integer indicial
roots. Hence the (6.30) holds.

∎

##  Appendix B Difference operators with arbitrary step sizes

In practical applications, one would probably like to consider those
$D$-modules like $\mathcal{O}_{\Delta}$ from Example 2.3 or
$\mathcal{O}_{\Delta d}$ from Example 2.9 with any step size
$h\in\mathbb{C}\setminus\\{0\\}$. So we gather some formulae about these
$D$-modules and the corresponding realizations in various function spaces.

######  Example B.1.

Let $\mathcal{O}$ be the space of analytic functions. Then $\mathcal{O}$
becomes a left $\mathcal{A}$-module with

$\begin{split}&(\partial
f)(x)=\frac{1}{h}\Delta_{h}f(x)=\frac{1}{h}\big{(}f(x+h)-f(x)\big{)},\\\
&(Xf)(x)=xf(x-h).\end{split}$

This left $\mathcal{A}$-module is denoted by $\mathcal{O}_{\Delta_{h}}$.

Clearly $\mathcal{O}_{\Delta_{h}}$ is the same as $\mathcal{O}_{\Delta}$ when
$h=1$. Similarly in $\mathcal{O}_{\Delta_{h}}$, if we denote

$(x)_{n,h}:=x(x-h)\cdots\big{(}x-(n-1)h\big{)}=h^{n}\dfrac{\Gamma(x/h+1)}{\Gamma(x/h-n+1)},$

then we have

$X^{n}\cdot 1=x(x-h)\cdots\big{(}x-(n-1)h\big{)}=(x)_{n,h}.$

As a result, the element $L=(X\partial)^{2}+X^{2}-\nu^{2}$ gives the
difference Bessel equation

$\begin{split}\Big{(}\frac{x+2h)^{2}}{h^{2}}-\nu^{2}\Big{)}f(x+2h)&-\frac{(x+2h)(2x+3h)}{h^{2}}f(x+h)\\\
&+2\Big{(}1+\frac{1}{h^{2}}\Big{)}(x+2h)(x+h)f(x)=0,\end{split}$ (B.1)

which converges to Bohner and Cuchta’s difference Bessel equation (3.54) with
step $h\to 1$. One can also recover the classical Bessel equation (1.19) when
the step $h\to 0$ with a more careful analysis. Classical solutions of (B.1)
are solutions of $\mathcal{D}/\mathcal{D}L$ in $\mathcal{O}_{\Delta_{h}}$. One
of them is

$\mathcal{D}/\mathcal{D}L\stackrel{{\scriptstyle\times
S_{\nu}}}{{\longrightarrow}}\mathcal{D}/\mathcal{D}(X\partial-\nu)\stackrel{{\scriptstyle\times(x)_{\nu,h}}}{{\longrightarrow}}\mathcal{O}_{\Delta_{h}},$
(B.2)

in which
$S_{\nu}=\displaystyle\sum_{k=0}^{\infty}\frac{(-1)^{k}X^{2k}}{2^{\nu+2k}k!\Gamma(\nu+k+1)}$,
and gives rise to the Newton series

$J^{\Delta_{h}}_{\nu}(x):=\sum_{k=0}^{\infty}\frac{(-1)^{k}h^{2k}}{2^{\nu+2k}k!\Gamma(\nu+k+1)}(x)_{\nu+2k,h}.$
(B.3)

Another solution in $\mathcal{O}_{\Delta_{h}}$ corresponding to $S_{-\nu}$
yields the Newton series

$J^{\Delta_{h}}_{-\nu}(x):=\sum_{k=0}^{\infty}\frac{(-1)^{k}h^{2k}}{2^{-\nu+2k}k!\Gamma(-\nu+k+1)}(x)_{-\nu+2k,h},$
(B.4)

which is “linearly independent” to the first one in case that $2\nu$ is not an
integer.

######  Example B.2.

Let $\mathcal{O}_{2}$ be the space of analytic functions in two variables. It
becomes a left $\mathcal{A}_{2}$-module when endowed with the structure

$\begin{array}[]{ll}\partial_{1}f(x,\,t)=\frac{1}{h}\big{(}f(x+h,\,t)-f(x,\,t)\big{)},&X_{1}f(x,\,t)=xf(x-h,\,t);\\\
\partial_{2}f(x,\,t)=f_{t}(x,\,t),&X_{2}f(x,\,t)=tf(x,\,t).\end{array}$ (B.5)

Such a left $\mathcal{A}_{2}$-module is denoted by
$\mathcal{O}_{\Delta_{h}d}$.

######  Example B.3.

Let $\mathcal{O}^{\mathbb{Z}}$ be the space of all bilateral sequences of
analytic functions with an appropriately imposed growth restriction. Let

$(f_{n})=(\cdots,\,f_{-1},\,f_{0},\,f_{1},\,\cdots)$

denote a bilateral infinite sequence of $f_{k}\in\mathcal{O}$. Then
$\mathcal{O}^{\mathbb{Z}}$ becomes a left $\mathcal{A}_{2}$-module by

$\begin{array}[]{ll}(\partial_{1}f)_{n}(x)=\frac{1}{h}\big{(}f_{n}(x+h)-f_{n}(x)\big{)},&(X_{1}f)_{n}(x)=xf_{n}(x-h),\\\
(\partial_{2}f)_{n}(x)=(n+1)f_{n+1}(x),&(X_{2}f)_{n}(x)=f_{n-1}(x),\end{array}$
(B.6)

for all bilateral sequences of analytic functions $(f_{n})$. Such a left
$\mathcal{A}_{2}$-module is sometimes denoted by
$\mathcal{O}_{\Delta_{h}}^{\mathbb{Z}}$.

We state without proof the following extensions of the Theorem 3.41 with an
arbitrary step size $h$:

######  Theorem B.4.

Let $\nu\in\mathbb{C}$.

  1. (1)

Let $(\mathscr{C}^{\Delta}_{n+\nu})_{n}$ be a bilateral sequence of analytic
functions which is a solution of the Bessel module $\mathcal{B}_{\nu}$ in
$\mathcal{O}_{\Delta}^{\mathbb{Z}}$. Then there exists a $1$-periodic function
$C_{\nu}$ such that

$e^{i\pi\nu}\frac{\sin(x/h-\nu)\pi}{\sin(\pi
x/h)}\,t^{-\nu}\big{[}\frac{h}{2}(t-\frac{1}{t})+1\big{]}^{x/h}\sim\sum_{n=-\infty}^{\infty}J^{\Delta_{h}}_{n+\nu}(x)\,t^{n},$
(B.7)

where the symbol $\sim$ means that the left-hand side is the Borel resummation
434343Definition 2.7 of the right-hand side whenever it diverges.

  2. (2)

Moreover, the manifestation of the holonomic system of PDEs (delay-
differential equations) (3.14) in $\mathcal{O}_{\Delta d}$ in Example B.2
defined by (2.4) is given by

$y(x,t)-y(x-h,t)=\frac{h}{2}(t-\frac{1}{t})y(x-h,t),\quad\nu
y(x,t)+ty_{t}(x,t)=\frac{x}{2}\big{(}t+\frac{1}{t}\big{)}y(x-h,t).$ (B.8)

We note that although the above result is valid for each fixed $h\not=0$, one
cannot take the limit $h\to 0$ so that the (B.8) becomes the generating
function (1.6) for the classical Bessel functions, unless when $\nu=0$, as in
the following theorem.

######  Theorem B.5.

  1. (1)

Let $(J^{\Delta_{h}}_{n}(x))$ and $(J_{n}(x))$ be sequences in
$\mathcal{O}_{\Delta_{h}}^{\mathbb{Z}}$ and ${\mathcal{O}_{d}}^{\mathbb{Z}}$.
Then we have the following limits

${\displaystyle\big{[}\frac{h}{2}(t-\frac{1}{t})+1\big{]}^{x/h}\qquad=}$${\displaystyle\sum_{n=-\infty}^{\infty}J^{\Delta_{h}}_{n}(x)\,t^{n}}$${\displaystyle\exp\big{[}\frac{x}{2}(t-\frac{1}{t})\big{]}\qquad=}$${\displaystyle\sum_{n=-\infty}^{\infty}J_{n}(x)\,t^{n}}$$\scriptstyle{h\to
0}$$\scriptstyle{h\to 0}$ (B.9)

  2. (2)

Moreover, the system of PDEs (B.5) with $\nu=0$, that is

$y(x,t)-y(x-h,t)=\frac{h}{2}(t-\frac{1}{t})y(x-h,t),\quad
ty_{t}(x,t)=\frac{x}{2}\big{(}t+\frac{1}{t}\big{)}y(x-h,t).$ (B.10)

becomes the system of PDEs (1.13) as $h\to 0$.

The case of the above limits when $\nu\not=0$, while likely to hold, is more
delicate that would be dealt with in a separate publication.

##  Appendix C List of tables

###  C.1. List of commonly used $D$-modules

$D$-module |  | “ground”
---
Weyl algebra
| underlying
---
space
Manifestation
$\mathcal{A}/\mathcal{A}L$ | $\mathcal{A}$ | $\mathcal{A}/\mathcal{A}L$ |  same as multiplication in $\mathcal{A}$
$\dfrac{\mathcal{A}_{2}}{\mathcal{A}_{2}L_{1}+\mathcal{A}_{2}L_{2}}$ | $\mathcal{A}_{2}$ | $\dfrac{\mathcal{A}_{2}}{\mathcal{A}_{2}L_{1}+\mathcal{A}_{2}L_{2}}$ |  same as multiplication in $\mathcal{A}_{2}$
$\mathbb{C}^{\mathbb{N}_{0}}$ | $\mathcal{A}$ | $\mathbb{C}^{\mathbb{N}_{0}}$ |  | $(\partial a)_{n}=a_{n+1}$
---
$(Xa)_{n}=na_{n-1}$
$\mathbb{C}^{\mathbb{Z}}$ | $\mathcal{A}$ | $\mathbb{C}^{\mathbb{Z}}$ |  | $(\partial a)_{n}=(n+1)a_{n+1},$
---
$(Xa)_{n}=a_{n-1}$
$\mathcal{O}_{d}$ | $\mathcal{A}$ | $\mathcal{O}$ |  | $\partial f(x)=f^{\prime}(x),$
---
$Xf(x)=xf(x)$
$\mathcal{O}_{\Delta}$ | $\mathcal{A}$ | $\mathcal{O}$ |  | $\partial f(x)=f(x+1)-f(x),$
---
$Xf(x)=xf(x-1)$
$\mathcal{O}_{dd}$ | $\mathcal{A}_{2}$ | $\mathcal{O}_{2}$ |  | $\partial_{1}f(x,t)=f_{x}(x,t),$
---
$X_{1}f(x,t)=xf(x,t)$
$\partial_{2}f(x,t)=f_{t}(x,t),$
$X_{2}f(x,t)=tf(x,t)$
$\mathcal{O}_{\Delta d}$ | $\mathcal{A}_{2}$ | $\mathcal{O}_{2}$ |  | $\partial_{1}f(x,t)=f(x+1,t)-f(x,t),$
---
$X_{1}f(x,t)=xf(x-1,t)$
$\partial_{2}f(x,t)=f_{t}(x,t),$
$X_{2}f(x,t)=tf(x,t)$
$\mathcal{O}_{d}^{\mathbb{N}_{0}}$ | $\mathcal{A}_{2}$ | $\mathcal{O}^{\mathbb{N}_{0}}$ |  | $(\partial_{1}f)_{n}(x)=f_{n}^{\prime}(x),$
---
$(X_{1}f)_{n}(x)=xf_{n}(x)$
$(\partial_{2}f)_{n}(x)=f_{n+1}(x),$
$(X_{2}f)_{n}(x)=nf_{n-1}(x)$
$\mathcal{O}_{d}^{\mathbb{Z}}$ | $\mathcal{A}_{2}$ | $\mathcal{O}^{\mathbb{Z}}$ |  | $(\partial_{1}f)_{n}(x)=f_{n}^{\prime}(x),$
---