# New results on the associated Meixner, Charlier, Laguerre, and Krawtchouk
polynomials

Khalid Ahbli

###### Abstract

We give new explicit representations as well as new generating functions for
the associated Meixner, Charlier, Laguerre, and Krawtchouk polynomials. The
obtained results are then used to derive new generating functions and
convolution-type formulas of the corresponding classical polynomials. Some
consequences of our results are also mentioned.

Faculty of Sciences of Agadir, Ibn Zohr University, Morocco

E-mail<EMAIL_ADDRESS>
Keywords: Explicit representation, Associated orthogonal polynomials,
Hypergeometric functions, Generating function.
Mathematics Subject Classification (2020): 33C45-05A15-33C20

##  1 Introduction

A set of polynomials $\\{P_{n}(x)\\}$ is orthogonal if $P_{n}(x)$ is a
polynomial of exact degree $n$ and there is a positive measure $d\mu$ on the
real line with an infinite number of points increase and for which all the
moments are finite such that
$\int_{\mathbb{R}}P_{n}(x)P_{m}(x)d\mu(x)=h_{n}\delta_{nm},\ n,m=0,1,\cdots,\
(h_{n}>0).$ A necessary and sufficient condition for orthogonality [21] is
that $\\{P_{n}(x)\\}$ satisfies the three-term recurrence relation

$\begin{split}&A_{n}P_{n+1}(x)=\left(B_{n}\,x+C_{n}\right)P_{n}(x)-D_{n}P_{n-1}(x),\quad
n=0,1,2,\cdots,\\\ &P_{-1}(x)=0,\text{ }P_{0}(x)=1,\end{split}$ (1.1)

where $A_{n},\,B_{n},\,C_{n},\,D_{n}$ are real coefficients such that

$A_{n-1}B_{n-1}B_{n}D_{n}>0,\quad n\geq 1.$ (1.2)

The associated orthogonal polynomials $\\{P_{n}(x;\gamma)\\}$ are defined by
(1.1)-(1.2) in terms of coefficients $A_{n+\gamma}$, $B_{n+\gamma}$,
$C_{n+\gamma}$ and $D_{n+\gamma}$ where $\gamma\geq 0$ is a real association
parameter. These polynomials have recently generated wide interest and some
work has been devoted to the task of obtaining explicit representation (or
closed-form expression) for these polynomials. This has been done successfully
for the associated Jacobi polynomials [1] and their special cases, including
the associated Gegenbauer [2], Laguerre and Hermite polynomials [3] and more
recently for the associated Meixner-Pollaczek polynomials [4]. Associated
orthogonal polynomials have many applications in diverse fields, such as
queuing and inventory models, chemical kinetics, population dynamics, and
quantum optics. For instance, it was shown in [10] that associated Meixner
polynomials are birth and death process polynomials (a stationary Markov
process) with rates $\lambda_{n}=c(n+\gamma+\beta)$, $\mu_{n}=n+\gamma$,
$0<c<1$. In [18, 19], associated Meixner-Pollaczek and Hermite polynomials
were used to construct some new sets of nonlinear coherent states which play
an important role in quantum optics. For other works on associated orthogonal
polynomials and their applications see the articles [5, 6, 7, 8, 9, 10, 11,
12, 13, 14, 15, 16, 17, 18, 19], the book [20] and their references.

In this paper, we will be concerned with the associated Meixner polynomials
(AMPs), denoted $\mathscr{M}_{n}(x;\beta,c,\gamma)$, that correspond to
coefficients
$A_{n+\gamma}=c,\,B_{n+\gamma}=c-1,\,C_{n+\gamma}=(c+1)(n+\gamma)+\beta
c,\,D_{n+\gamma}=(n+\gamma)(n+\gamma+\beta-1)$. These polynomials were first
studied by Ismail et al. in [10] where a generating function was derived. They
proved that the generating function of these polynomials has the following
integral representation:

$\sum_{n=0}^{+\infty}\frac{(c\,t)^{n}}{(\gamma+1)_{n}}\mathscr{M}_{n}(x;\beta,c,\gamma)=\gamma\left(1-c\,t\right)^{-\beta-x}\left(1-t\right)^{x}\int_{0}^{1}u^{\gamma-1}\left(1-c\,t\,u\right)^{x+\beta-1}\left(1-t\,u\right)^{-x-1}d\,u,$
(1.3)

valid for $\beta>0$, $\gamma\geq 0$ and $c\neq 1$ (see [10, p.345(4.4)]). We
will show how these polynomials are connected to the associated Meixner-
Pollaczek polynomials, $\mathscr{P}_{n}^{(\nu)}(x;\varphi,\gamma)$, obtained
in [22]. Next, we use this connection relation (see Eq.(2.3) below) and the
explicit representation of the polynomials
$\mathscr{P}_{n}^{(\nu)}(x;\varphi,\gamma)$, recently obtained by Luo and
Raina in [4], to derive a new explicit expression for the polynomials
$\mathscr{M}_{n}(x;\beta,c,\gamma)$. The resulting formula will then be used
to obtain a new generating function of these polynomials. Furthermore, we
exploit the obtained results to derive new explicit representations as well as
generating functions for the associated Charlier, Laguerre, and Krawtchouk
polynomials. Different interesting identities, generating functions and
convolution-type formulas (in terms of classical polynomials) are also
established. Some proofs of the stated results are postponed to Sect. 7. A
conclusion is also given at the end of this paper.

##  2 A new explicit formulae and generating function for the AMPs

In this section, we will give a new explicit representation for the AMPs in
terms of terminating ${}_{4}F_{3}(1)$-series which we deduced from the one of
the associated Meixner-Pollaczek polynomials obtained in [4]. The derived
representation is then used to compute a new generating function of these
polynomials.

###  2.1 Explicit representation of the AMPs

The AMPs $\mathscr{M}_{n}(x;\beta,c,\gamma)$ satisfy the three-term recurrence
relation:

$\begin{split}c\mathscr{M}_{n+1}(x;\beta,c,\gamma)=[(c-1)x+(c+1)(n+\gamma)&+\beta
c]\mathscr{M}_{n}(x;\beta,c,\gamma)\\\
&-(n+\gamma)(n+\gamma+\beta-1)\mathscr{M}_{n-1}(x;\beta,c,\gamma),\end{split}$
(2.1)

with initial conditions $\mathscr{M}_{-1}(x;\beta,c,\gamma)=0$ and
$\mathscr{M}_{0}(x;\beta,c,\gamma)=1$. The conditions for orthogonality (1.2)
require that $c>0,\,c\neq 1$ and $\gamma+\beta>0$. Now, we proceed to
establish an explicit representation for the AMPs. Our main tools here are the
explicit representation for associated Meixner-Pollaczek polynomials
$\mathscr{P}_{n}^{(\nu)}(x;\varphi,\gamma)$ obtained Luo and Raina in [4,
p.3(1.8)],

$\displaystyle\begin{split}\mathscr{P}_{n}^{(\nu)}(x;\varphi,\gamma)=\frac{(2\nu+\gamma)_{n}}{n!}\sum_{k=0}^{n}e^{i(n-k)\varphi}\left(e^{i\varphi}-e^{-i\varphi}\right)^{k}\frac{(-n)_{k}(\gamma+\nu+ix)_{k}}{(\gamma+1)_{k}(\gamma+2\nu)_{k}}\\\
\times{}_{4}F_{3}\left(\begin{array}[]{c}k-n,\,\gamma+\nu+ix+k,\,\gamma+2\nu-1,\,\gamma\\\
\gamma+\nu+ix,\,\gamma+2\nu+k,\,\gamma+1+k\end{array}\big{|}1\right),\end{split}$
(2.2)

and the following connection relation:

$\mathscr{P}_{n}^{(\nu)}(x;\varphi,\gamma)=\frac{e^{-in\varphi}}{(\gamma+1)_{n}}\mathscr{M}_{n}(ix-\nu;2\nu,e^{-2i\varphi},\gamma),$
(2.3)

which can be checked by comparing the recurrence relation (2.1) of
$\mathscr{M}_{n}(x;\beta,c,\gamma)$ with the recurrence relation of
$\mathscr{P}_{n}^{(\nu)}(x;\varphi,\gamma)$ given in [22, p.2256] by

$\displaystyle\begin{split}(n+\gamma+1)\mathscr{P}_{n+1}^{(\nu)}(x;\varphi,\gamma)=2[(n+\gamma+\nu)\cos\varphi+&x\sin\varphi]\mathscr{P}_{n}^{(\nu)}(x;\varphi,\gamma)\\\
&-(n+\gamma+2\nu-1)\mathscr{P}_{n-1}^{(\nu)}(x;\varphi,\gamma).\end{split}$
(2.4)

Here, $(a)_{k}$ is the Pochhammer symbol defined by $(a)_{0}=1$,
$(a)_{k}=a(a+1)\cdots(a+k-1)$, $k\geq 1$. We are now in position to state the
following result.

######  Proposition 2.1.

We have the following explicit representation for the AMPs

$\displaystyle\begin{split}\mathscr{M}_{n}(x;\beta,c,\gamma)=c^{-n}\frac{(\gamma+1)_{n}(\gamma+\beta)_{n}}{n!}\sum_{k=0}^{n}(1-c)^{k}\frac{(-n)_{k}(\gamma+\beta+x)_{k}}{(\gamma+1)_{k}(\gamma+\beta)_{k}}\\\
\times{}_{4}F_{3}\left(\begin{array}[]{c}k-n,\,\gamma+\beta+x+k,\,\gamma+\beta-1,\,\gamma\\\
\gamma+\beta+x,\,\gamma+\beta+k,\,\gamma+1+k\end{array}\big{|}1\right),\end{split}$
(2.5)

valid for $c>0,\,c\neq 1$ and $\gamma+\beta>0$.

In view of the relation
$\mathscr{M}_{n}(x;\beta,c,\gamma)=c^{-n}\mathscr{M}_{n}(-\beta-x;\beta,c^{-1},\gamma)$,
which follows directly from the recurrence relation, we obtain from (2.5)
another expression of the AMPs given by

$\displaystyle\begin{split}\mathscr{M}_{n}(x;\beta,c,\gamma)=\frac{(\gamma+1)_{n}(\gamma+\beta)_{n}}{n!}\sum_{k=0}^{n}\tilde{c}^{k}\frac{(-n)_{k}(\gamma-x)_{k}}{(\gamma+1)_{k}(\gamma+\beta)_{k}}{}_{4}F_{3}\left(\begin{array}[]{c}k-n,\,\gamma-x+k,\,\gamma+\beta-1,\,\gamma\\\
\gamma-x,\,\gamma+\beta+k,\,\gamma+1+k\end{array}\big{|}1\right)\end{split}$
(2.6)

where $\tilde{c}=\frac{c-1}{c}$. The case $\gamma=0$ in (2.6) corresponds to
the Meixner polynomials [21, p.176 (3.5)]:

$M_{n}(x;\beta,c)=(\beta)_{n}\,{}_{2}F{}_{1}\left(\begin{array}[]{c}-n,\,-x\\\
\beta\end{array}\left|1-\frac{1}{c}\right.\right).$ (2.7)

In the special case $\gamma+\beta=1$, the polynomials
$\mathscr{M}_{n}(x;\beta,c,\gamma)$ reduce to a particular case of Meixner
polynomials. In fact, when $\gamma+\beta=1$ Eq.(2.6) reduces to

$\displaystyle\mathscr{M}_{n}(x;\beta,c,1-\beta)=(2-\beta)_{n}\sum_{k=0}^{n}\frac{(-n)_{k}(1-\beta-x)_{k}}{(2-\beta)_{k}}\frac{\tilde{c}^{k}}{k!}=(2-\beta)_{n}\,{}_{2}F{}_{1}\left(\begin{array}[]{c}-n,\,1-\beta-x\\\
2-\beta\end{array}\left|\tilde{c}\right.\right),$ (2.10)

and hence

$\mathscr{M}_{n}(x;\beta,c,1-\beta)=M_{n}(x+\beta-1;2-\beta,c).$ (2.11)

The above connection formula is known; see [23, Eq.(11)].

######  Corollary 2.1.

The following finite sum

$\displaystyle\begin{split}&\sum_{k=0}^{n}t^{k}\frac{(-n)_{k}(a+y)_{k}}{(a+1)_{k}(b+1)_{k}}{}_{4}F_{3}\left(\begin{array}[]{c}k-n,\,a+y+k,\,a,\,b\\\
a+y,\,b+1+k,\,a+1+k\end{array}\big{|}1\right)&\\\
&=\frac{n!}{b-a}\left\\{\frac{b}{(a+1)_{n}}\,{}_{2}F_{1}\left(\begin{array}[]{c}1-y,a\\\
a-b+1\end{array}\big{|}t\right){}_{2}F_{1}\left(\begin{array}[]{c}y,-n-a\\\
b-a+1\end{array}\big{|}t\right)\right.&\\\
&\left.-(1-t)^{n+1}\frac{a}{(b+1)_{n}}\,{}_{2}F_{1}\left(\begin{array}[]{c}1-y,n+a+1\\\
a-b+1\end{array}\big{|}t\right){}_{2}F_{1}\left(\begin{array}[]{c}y,1-a\\\
b-a+1\end{array}\big{|}t\right)\right\\}&\end{split}$ (2.12)

is valid for $a>0,\,b>-1,\,b\neq 0$ and $b-a\neq 0,1,2,...$ .

For special values of parameters ($y=1$ or $t=0$), we obtain:

$\begin{split}\sum_{k=0}^{n}t^{k}\frac{(-n)_{k}}{(b+1)_{k}}{}_{3}F_{2}\left(\begin{array}[]{c}k-n,\,a,\,b\\\
a+1,\,b+1+k\end{array}\big{|}1\right)=&\frac{n!}{b-a}\left\\{\frac{b}{(a+1)_{n}}\,{}_{2}F_{1}\left(\begin{array}[]{c}1,-n-a\\\
b-a+1\end{array}\big{|}t\right)\right.\\\
&\left.-(1-t)^{n+1}\frac{a}{(b+1)_{n}}\,{}_{2}F_{1}\left(\begin{array}[]{c}1,1-a\\\
b-a+1\end{array}\big{|}t\right)\right\\},\end{split}$

and

${}_{3}F_{2}\left(\begin{array}[]{c}-n,\,a,\,b\\\
a+1,\,b+1\end{array}\big{|}1\right)=\frac{n!}{b-a}\left\\{\frac{b}{(a+1)_{n}}-\frac{a}{(b+1)_{n}}\right\\}.$
(2.13)

######  Remark 2.1.

Note that the only known expression of these polynomials was obtained in a
quadratic form (cross products) made out of Gauss hypergeometric functions
${}_{2}F_{1}$ as (see [23, Eq(21)]):

$\displaystyle\begin{split}\mathscr{M}_{n}(x;\beta,c,\gamma)=\frac{1}{\beta-1}\left\\{(\gamma+\beta-1)_{n+1}\,{}_{2}F_{1}\left(\begin{array}[]{c}x+1,\gamma\\\
2-\beta\end{array}\big{|}\tilde{c}\right){}_{2}F_{1}\left(\begin{array}[]{c}-x,-n-\gamma\\\
\beta\end{array}\big{|}\tilde{c}\right)\right.\\\
\left.-(\gamma)_{n+1}\,{}_{2}F_{1}\left(\begin{array}[]{c}x+\beta,\gamma+\beta-1\\\
\beta\end{array}\big{|}\tilde{c}\right){}_{2}F_{1}\left(\begin{array}[]{c}1-\beta-x,-n-\gamma-\beta+1\\\
2-\beta\end{array}\big{|}\tilde{c}\right)\right\\}\end{split}$ (2.14)

where $\tilde{c}=\frac{c-1}{c}$. This relation is valid under restrictions
$\gamma>0$, $\gamma+\beta\in\mathbb{R}_{+}^{*}\backslash\\{1\\}$ and
$\beta\notin\mathbb{N}^{*}$.

###  2.2 A generating function for the AMPs

Let us first notice that the generating function (1.3) may be deduced from the
generating function for the associated Meixner-Pollaczek polynomials derived
in [4, p.3(1.7)] with the help of the connection relation (2.3). Precisely, we
have

$\displaystyle\sum_{n=0}^{+\infty}\frac{(c\,t)^{n}}{(\gamma+1)_{n}}\mathscr{M}_{n}(x;\beta,c,\gamma)=\left(1-c\,t\right)^{-\beta-x}\left(1-t\right)^{x}F_{1}\left[\gamma,1-\beta-x,1+x;\gamma+1;c\,t,t\right],$
(2.15)

where $F_{1}$ is the first Appell hypergeometric function defined by (7.11).
Then, the integral in the R.H.S of (1.3) is just the integral representation
of the Appell function $F_{1}$ (see [27, p.77(4)]). In addition, if we put
$\gamma=0$ in (2.15), we recover the following generating function of Meixner
polynomials

$\displaystyle\sum_{n=0}^{+\infty}\frac{(c\,t)^{n}}{n!}M_{n}(x;\beta,c)=\left(1-c\,t\right)^{-\beta-x}\left(1-t\right)^{x}.$
(2.16)

Substitute (2.16) into (1.3) with $(t,\beta,x)=(tu,2-\beta,-x-1)$, integrate
the obtained expression, and then use the Rainville formula [25, p.101],

$\sum_{n=0}^{\infty}\sum_{k=0}^{\infty}A(k,n)=\sum_{n=0}^{\infty}\sum_{k=0}^{n}A(k,n-k),$
(2.17)

to get an expression of $\mathscr{M}_{n}(x;\beta,c,\gamma)$ in terms of
$M_{n}(x;\beta,c)$ as follows .

######  Proposition 2.2.

Let $\gamma>0$. Then, we have the convolution-type formula

$\frac{n!\,\mathscr{M}_{n}(x;\beta,c,\gamma)}{(\gamma+1)_{n}}=\sum_{k=0}^{n}\displaystyle\binom{n}{k}\frac{\gamma}{(k+\gamma)}M_{n-k}(x;\beta,c)M_{k}(-x-1;2-\beta,c).$
(2.18)

Moreover, we have the generating function

$\sum_{n=0}^{+\infty}\frac{\gamma\,(c\,t)^{n}}{(n+\gamma)n!}M_{n}(x;\beta,c)=F_{1}\left[\gamma,x+\beta,-x;\gamma+1;c\,t,t\right],\quad|t|<|c|^{-1},\,(\gamma\text{
arbitrary}).$ (2.19)

The last formula is obtained by direct computation with the help of generating
functions (2.15) and (2.16).

We end this subsection by stating a new generating function for the AMPs (see
Sect. 7 for a proof).

######  Theorem 2.1.

A generating function for the AMPs is given by

$\displaystyle\sum_{n=0}^{+\infty}\frac{(c\,t)^{n}}{(\gamma+\beta)_{n}}\mathscr{M}_{n}(x;\beta,c,\gamma)=\left(1-c\,t\right)^{-1}F_{1}\left[1,\gamma,-x;\gamma+\beta;t,\frac{t(1-c)}{1-ct}\right]$
(2.20)

for $|t|<\min\\{1,\,c^{-1}\\}$, in terms of Appell function $F_{1}$ defined by
(7.11). In particular, we have

$\displaystyle\sum_{n=0}^{+\infty}M_{n}(x;\beta,c)t^{n}=\left(1-t\right)^{-1}{}_{2}F_{1}\left(\begin{array}[]{c}1,\,-x\\\
\beta\end{array}\left|\frac{t(1-c)}{c(1-t)}\right.\right).$ (2.23)

##  3 The associated Charlier polynomials

The associated Charlier polynomials (ACPs) are defined by the three-term
recurrence relation:

$\displaystyle
a\,\mathscr{C}_{n+1}(x;a,\gamma)=(n+\gamma+a-x)\mathscr{C}_{n}(x;a,\gamma)-(n+\gamma)\mathscr{C}_{n-1}(x;a,\gamma)$
(3.1)

with initial conditions $\mathscr{C}_{-1}(x;a,\gamma)=0$ and
$\mathscr{C}_{0}(x;a,\gamma)=1$. Applying the criterion (1.2) we find that
$\mathscr{C}_{n}(x;a,\gamma)$ are orthogonal if $a>0$. By comparing the
recurrence relations of the AMPs and ACPs, it is easy to check the limiting
relation (see [23, (35)]):

$\mathscr{C}_{n}(x;a,\gamma)=\lim\limits_{\beta\rightarrow\infty}\frac{1}{(\gamma+\beta)_{n}}\mathscr{M}_{n}\left(x;\beta,\frac{a}{a+\beta},\gamma\right).$
(3.2)

Representation (2.6) provides for ACPs two interesting explicit formulas. The
first one is obtained thanks to the relation (3.2), upon using the limit

$\lim\limits_{a\to\infty}\frac{(ax)^{k}}{(ay+b)_{k}}=\left(\frac{x}{y}\right)^{k},\
k\geq 1,$ (3.3)

where $x,y,b$ are fixed, and is given by

$\displaystyle\mathscr{C}_{n}(x;a,\gamma)=\frac{(\gamma+1)_{n}}{n!}\sum_{k=0}^{n}(-a)^{-k}\frac{(-n)_{k}(\gamma-x)_{k}}{(\gamma+1)_{k}}{}_{3}F_{2}\left(\begin{array}[]{c}k-n,\,\gamma-x+k,\,\gamma\\\
\gamma-x,\,\gamma+k+1\end{array}\big{|}1\right),\ a>0.$ (3.6)

The second formula is obtained from (3.6) by applying the transformation ([26,
p.142]):

$\displaystyle{}_{3}F{}_{2}\left(\begin{array}[]{c}-m,\,a,\,b\\\
c,\,d\end{array}\big{|}1\right)=\frac{(c-a)_{m}}{(c)_{m}}{}_{3}F_{2}\left(\begin{array}[]{c}-m,\,a,\,d-b\\\
a-c+1-m,\,d\end{array}\big{|}1\right).$ (3.11)

It is given by

$\displaystyle\mathscr{C}_{n}(x;a,\gamma)=\sum_{k=0}^{n}(-a)^{-k}\frac{(-n)_{k}(\gamma-x)_{k}}{k!}\
{}_{3}F_{2}\left(\begin{array}[]{c}-k,\,\gamma,\,k-n\\\
-n,\,\gamma-x\end{array}\big{|}1\right).$ (3.14)

Following the same lines as for the AMPs, we derive the generating function
for the ACPs. Precisely, we have the following result (see Sect. 7 for a
proof).

######  Proposition 3.1.

A generating function for the ACPs is given by

$\displaystyle\sum_{n=0}^{+\infty}\frac{t^{n}}{(\gamma+1)_{n}}\mathscr{C}_{n}(x;a,\gamma)=e^{t}\left(1-\frac{t}{a}\right)^{x}\Phi_{1}\left[\gamma,x+1;\gamma+1;\frac{t}{a},-t\right],\quad|t|<|a|,$
(3.15)

where $\Phi_{1}$ is the Humbert’s confluent hypergeometric function defined by
(7.24).
In particular, we have

$\displaystyle\sum_{n=0}^{+\infty}\frac{t^{n}}{n!}C_{n}(x;a)=e^{t}\left(1-\frac{t}{a}\right)^{x}.$
(3.16)

Let us now apply an alternative method to compute the generating function of
$\mathscr{C}_{n}(x;a,\gamma)$. Multiplying both the sides of (3.1) by
$t^{n}/(\gamma+1)_{n}$ and then summing with respect to $n$ from $0$ to
$+\infty$, to get the following first-order differential equation

$t(a-t)\frac{\partial}{\partial
t}\mathscr{G}_{\gamma}(x,t)+\left[t^{2}+(x-a-\gamma)t+a\gamma\right]\mathscr{G}_{\gamma}(x,t)=a\gamma$
(3.17)

where $\mathscr{G}_{\gamma}(x,t)$ denotes the L.H.S of (3.15). Next, we make
the substitution
$\mathscr{G}_{\gamma}(x,t)=e^{t}\left(1-t/a\right)^{x}\mathscr{H}_{\gamma}(x,t)$
and obtain for $\mathscr{H}_{\gamma}(x,t)$ the following differential
equation:

$t\frac{\partial}{\partial
t}\mathscr{H}_{\gamma}(x,t)+\gamma\mathscr{H}_{\gamma}(x,t)=\gamma
e^{-t}\left(1-\frac{t}{a}\right)^{-x-1}.$ (3.18)

The solution of the above equation, after taking into account the condition
$\mathscr{G}_{\gamma}(x,0)=1$, is

$\mathscr{H}_{\gamma}(x,t)=\gamma\int_{0}^{1}u^{\gamma-1}e^{-u\,t}\left(1-\frac{u\,t}{a}\right)^{-x-1}du,\quad|t|<\min\\{1,\,|a|\\}.$
(3.19)

Thus, $\mathscr{G}_{\gamma}(x,t)$ has the following integral representation

$\mathscr{G}_{\gamma}(x,t)=\gamma
e^{t}\left(1-\frac{t}{a}\right)^{x}\int_{0}^{1}u^{\gamma-1}e^{-t\,u}\left(1-\frac{t\,u}{a}\right)^{-x-1}du.$
(3.20)

Again, substitute (3.16) into (3.20) with $(t,\beta,x)=(-tu,-a,-x-1)$,
integrate the obtained expression, and then use (2.17), to get the following
corollary.

######  Corollary 3.1.

Let $\gamma>0$. Then we have

$\frac{n!\,\mathscr{C}_{n}(x;a,\gamma)}{(\gamma+1)_{n}}=\sum\limits_{k=0}^{n}\displaystyle\binom{n}{k}\frac{\gamma(-1)^{k}}{(\gamma+k)}C_{n-k}(x;a)C_{k}(-x-1;-a).$
(3.21)

Consequently, we get the following generating function

$\displaystyle\sum_{n=0}^{+\infty}\frac{\gamma}{\gamma+n}C_{n}(x;a)\frac{t^{n}}{n!}=\Phi_{1}\left[\gamma,-x;\gamma+1;\frac{t}{a},t\right],\quad|t|<|a|,\,(\gamma\text{
arbitrary}).$ (3.22)

##  4 The associated Laguerre polynomials

The associated Laguerre polynomials (ALPs) can be defined by the three-term
recurrence relation:

$\displaystyle\begin{split}(n+\gamma+1)\mathscr{L}_{n+1}^{(\alpha)}(x;\gamma)=[2(n+\gamma)+\alpha+1-x]\mathscr{L}_{n}^{(\alpha)}(x;\gamma)-(n+\gamma+\alpha)\mathscr{L}_{n-1}^{(\alpha)}(x;\gamma)\end{split}$
(4.1)

with initial conditions $\mathscr{L}_{-1}^{(\alpha)}(x;\gamma)=0$ and
$\mathscr{L}_{0}^{(\alpha)}(x;\gamma)=1$. Again, with the help of the
criterion (1.2), we show that $\mathscr{L}_{n}^{(\alpha)}(x;\gamma)$ are
orthogonal if and only if $\alpha+\gamma>-1$. By comparing (2.1) with (4.1) it
is easy to check the limit relation (see [23, (39)]):

$\mathscr{L}_{n}^{(\alpha)}(x;\gamma)=\lim\limits_{c\rightarrow
1}\frac{1}{(\gamma+1)_{n}}\mathscr{M}_{n}\left(\frac{x}{1-c};\alpha+1,c,\gamma\right).$
(4.2)

This relation allows us to write an explicit formula for the ALPs as follows

$\mathscr{L}_{n}^{(\alpha)}(x;\gamma)=\frac{(\gamma+\alpha+1)_{n}}{n!}\sum_{k=0}^{n}\frac{(-n)_{k}\,x^{k}}{(\gamma+1)_{k}(\gamma+\alpha+1)_{k}}{}_{3}F_{2}\left(\begin{array}[]{c}k-n,\,\gamma+\alpha,\,\gamma\\\
\gamma+\alpha+k+1,\,\gamma+k+1\end{array}\big{|}1\right)$ (4.3)

where $\alpha>-1$. Using the transformation (3.11), these polynomials can be
rewritten as follows (see also [13, Eq(1.34)]):

$\mathscr{L}_{n}^{(\alpha)}(x;\gamma)=\frac{(\alpha+1)_{n}}{n!}\sum_{k=0}^{n}\frac{(-n)_{k}\,x^{k}}{(\gamma+1)_{k}(\alpha+1)_{k}}{}_{3}F_{2}\left(\begin{array}[]{c}k-n,\,1-\alpha+k,\,\gamma\\\
-\alpha-n,\,\gamma+k+1\end{array}\big{|}1\right).$ (4.4)

We now give a generating function for the ALPs. It was already found in [3,
p.25] by using the method of differential equations. The alternative proof we
present here (see Sect.7 below) is direct and simpler.

######  Proposition 4.1.

A generating function of the ALPs is

$\displaystyle\sum_{n=0}^{+\infty}t^{n}\mathscr{L}_{n}^{(\alpha)}(x;\gamma)=(1-t)^{-\gamma-\alpha-1}\exp\left(\frac{x\,t}{t-1}\right)\Phi_{1}\left[\gamma,\,\gamma+\alpha,\,\gamma+1;\,\frac{t}{t-1};\,\frac{-x\,t}{t-1}\right]$
(4.5)

where $|t|<\frac{1}{2},\,x\in\mathbb{R}$ and $\Phi_{1}$ is the Humbert
confluent hypergeometric function defined by (7.24).
In particular, for $\gamma=0$, we recover the generating function of classical
Laguerre polynomials

$\displaystyle\sum_{n=0}^{+\infty}t^{n}L_{n}^{(\alpha)}(x)=(1-t)^{-\alpha-1}\exp\left(\frac{x\,t}{t-1}\right),\quad|t|<1,\,|x|<+\infty.$
(4.6)

Substituting the connection formula ([3, p.24(2.20)])
$\mathscr{L}_{n}^{(\alpha)}(x;\gamma)=\sum_{k=0}^{n}\frac{\gamma}{k+\gamma}L_{n-k}^{(\alpha)}(x)L_{k}^{(-\alpha)}(-x)$
in (4.5), applying (2.17) to the obtained expression, and then identifying the
generating function (4.6) in the final expression to get the following result.

######  Corollary 4.1.

A generating function for Laguerre polynomials

$\displaystyle\sum_{n=0}^{+\infty}\frac{\gamma}{n+\gamma}L_{n}^{(\alpha)}(x)t^{n}=(1-t)^{-\gamma}\Phi_{1}\left[\gamma,\,\gamma-\alpha,\,\gamma+1;\,\frac{t}{t-1};\,\frac{x\,t}{t-1}\right]\quad(\gamma\text{
arbitrary}).$ (4.7)

The above generating function generalizes the one given in [24, Eq(9.12.12)],
which we obtain by setting $\gamma=\alpha$ in (4.7).

######  Remark 4.1.

Notice that, by a different method, the explicit polynomial form (4.3) was
found by Askey and Wimp in [3, p.22, Eq(2.8)] and it can also be obtained from
the associated Meixner-Pollaczek polynomials by using the limit relation:

$\mathscr{L}_{n}^{(\alpha)}(x;\gamma)=\lim\limits_{\varphi\to
0}\mathscr{P}_{n}^{((\alpha+1)/2)}\left(\frac{-x}{2\sin\varphi};\,\varphi,\,\gamma\right),$

as was explained by Rahman in [13, p.7].

##  5 The associated Krawtchouk polynomials

The associated Krawtchouk polynomials (AKPs) can be defined by the three-term
recurrence relation:

$\displaystyle\begin{split}p(N-n-\gamma)\mathscr{K}_{n+1}(x;p,N,\gamma)=\left[pN+(n+\gamma)(1-2p)-x\right]\mathscr{K}_{n}(x;p,N,\gamma)\\\
-(n+\gamma)(1-p)\mathscr{K}_{n-1}(x;p,N,\gamma),\end{split}$ (5.1)

with initial conditions $\mathscr{K}_{-1}(x;p,N,\gamma)=0$ and
$\mathscr{K}_{0}(x;p,N,\gamma)=1$. Applying the criterion (1.2) to (5.1) we
see that the orthogonality is obtained in the following cases:

  1. (i)

$p<0$, $\quad N-\gamma<0$

  2. (ii)

$0<p<1$, $\quad n=0,1,\cdots,\lfloor N-\gamma\rfloor$

  3. (iii)

$p>1$, $\quad N-\gamma<0$.

The notation $\lfloor x\rfloor$ stands for the largest integer less than or
equal to $x$. The AKPs are related to the AMPs in the following way:

$\mathscr{K}_{n}(x;p,N,\gamma)=\frac{\mathscr{M}_{n}(x;-N,p/(p-1),\gamma)}{(-N+\gamma)_{n}}.$
(5.2)

From (2.5) and (5.2) follows the explicit representation for the AKPs

$\displaystyle\begin{split}\mathscr{K}_{n}(x;p,N,\gamma)=\left(\frac{p-1}{p}\right)^{n}\frac{(\gamma+1)_{n}}{n!}\sum_{k=0}^{n}(1-p)^{-k}\frac{(-n)_{k}(\gamma-N+x)_{k}}{(\gamma+1)_{k}(\gamma-N)_{k}}\\\
\times{}_{4}F_{3}\left(\begin{array}[]{c}k-n,\,\gamma-N+x+k,\,\gamma-N-1,\,\gamma\\\
\gamma-N+x,\,\gamma-N+k,\,\gamma+1+k\end{array}\big{|}1\right)\end{split}$
(5.3)

valid under restrictions $(i),\ (ii),(iii)$ given above. An alternative form
of $\mathscr{K}_{n}(x;p,N,\gamma)$ stems from (2.6) and is given by

$\displaystyle\begin{split}\mathscr{K}_{n}(x;p,N,\gamma)=\frac{(\gamma+1)_{n}}{n!}\sum_{k=0}^{n}p^{-k}\frac{(-n)_{k}(\gamma-x)_{k}}{(\gamma+1)_{k}(\gamma-N)_{k}}{}_{4}F_{3}\left(\begin{array}[]{c}k-n,\,\gamma-x+k,\,\gamma-N-1,\,\gamma\\\
\gamma-x,\,\gamma-N+k,\,\gamma+1+k\end{array}\big{|}1\right).\end{split}$
(5.4)

Next, we establish the following generating function for the AKPs.

######  Theorem 5.1.

For $x=x_{j}$, $j=\\{0,1,\cdots,M\\}$ (where it is possible that $M=\infty$),
we have

$\displaystyle\sum_{n=0}^{+\infty}t^{n}\mathscr{K}_{n}(x;p,N,\gamma)=\left(1-t\right)^{-1}F_{1}\left[1,\gamma,-x;\gamma-N;\frac{t(p-1)}{p},\frac{t}{p(t-1)}\right],$
(5.5)

valid for
$\max\left\\{\left|\frac{t(p-1)}{p}\right|,\left|\frac{t}{p(t-1)}\right|\right\\}<1$.

It should be noted here that the case when $n=N-\gamma\in\mathbb{N}$ must be
understood by continuity. In fact, in this case the hypergeometric
${}_{4}F_{3}(1)$ in the expression of $\mathscr{K}_{n}(x;p,N,\gamma)$ reduces
to ${}_{3}F_{2}(1)$ and we still have a polynomial of degree $N-\gamma$. For
instance, if we take $n=N-\gamma$ in (5.4) we get

$\displaystyle\begin{split}\mathscr{K}_{N-\gamma}(x;p,N,\gamma)=\frac{(\gamma+1)_{N-\gamma}}{(N-\gamma)!}\sum_{k=0}^{N-\gamma}p^{-k}\frac{(\gamma-x)_{k}}{(\gamma+1)_{k}}{}_{3}F_{2}\left(\begin{array}[]{c}\gamma-x+k,\,\gamma-N-1,\,\gamma\\\
\gamma-x,\,\gamma+1+k\end{array}\big{|}1\right).\end{split}$ (5.6)

Therefore, in the case when $N-\gamma\in\mathbb{N}$, we will need a special
notation for the generating function of the AKPs. We define the $N$-th partial
sum of a power series in $t$ by

$\left[f(t)\right]_{N}:=\sum\limits_{k=0}^{N}\frac{f^{(k)}(0)}{k!}t^{k},$
(5.7)

for every function $f$ for which $f^{(k)}(0)$, $k=0,1,\cdots,N$ exists. Then,
for $x=0,1,2,\cdots,N-\gamma$, we have the following generating function for
the AKPs

$\displaystyle\sum_{n=0}^{N-\gamma}t^{n}\mathscr{K}_{n}(x;p,N,\gamma)=\left[\left(1-t\right)^{-1}F_{1}\left[1,\gamma,-x;\gamma-N;\frac{t(p-1)}{p},\frac{t}{p(t-1)}\right]\right]_{N-\gamma}.$
(5.8)

Taking $\gamma=0$ in the above relation gives the generating function of the
classical Krawtchouk polynomials, $K_{n}(x;p,N)=\mathscr{K}_{n}(x;p,N,0)$,
obtained for $x=0,1,\cdots,N$ in [24, (1.10.13)]:

$\displaystyle\sum_{n=0}^{N}t^{n}K_{n}(x;p,N)=\left[\left(1-t\right)^{-1}{}_{3}F_{2}\left(\begin{array}[]{c}1,\,-x\\\
-N\end{array}\big{|}\frac{t}{p(t-1)}\right)\right]_{N},$ (5.11)

where $N$ is a nonnegative integer.

##  6 Concluding remark

In this paper, we have established explicit representation as well as
generating function of the AMPs. These results are then used to derive similar
relations for a sequence of associated orthogonal polynomials which belong to
the same family by limiting procedures or connection formula. We recall that
orthogonality measures of AMPs, ACPs, and AKPs are unknown. An attempt to get
the orthogonality measures for the AMPs and ACPs was made in [10] where the
authors compute the Stieljes transform of the orthogonality measures of these
two polynomials. They showed that these measures are unique. As is well known,
if elements of one set of orthogonal polynomials converge to those of another
set, and the measures are uniquely determined, then there must be the same
corresponding limit relation for the measures. This means that it suffices to
find the orthogonality measure of the AMPs to deduce those of the other
related polynomials. This question may be the subject of forthcoming works.

##  7 Proofs

This section is devoted to some technical proofs.

###### Proof of corollary 2.1.

By combining (2.14) and (2.6) then making the modifications
$(t,a,b,y)\leftarrow(\tilde{c},\gamma,\gamma+\beta-1,-x)$, we can readily
obtain the following finite sum formula for ${}_{4}F_{3}(1)$:

$\displaystyle\begin{split}&\frac{(a+1)_{n}(b+1)_{n}}{n!}\sum_{k=0}^{n}t^{k}\frac{(-n)_{k}(a+y)_{k}}{(a+1)_{k}(b+1)_{k}}{}_{4}F_{3}\left(\begin{array}[]{c}k-n,\,a+y+k,\,a,\,b\\\
a+y,\,b+1+k,\,a+1+k\end{array}\big{|}1\right)\\\
&=\frac{1}{b-a}\left\\{(b)_{n+1}\,{}_{2}F_{1}\left(\begin{array}[]{c}1-y,a\\\
a-b+1\end{array}\big{|}t\right){}_{2}F_{1}\left(\begin{array}[]{c}y,-n-a\\\
b-a+1\end{array}\big{|}t\right)\right.\\\
&\left.-(a)_{n+1}\,{}_{2}F_{1}\left(\begin{array}[]{c}b-a+1-y,b\\\
b-a+1\end{array}\big{|}t\right){}_{2}F_{1}\left(\begin{array}[]{c}a-b+y,-n-b\\\
a-b+1\end{array}\big{|}t\right)\right\\}.\end{split}$ (7.1)

Next, to ${}_{2}F_{1}$ in the second line of the RHS of the above equation, we
apply the Euler transformation [25, p.33(21)]:

${}_{2}F{}_{1}\left(\begin{array}[]{c}A,B\\\
C\end{array}\big{|}z\right)=(1-z)^{C-A-B}{}_{2}F_{1}\left(\begin{array}[]{c}C-A,C-B\\\
C\end{array}\big{|}z\right),$ (7.2)

where $C\neq 0,-1,-2,...,|\text{arg}(1-z)|<\pi$, for $A=b-a+1-y,\ B=b,\
C=b-a+1,z=t$ in the first ${}_{2}F_{1}$ and $A=a-b+y,\ B=-n-b,\ C=a-b+1,z=t$
to get the formula in (2.12). ∎

###### Proof of Theorem 2.1.

Denoting the left-hand side of (2.20) by $\Upsilon(x,t)$ and substituting the
expression of $\mathscr{M}_{n}(x;\beta,c,\gamma)$, we obtain

$\displaystyle\begin{split}\Upsilon(x,t)&=\sum_{n=0}^{+\infty}\frac{t^{n}(\gamma+1)_{n}}{n!}\sum_{k=0}^{n}(1-c)^{k}\frac{(-n)_{k}(\gamma+\beta+x)_{k}}{(\gamma+1)_{k}(\gamma+\beta)_{k}}{}_{4}F_{3}\left(\begin{array}[]{c}k-n,\,\gamma+\beta+x+k,\,\gamma+\beta-1,\,\gamma\\\
\gamma+\beta+x,\,\gamma+\beta+k,\,\gamma+1+k\end{array}\big{|}1\right)\\\
&=\sum_{n=0}^{+\infty}\sum_{k=0}^{n}\sum_{j=0}^{n-k}\frac{t^{n}(1-c)^{k}}{n!j!}\frac{(-n)_{k}(-n-k)_{j}(\gamma+1)_{n}(\gamma+\beta+x)_{k}(\gamma+\beta+x+k)_{j}(\gamma+\beta-1)_{j}(\gamma)_{j}}{(\gamma+1)_{k}(\gamma+\beta)_{k}(\gamma+\beta+x)_{j}(\gamma+\beta+k)_{j}(\gamma+1+k)_{j}}\\\
&=\sum_{n=0}^{+\infty}\sum_{k=0}^{+\infty}\sum_{j=0}^{+\infty}\frac{t^{n}(t(c-1))^{k}(-t)^{j}}{n!j!}\frac{(\gamma+1+k+j)_{n}(\gamma+\beta+x)_{k}(\gamma+\beta+x+k)_{j}(\gamma+\beta-1)_{j}(\gamma)_{j}}{(\gamma+\beta)_{k}(\gamma+\beta+x)_{j}(\gamma+\beta+k)_{j}},\end{split}$

where we have used, respectively, the series transformation [25, p.102(17)]:

$\sum_{n=0}^{\infty}\sum_{k=0}^{n}\sum\limits_{j=0}^{n-k}A(j,k,n)=\sum_{n=0}^{\infty}\sum_{k=0}^{\infty}\sum\limits_{j=0}^{\infty}A(j,k,n+k+j),$
(7.3)

and the identities

$\frac{(-n-k-j)_{k}(-n-j)_{j}}{(n+k+j)!}=\frac{(-1)^{k+j}}{n!},$ (7.4)
$\frac{(\gamma+1)_{n+k+j}}{(\gamma+1)_{k}(\gamma+1+k)_{j}}=(\gamma+1+k+j)_{n},$
(7.5)

obtained by virtue of relations

$\displaystyle(-n)_{k}=(-1)^{k}\frac{n!}{(n-k)!},\quad k=0\leq k\leq n,$ (7.6)
$\displaystyle(a)_{n+m}=(a)_{n}(a+n)_{m}=(a)_{m}(a+m)_{n},\ a\in\mathbb{C},\
n,m\in\mathbb{N}.$ (7.7)

After summation over $n$ in the last expression of $\Upsilon(x,t)$, we obtain

$\displaystyle\Upsilon(x,t)=(1-t)^{-\gamma-1}\sum_{k=0}^{+\infty}\sum_{j=0}^{+\infty}\left(\frac{t(c-1)}{1-t}\right)^{k}\frac{\left(\frac{t}{t-1}\right)^{j}}{j!}\frac{(\gamma+\beta+x)_{k}(\gamma+\beta+x+k)_{j}(\gamma+\beta-1)_{j}(\gamma)_{j}}{(\gamma+\beta)_{k}(\gamma+\beta+x)_{j}(\gamma+\beta+k)_{j}}.$

We apply again the identity (7.7) to get, after simplifications,

$\displaystyle\Upsilon(x,t)$ $\displaystyle=$
$\displaystyle(1-t)^{-\gamma-1}\sum_{k=0}^{+\infty}\sum_{j=0}^{+\infty}\left(\frac{t(c-1)}{1-t}\right)^{k}\frac{\left(\frac{t}{t-1}\right)^{j}}{j!}\frac{(\gamma+\beta+x+j)_{k}(\gamma+\beta-1)_{j}(\gamma)_{j}}{(\gamma+\beta)_{j}(\gamma+\beta+j)_{k}}$
$\displaystyle=$
$\displaystyle(1-t)^{-\gamma-1}\sum_{j=0}^{+\infty}\frac{(\gamma+\beta-1)_{j}(\gamma)_{j}}{(\gamma+\beta)_{j}}{}_{2}F_{1}\left(\begin{array}[]{c}\gamma+\beta+x+j,\,1\\\
\gamma+\beta+j\end{array}\left|\frac{t(c-1)}{1-t}\right.\right)\frac{\left(\frac{t}{t-1}\right)^{j}}{j!}.$

Next, to ${}_{2}F_{1}$ in the above equation, we apply the Euler
transformation [25, p.33(21)]:

${}_{2}F{}_{1}\left(\begin{array}[]{c}a,b\\\
e\end{array}\big{|}z\right)=(1-z)^{e-a-b}{}_{2}F_{1}\left(\begin{array}[]{c}e-a,e-b\\\
e\end{array}\big{|}z\right),$ (7.9)

where $e\neq 0,-1,-2,...,|\text{arg}(1-z)|<\pi$, to get

$\displaystyle\Upsilon(x,t)=(1-t)^{x-\gamma}(1-ct)^{-x-1}\sum_{j=0}^{+\infty}\frac{(\gamma+\beta-1)_{j}(\gamma)_{j}}{(\gamma+\beta)_{j}}{}_{2}F_{1}\left(\begin{array}[]{c}-x,\,\gamma+\beta-1+j\\\
\gamma+\beta+j\end{array}\left|\frac{t(c-1)}{1-t}\right.\right)\frac{\left(\frac{t}{t-1}\right)^{j}}{j!}.$

We identify the infinite series as the Appell hypergeometric function $F_{1}$
defined by ([25, p.53(4)]):

$\displaystyle\begin{split}F_{1}\left[\alpha,\,\beta_{1},\,\beta_{2};\,\sigma;\,x,\,y\right]&=\sum\limits_{m,n=0}^{+\infty}\frac{(\alpha)_{m+n}(\beta_{1})_{m}(\beta_{2})_{n}}{(\sigma)_{m+n}}\frac{x^{m}}{m!}\frac{y^{n}}{n!}\\\
&=\sum\limits_{m=0}^{+\infty}\frac{(\alpha)_{m}(\beta_{1})_{m}}{(\sigma)_{m}}{}_{2}F_{1}\left(\begin{array}[]{c}\alpha+m,\beta_{2}\\\
\sigma+m\end{array}\big{|}y\right)\frac{x^{m}}{m!},\quad\max\\{|x|,|y|\\}<1.\end{split}$
(7.11)

Thus, we have

$\displaystyle\Upsilon(x,t)=(1-t)^{x-\gamma}(1-ct)^{-x-1}F_{1}\left[\gamma+\beta-1;\gamma,-x;\gamma+\beta;\frac{t}{t-1},\frac{t(1-c)}{t-1}\right].$

To get (2.20), it suffices to apply the transformation (see [27, p.78]):

$\displaystyle
F_{1}\left[\alpha,\,\beta_{1},\,\beta_{2};\,\sigma;\,x,\,y\right]=(1-x)^{-\beta_{1}}(1-y)^{-\beta_{2}}F_{1}\left[\sigma-\alpha,\,\beta_{1},\,\beta_{2};\,\sigma;\,\frac{x}{x-1},\,\frac{y}{y-1}\right]$
(7.12)

Finally, the formula (2.23) is a particular case of [24, Eq(9.10.13)] and it
is obtained here directly by putting $\gamma=0$ in (2.20). The proof is
complete. ∎

###### Proof of Proposition 3.1.

In fact, recalling the expression of (3.6) and denoting the left-hand side of
(3.15) by $\mathscr{G}(x,t)$, we obtain

$\displaystyle\mathscr{G}(x,t)$ $\displaystyle=$
$\displaystyle\sum_{n=0}^{+\infty}\frac{t^{n}}{n!}\sum_{k=0}^{n}(-a)^{-k}\frac{(-n)_{k}(\gamma-x)_{k}}{(\gamma+1)_{k}}{}_{3}F_{2}\left(\begin{array}[]{c}k-n,\,\gamma-x+k,\,\gamma\\\
\gamma-x,\,\gamma+k+1\end{array}\big{|}1\right)$ (7.15) $\displaystyle=$
$\displaystyle\sum_{n=0}^{+\infty}\sum_{k=0}^{n}\sum_{j=0}^{n-k}\frac{t^{n}(-a)^{-k}}{n!j!}\frac{(-n)_{k}(k-n)_{j}(\gamma-x)_{k}(\gamma-x+k)_{j}(\gamma)_{j}}{(\gamma+1)_{k}(\gamma-x)_{j}(\gamma+k+1)_{j}}$
(7.16) $\displaystyle=$
$\displaystyle\sum_{n=0}^{+\infty}\sum_{k=0}^{+\infty}\sum_{j=0}^{+\infty}\frac{t^{n+k+j}(-a)^{-k}}{(n+k+j)!j!}\frac{(-n-k-j)_{k}(-n-j)_{j}(\gamma-x)_{k}(\gamma-x+k)_{j}(\gamma)_{j}}{(\gamma+1)_{k}(\gamma-x)_{j}(\gamma+k+1)_{j}}.$
(7.17)

These calculations can be done by applying the identity (7.3). Next, applying
(7.4) and the relation
$(\gamma+y)_{k}(\gamma+y+k)_{j}=(\gamma+y)_{j}(\gamma+y+j)_{k}$, for $y=-x,\
1$, to the last expression of $\mathscr{G}(x,t)$ to get

$\displaystyle\mathscr{G}(x,t)$ $\displaystyle=$ $\displaystyle
e^{t}\sum_{k=0}^{+\infty}\sum_{j=0}^{+\infty}\frac{(\gamma-x+j)_{k}(\gamma)_{j}}{(\gamma+1+j)_{k}(\gamma+1)_{j}}\left(\frac{t}{a}\right)^{k}\frac{(-t)^{j}}{j!}$
(7.20) $\displaystyle=$ $\displaystyle
e^{t}\sum_{j=0}^{+\infty}\frac{(\gamma)_{j}}{(\gamma+1)_{j}}{}_{2}F_{1}\left(\begin{array}[]{c}\gamma-x+j,\,1\\\
\gamma+1+j\end{array}\left|\frac{t}{a}\right.\right)\frac{(-t)^{j}}{j!}.$

Next, to ${}_{2}F_{1}$ in (7.20) we apply the Euler transformation (7.9), to
get

$\displaystyle\mathscr{G}(x,t)=e^{t}\left(1-\frac{t}{a}\right)^{x}\sum_{j=0}^{+\infty}\frac{(\gamma)_{j}}{(\gamma+1)_{j}}{}_{2}F_{1}\left(\begin{array}[]{c}\gamma+j,\,x+1\\\
\gamma+1+j\end{array}\left|\frac{t}{a}\right.\right)\frac{(-t)^{j}}{j!}.$
(7.23)

By recognizing the Humbert confluent hypergeometric function $\Phi_{1}$
defined by [25, p.58(36)]:

$\displaystyle\begin{split}\Phi_{1}\left[\alpha_{1},\,\lambda;\,\alpha_{2};\,x,\,y\right]&=\sum\limits_{m,n=0}^{+\infty}\frac{(\alpha_{1})_{m+n}(\lambda)_{m}}{(\alpha_{2})_{m+n}}\frac{x^{m}}{m!}\frac{y^{n}}{n!}\\\
&=\sum\limits_{n=0}^{+\infty}\frac{(\alpha_{1})_{n}}{(\alpha_{2})_{n}}{}_{2}F_{1}\left(\begin{array}[]{c}\alpha_{1}+n,\
\lambda\\\
\alpha_{2}+n\end{array}\big{|}x\right)\frac{y^{n}}{n!},\quad|x|<1,\,|y|<+\infty.\end{split}$
(7.24)

in the right-hand side of (7.23), we complete the proof of the Proposition
3.1. ∎

###### Proof of Proposition 4.1.

Denote the left-hand side of (4.5) by $\Lambda(x,t)$. Similar calculations to
the ones of the proof of Proposition 3.1 give

$\displaystyle\Lambda(x,t)$ $\displaystyle=$
$\displaystyle\sum_{n=0}^{+\infty}\frac{t^{n}}{n!}\sum_{k=0}^{n}\frac{(\gamma+\alpha+1)_{n}(-n)_{k}\,x^{k}}{(\gamma+1)_{k}(\gamma+\alpha+1)_{k}}{}_{3}F_{2}\left(\begin{array}[]{c}k-n,\,\gamma+\alpha,\,\gamma\\\
\gamma+\alpha+k+1,\,\gamma+k+1\end{array}\big{|}1\right)$ (7.27)
$\displaystyle=$
$\displaystyle\sum_{n=0}^{+\infty}\sum_{k=0}^{+\infty}\sum_{j=0}^{+\infty}\frac{(\gamma+\alpha+1)_{n+k+j}t^{n}}{n!}\frac{(-xt)^{k}(-t)^{j}(\gamma+\alpha)_{j}(\gamma)_{j}}{j!(\gamma+1)_{k+j}(\gamma+\alpha+1)_{k+j}}$
(7.28) $\displaystyle=$
$\displaystyle\sum_{n=0}^{+\infty}\sum_{k=0}^{+\infty}\sum_{j=0}^{+\infty}\frac{(\gamma+\alpha+1+k+j)_{n}t^{n}}{n!}\frac{(-xt)^{k}(-t)^{j}(\gamma+\alpha)_{j}(\gamma)_{j}}{j!(\gamma+1)_{k+j}}$
(7.29) $\displaystyle=$
$\displaystyle(1-t)^{-\gamma-\alpha-1}\sum_{k=0}^{+\infty}\sum_{j=0}^{+\infty}\frac{(\frac{xt}{t-1})^{k}(\frac{t}{t-1})^{j}(\gamma+\alpha)_{j}(\gamma)_{j}}{j!(\gamma+1)_{k+j}}$
(7.30) $\displaystyle=$
$\displaystyle(1-t)^{-\gamma-\alpha-1}\sum_{j=0}^{+\infty}\frac{(\frac{t}{t-1})^{j}(\gamma+\alpha)_{j}(\gamma)_{j}}{j!(\gamma+1)_{j}}{}_{1}F_{1}\left(\begin{array}[]{c}1\\\
\gamma+1+j\end{array}\left|\frac{xt}{t-1}\right.\right).$ (7.33)

Next, we use the Kummer transformation [25, p.37(7)]:

${}_{1}F_{1}\left(a;b;z\right)=e^{z}\,{}_{1}F_{1}\left(b-a;b;-z\right),$
(7.34)

to get

$\displaystyle\Lambda(x,t)=(1-t)^{-\gamma-\alpha-1}\exp\left(\frac{xt}{t-1}\right)\sum_{j=0}^{+\infty}\frac{(\frac{t}{t-1})^{j}(\gamma+\alpha)_{j}(\gamma)_{j}}{j!(\gamma+1)_{j}}{}_{1}F_{1}\left(\begin{array}[]{c}\gamma+j\\\
\gamma+1+j\end{array}\left|\frac{-xt}{t-1}\right.\right).$ (7.37)

Identification of the summation over $j$ as the Humbert confluent
hypergeometric function $\Phi_{1}$ then completes the proof of Proposition
4.1. ∎

### Acknowledgments

I would like to thank Professor Zouhaïr Mouayn for his comments and invaluable
suggestions that improved the presentation of this manuscript. I would like
also to thank the Moroccan Association of Harmonic Analysis and Spectral
Geometry.

### Data availability statement

All data generated or analyzed during this study are included in this
published article.

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