# Classical orthogonal polynomials revisited111My friend and coauthor
Professor J. Petronilho passed away on August 27, 2021. In the academic years
2016/17 and 2020/21 we taught a PhD course on Orthogonal Polynomials and
Special Functions at the Universities of Coimbra and Porto (Portugal). This
manuscript go back to the beginning of 2015 and it was later including in a
monograph [12] on the subject that Professor Petronilho finished in 2016. (His
monographs was during all these years actively used by students and
colleagues, mainly from Portugal and Spain.) Probably, for this reason or the
fact that we had started working on [3], which allow us to rewrite many
results in a more general framework, we shelved this project. And this would
have been its final destination if I had never read some recent publications
where the same topic is addressed. But after that, I felt that is time to dust
off this manuscript, allowing active readers the freedom to draw their own
conclusions. Finally, I thank Professor Petronilho’s youngest daughter, Inês,
who found these old files and sent them to me.

K. Castillo<EMAIL_ADDRESS>J. Petronilho  CMUC, University of Coimbra,
Department of Mathematics.
EC Santa Cruz, 3001-501 Coimbra, Portugal.

###### Abstract

This manuscript contains a small portion of the algebraic theory of orthogonal
polynomials developed by Maroni and their applicability to the study and
characterization of the classical families, namely Hermite, Laguerre, Jacobi,
and Bessel polynomials. It is presented a cyclical proof of some of the most
relevant characterizations, particularly those due to Al-Salam and Chihara,
Bochner, Hahn, Maroni, and McCarthy. Two apparently new characterizations are
also added. Moreover, it is proved through an equivalence relation that, up to
constant factors and affine changes of variables, the four families of
polynomials named above are the only families of classical orthogonal
polynomials.

###### keywords:

Moment linear functionals , classical orthogonal polynomials , algebraic
theory of orthogonal polynomials

###### MSC:

[2010] 42C05 , 33C45

##  1 Orthogonal polynomials: The algebraic theory

Denote by $\mathrm{M}_{m}(\mathrm{A})$ the ring of all square matrices of
order $m$ over the commutative ring with unit $\mathrm{A}$. Consider the free
modulo $\mathcal{M}=\mathrm{M}_{m}(\mathrm{A})[X]$ of all “polynomials” in one
indeterminate $X$ with coefficients in $\mathrm{M}_{m}(\mathrm{A})$. (Assume
that the indeterminate is contained in the centre of $\mathcal{M}$.) Any free
system of polynomials over $\mathrm{M}_{m}(\mathrm{A})$ is basis of
$\mathcal{M}$. Moreover, the dual of a finite generated free module is a
finitely generated free module. However, the dual system associated with a
free sequence $(p_{n})_{n\geq 0}$ in $\mathcal{M}$, $(\mathbf{a}_{n})_{n\geq
0}$, is not a system of generators of its algebraic dual, $\mathcal{M}^{*}$,
in general $($cf. [1, VII, §3, Exercise 10]$)$. But if we consider, for
instance, $\mathcal{P}=\mathbb{C}[X]$, instead of $\mathcal{M}$, endowed with
an “appropriate” topology, then $\mathcal{P}^{*}=\mathcal{P}^{\prime}$ $($cf.
[15, Exercise 13.1, p. 134]$)$, $\mathcal{P}^{\prime}$ being the topological
dual $\mathcal{P}$. Hence

$\displaystyle\mathbf{u}=\sum_{n=0}^{\infty}\left\langle{p_{n},\mathbf{u}}\right\rangle\mathbf{a}_{n},$
(1.1)

for all $\mathbf{u}\in\mathcal{P}^{\prime}$. (Recall that for every pair of
elements $x\in\mathrm{E}$ and $\mathbf{x}\in\mathrm{E}^{*}$, the element
$\mathbf{x}(x)$ of $\mathrm{E}$ is denote by
$\left\langle{x,\mathbf{x}}\right\rangle$, where $\mathrm{E}$ is a left
$\mathrm{A}$-module whose domain of operators is $\mathrm{A}$.) This
observation is the cornerstone of Rota’s umbral calculus $($cf. [14]$)$ and
the algebraic theory of orthogonal polynomials (OP) founded by Maroni $($cf.
[6, 7, 8]$)$. For further reading on how to re-establish the “symmetry”
between an infinite dimensional vector space and its dual see [2, II, §6].

This manuscript contains a short exposition of Maroni’s approach on OP and
their applicability to the study and characterization of the classical
families of OP. A must read on this topic are [9, 10, 11]. Even a proof of
Theorem 3.1 can be found, in one way or another, in the works of Maroni.
However, among other questions of pedagogical nature, in this survey we
present a cyclical proof of our main results, Theorem 3.1. Moreover, the
characterizations (C4) and (C4’) are apparently new ones. While is true that
any classical functional is equivalent to one of the canonical forms given in
Table 1 below, Theorem 3.2 rigorously reflects this property and, as far as we
know, it is not available in the literatura. Many results are stated without
proof, either because they are simple to prove or because they can be easily
found in the literature. Other results that fit in these two categories are,
however, proved when the proof methods are different or more attractive, from
our point of view, to the existing one. The positive definite case, widely
discussed in the literature, is intentionally omitted. Finally, as the reader
will notice, some results are “purely” algebraic. In any case, we will not
make distinction between $\mathcal{P}^{*}$ and $\mathcal{P}^{\prime}$ after
Proposition 1.1 below.

###  1.1 The spaces $\mathcal{P}$ and $\mathcal{P}^{\prime}$

It is useful to consider OP as test functions living in an appropriate locally
convex space (LCS), which we denote by $\mathcal{P}$. This LCS is the set of
all polynomials (with real or complex coefficients) endowed with a strict
inductive limit topology, so that

$\mathcal{P}=\bigcup_{n=0}^{\infty}\mathcal{P}_{n}=\mbox{\rm
ind\,lim}_{n}\,\mathcal{P}_{n}\;,$ (1.2)

where $\mathcal{P}_{n}$ is the space of all polynomials of degree at most $n$.
(For the sake of simplicity, we do not distinguish between polynomial and
polynomial function.) Since $\mathcal{P}_{n}$ is a finite dimensional vector
space, all its norms are equivalent, so there is no need to specify any
particular one. For the development of the theory to be presented here it is
not important to know much about the topology (the definition and basic
properties of LCS, including inductive limit topologies, can be found, e.g.,
in Chapter V of Reed and Simon’s book [13], but the reader should keep in mind
that the reason why such topology is introduced is because it implies the
following fundamental property:

######  Proposition 1.1.

Let $\mathcal{P}=\mbox{\rm ind\,lim}_{n}\,\mathcal{P}_{n}$, as in
$(\ref{P=indlimPn})$, and let $\mathcal{P}^{*}$ and $\mathcal{P}^{\prime}$ be
the algebraic and the topological duals of $\mathcal{P}$, respectively. Then

$\mathcal{P}^{\prime}=\mathcal{P}^{*}\,.$ (1.3)

###### Proof.

Obviously, $\mathcal{P}^{\prime}\subseteq\mathcal{P}^{*}$. To prove that
$\mathcal{P}^{*}\subseteq\mathcal{P}^{\prime}$, take ${\bf
u}\in\mathcal{P}^{*}$. From the basic properties of the inductive limit
topologies, to prove that ${\bf u}\in\mathcal{P}^{\prime}$ it suffices to show
that the restriction ${\bf u}|\mathcal{P}_{n}$ is continuous for every $n$.
But this is a trivial assertion, since ${\bf u}|\mathcal{P}_{n}$ is a linear
functional defined on a finite dimensional normed space. ∎

Equality (1.3) means that every linear functional defined in $\mathcal{P}$ is
continuous (for the strict inductive limit topology in $\mathcal{P}$). This is
a “curious” property, because, for instance, (1.3) is not true for a normed
vector space $N$. Indeed, $N^{\prime}=N^{*}$ if ${\rm dim}\,N<\infty$, whilst
$N^{\prime}\neq N^{*}$ whenever ${\rm dim}\,N=\infty$. Note that being a
strict inductive limit of the spaces $\mathcal{P}_{n}$, and taking into
account that each $\mathcal{P}_{n}$ is a proper closed subspace of
$\mathcal{P}_{n+1}$ (so that $\mathcal{P}$ is indeed an hyper strict inductive
limit of the spaces $\mathcal{P}_{n}$), the general theory of LCS ensures that
$\mathcal{P}$ cannot be a metrizable space, and so _a fortiori_ it is not a
normed space—or, to be more precise, it is not possible to provide
$\mathcal{P}$ with a norm that generates in it the above inductive limit
topology.

In $\mathcal{P}^{\prime}$ we consider the weak dual topology, which, by
definition, is generated by the family of semi-norms
$s_{p}:\mathcal{P}^{\prime}\to[0,+\infty[$, $p\in\mathcal{P}$, defined by

$s_{p}\big{(}{\bf u}\big{)}=|\langle{\bf u},p\rangle|\;,\quad{\bf
u}\in\mathcal{P}^{\prime}\;.$ (1.4)

It turns out that this family of semi-norms $s_{p}$ is equivalent to the
family of semi-norms $|\cdot|_{n}:\mathcal{P}^{\prime}\to[0,+\infty[$,
$n\in\mathbb{N}_{0}$, defined by

$|{\bf u}|_{n}=\max_{0\leq k\leq n}|\langle{\bf u},x^{k}\rangle|\;,\quad{\bf
u}\in\mathcal{P}^{\prime}\;.$ (1.5)

Indeed, the following proposition holds.

######  Proposition 1.2.

$\mathcal{S}=\\{s_{p}:p\in\mathcal{P}\\}$ and
$\mathcal{S}_{\sharp}=\\{|\cdot|_{n}:n\in\mathbb{N}_{0}\\}$, with $s_{p}$ and
$|\cdot|_{n}$ given by $(\ref{semip1})$–$(\ref{sharp1})$, are equivalent
families of seminorms in $\mathcal{P}^{\prime}$, provided
$\mathcal{P}=\mbox{\rm ind\,lim}_{n}\,\mathcal{P}_{n}$.

###### Proof.

Given $p\in\mathcal{P}$, putting $p(x)=\sum_{j=0}^{n}a_{j}x^{j}$ and
$C(p)=\sum_{j=0}^{n}|a_{j}|$, we have

$s_{p}({\bf u})=|\langle{\bf
u},p\rangle|=\Big{|}\sum_{j=0}^{n}a_{j}\langle{\bf u},x^{j}\rangle\Big{|}\leq
C(p)|{\bf u}|_{n}\,,\quad\forall{\bf u}\in\mathcal{P}^{\prime}\,.$

On the other hand, given $n\in\mathbb{N}_{0}$, setting $p_{j}(x)=x^{j}$
($j=0,1,\ldots,n$), we have

$|{\bf u}|_{n}=\max_{0\leq j\leq n}|\langle{\bf
u},x^{j}\rangle|\leq\sum_{j=0}^{n}|\langle{\bf
u},x^{j}\rangle|=\sum_{j=0}^{n}s_{p_{j}}({\bf u})\,,\quad\forall{\bf
u}\in\mathcal{P}^{\prime}\,.$

Thus, $\mathcal{S}$ and $\mathcal{S}_{\sharp}$ are equivalent families of
semi-norms (see [13, p.126]). ∎

######  Remark 1.1.

Since $\mathcal{S}_{\sharp}$ is a countable family of seminorms, then
$\mathcal{P}^{\prime}$ is a metrizable space, a metric being given by

$\varrho({\bf u},{\bf v})=\sum_{n=0}^{\infty}\frac{1}{2^{n}}\frac{|{\bf
u}-{\bf v}|_{n}}{1+|{\bf u}-{\bf v}|_{n}}\;,\quad{\bf u},{\bf
v}\in\mathcal{P}^{\prime}\;.$

Moreover, $\mathcal{P}^{\prime}$ is a Fréchet space.

###  1.2 Dual basis in $\mathcal{P}^{*}$

In $\mathcal{P}^{*}$, addition and multiplications by scalars can be defined
by

$\displaystyle\left\langle{\mathbf{u}+\mathbf{v},x^{n}}\right\rangle$
$\displaystyle=\left\langle{\mathbf{u},x^{n}}\right\rangle+\left\langle{\mathbf{v},x^{n}}\right\rangle,\quad\mathbf{v}\in\mathcal{P}^{*},$
$\displaystyle\left\langle{\alpha\mathbf{u},x^{n}}\right\rangle$
$\displaystyle=\alpha\left\langle{\mathbf{u},x^{n}}\right\rangle$

for all $n\in\mathbb{N}_{0}$. $\mathcal{P}^{*}$, endowed with these
operations, is a vector space over $\mathbb{C}$. In the vector space
$\mathcal{P}^{*}$, the identity for the additivity is denoted by $\mathbf{0}$
and called the zero (or the null element). The zero is therefore defined by
the relation $\left\langle{\mathbf{0},x^{n}}\right\rangle=0$ for all
$n\in\mathbb{N}_{0}$. Of course, the elements of $\mathcal{P}^{*}$ can not
only be added, but also multiplied (the Cauchy product) in order to make the
vector space $\mathcal{P}^{*}$ into an algebra. Since we work mainly on
$\mathcal{P}^{*}$ instead of $\mathcal{P}$, it would be explicitly build bases
in $\mathcal{P}^{*}$. This makes sense, since (1.3) allow us writing
expansions (finite or infinite sums) of the elements of $\mathcal{P}^{*}$ in
terms of the elements of a given basis, in the sense of the weak dual
topology. Such basis in $\mathcal{P}^{*}$ may be achieved in a natural way, by
analogy with the case of finitely generated free modules. A simple set in
$\mathcal{P}$ is a sequence of polynomials, $\\{R_{n}\\}_{n\geq 0}$, such that
${\rm deg}\,R_{n}=n$ for every $n\in\mathbb{N}_{0}$ (where
$R_{0}\equiv\mbox{\rm const.}\neq 0$). To any simple set in $\mathcal{P}$,
$\\{R_{n}\\}_{n\geq 0}$, we may associate a dual basis, which, by definition,
is a sequence of linear functionals $\\{{\bf a}_{n}\\}_{n\geq 0}$, being ${\bf
a}_{n}:\mathcal{P}\to\mathbb{C}$, such that

$\langle{\bf a}_{n},R_{k}\rangle=\delta_{n,k},\quad n,k=0,1,2,\dots\;,$

where $\delta_{n,k}$ represents the Kronecker symbol ($\delta_{n,k}=1$ if
$n=k$; $\delta_{n,k}=0$ if $n\neq k$).

######  Proposition 1.3.

Let $\\{R_{n}\\}_{n\geq 0}$ be a simple set in $\mathcal{P}$ and $\\{{\bf
a}_{n}\\}_{n\geq 0}$ the associated dual basis. Let ${\bf
u}\in\mathcal{P}^{*}$. Then

${\bf u}=\sum_{n=0}^{\infty}\langle{\bf u},R_{n}\rangle\,{\bf a}_{n}\,,$ (1.6)

in the sense of the weak dual topology in $\mathcal{P}^{\prime}$.

###### Proof.

Notice first that the assertion makes sense, according with (1.3). To prove
it, fix $N\in\mathbb{N}$ and let

${\bf s}_{N}=\sum_{n=0}^{N-1}\lambda_{n}{\bf
a}_{n},\quad\lambda_{n}=\langle{\bf u},R_{n}\rangle$

be the partial sum of order $N$ of the series appearing in (1.6). We need to
show that

$\lim_{N\to\infty}\langle{\bf s}_{N}-{\bf u},p\rangle=0\;,\quad\forall
p\in\mathcal{P}\;.$

Clearly, it suffices to prove that this equality holds for
$p\in\\{R_{0},R_{1},R_{2},\dots\\}$. Indeed, fix $k\in\mathbb{N}_{0}$. Then,
for $N>k$,

$\langle{\bf s}_{N}-{\bf u},R_{k}\rangle=\sum_{n=0}^{N-1}\langle{\bf
u},R_{n}\rangle\langle{\bf a}_{n},R_{k}\rangle-\langle{\bf
u},R_{k}\rangle=0\;,$

hence $\;\lim_{N\to\infty}\langle{\bf s}_{N}-{\bf u},R_{k}\rangle=0$. ∎

###  1.3 Basic operations in $\mathcal{P}$ and $\mathcal{P}^{\prime}$

Given a functional ${\bf u}\in\mathcal{P}^{\prime}$, we will denote by

$u_{n}=\langle{\bf u},x^{n}\rangle\;,\quad n\in\mathbb{N}_{0}\;,$

the moment of order $n$ of ${\bf u}$. Clearly, if ${\bf u}$ and ${\bf v}$ are
two functionals in $\mathcal{P}^{\prime}$ such that the corresponding
sequences of moments satisfy $u_{n}=v_{n}$ for all $n\in\mathbb{N}_{0}$, then
${\bf u}={\bf v}$. Therefore, each functional ${\bf u}\in\mathcal{P}^{\prime}$
is uniquely determined by its sequence of moments. Define operators $M_{\phi}$
and $T$, from $\mathcal{P}$ into $\mathcal{P}$, by

$M_{\phi}p(x)=\phi(x)p(x)\;,\quad Tp(x)=-p^{\prime}(x)\;,$ (1.7)

where $\phi\in\mathcal{P}$ (fixed) and ′ denotes derivative with respect to
$x$. Let $M_{\phi}^{\prime}$ and $T^{\prime}$ be the corresponding dual
operators. For each ${\bf u}\in\mathcal{P}^{\prime}$, the images
$M_{\phi}^{\prime}{\bf u}$ and $T^{\prime}{\bf u}$ are elements (functionals)
in $\mathcal{P}^{\prime}$, hereafter denoted by $\phi\,{\bf u}$ and $D{\bf
u}$.

######  Definition 1.1.

Let ${\bf u}\in\mathcal{P}^{\prime}$, $\phi\in\mathcal{P}$, and
$c\in\mathbb{C}$.

  1. (i)

the left multiplication of ${\bf u}$ by $\phi$, denoted by $\phi{\bf u}$, is
the functional in $\mathcal{P}^{\prime}$ defined by

$\langle\phi{\bf u},p\rangle=\langle{\bf u},\phi p\rangle\;,\quad
p\in\mathcal{P}\;;$

  2. (ii)

the derivative of ${\bf u}$, denoted by $D{\bf u}$, is the functional in
$\mathcal{P}^{\prime}$ defined by

$\langle D{\bf u},p\rangle=-\langle{\bf u},p^{\prime}\rangle\;,\quad
p\in\mathcal{P}\;;$

Note that these definitions, introduced by duality with respect to the
operators defined in (1.7), are in accordance with those usually given in the
Theory of Distributions (this explains the minus sign appearing in the second
definition). Note also that

$D(\phi{\bf u})=\phi^{\prime}{\bf u}+\phi\,D{\bf u}\,,\quad{\bf
u}\in\mathcal{P}^{\prime}\;,\;\phi\in\mathcal{P}\;.$

######  Definition 1.2 (translation operators).

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

  1. (i)

The translator operator on $\mathcal{P}$ is
$\tau_{b}:\mathcal{P}\to\mathcal{P}$ $(p\mapsto\tau_{b}p)$ defined by

$\tau_{b}p(x)=p(x-b)\;,\quad p\in\mathcal{P}\;;$ (1.8)

  2. (ii)

The translator operator on $\mathcal{P}^{\prime}$ is
$\bm{\tau}_{b}=\tau_{-b}^{\,\prime}$, i.e.,
$\bm{\tau}_{b}:\mathcal{P}^{\prime}\to\mathcal{P}^{\prime}$ is the dual
operator of $\tau_{-b}$, so that

$\langle\bm{\tau}_{b}{\bf u},p\rangle=\langle{\bf
u},\tau_{-b}p\rangle=\langle{\bf u},p(x+b)\rangle\;,\quad{\bf
u}\in\mathcal{P}^{\prime}\;,\quad p\in\mathcal{P}\;.$ (1.9)

######  Definition 1.3 (homothetic operators).

Let $a\in\mathbb{C}\setminus\\{0\\}$.

  1. (i)

The homothetic operator on $\mathcal{P}$ is $h_{a}:\mathcal{P}\to\mathcal{P}$
$(p\mapsto h_{a}p)$ defined by

$h_{a}p(x)=p(ax)\;,\quad p\in\mathcal{P}\;.$ (1.10)

  2. (ii)

The homothetic operator on $\mathcal{P}^{\prime}$ is
$\bm{h}_{a}=h_{a}^{\,\prime}$, i.e.,
$\bm{h}_{a}:\mathcal{P}^{\prime}\to\mathcal{P}^{\prime}$ is the dual operator
of $h_{a}$, so that

$\langle\bm{h}_{a}{\bf u},p\rangle=\langle{\bf u},h_{a}p\rangle=\langle{\bf
u},p(ax)\rangle\;,\quad{\bf u}\in\mathcal{P}^{\prime}\;,\quad
p\in\mathcal{P}\;.$ (1.11)

######  Proposition 1.4.

Let $\\{P_{n}\\}_{n\geq 0}$ be a simple set in $\mathcal{P}$ and $\\{{\bf
a}_{n}\\}_{n\geq 0}$ its associated dual basis. Let
$a\in\mathbb{C}\setminus\\{0\\}$ and $b\in\mathbb{C}$. Define

$Q_{n}=a^{-n}\big{(}h_{a}\circ\tau_{-b}\big{)}P_{n}\;,\quad n=0,1,2,\ldots.$
(1.12)

Then $\\{Q_{n}\\}_{n\geq 0}$ is a simple set in $\mathcal{P}$, and its dual
basis, $\\{{\bf b}_{n}\\}_{n\geq 0}$, is given by

${\bf b}_{n}=a^{n}\big{(}\bm{h}_{a^{-1}}\circ\bm{\tau}_{-b}\big{)}{\bf
a}_{n}\;,\quad n=0,1,2,\ldots\;.$ (1.13)

######  Remark 1.2.

The polynomial $Q_{n}$ in (1.12) is indeed

$Q_{n}(x)=a^{-n}P_{n}(ax+b)\;,\quad n=0,1,2,\ldots\;,$ (1.14)

so that $Q_{n}$ is obtained from $P_{n}$ by an affine change of the variable,
being $Q_{n}$ normalized so that it becomes a monic polynomial whenever
$P_{n}$ is monic.

######  Proposition 1.5.

Let ${\bf u}\in\mathcal{P}^{\prime}$ and $p,q\in\mathcal{P}\setminus\\{0\\}$,
and denote by $Z_{p}$ and $Z_{q}$ the zeros of $p$ and $q$, respectively. Then
the following property holds:

$Z_{p}\cap Z_{q}=\emptyset\quad\wedge\quad p{\bf u}=q{\bf u}={\bf
0}\qquad\Rightarrow\qquad{\bf u}={\bf 0}\;.$ (1.15)

###### Proof.

Assume that $p\mathbf{u}=q\mathbf{u}=\mathbf{0}$. Since $p$ and $q$ are
relative prime, there exist polynomials $a,b$ for which $a\,p+b\,q=1$. Since,
for all $n$,
$\left\langle{p\mathbf{u},a\,x^{n}}\right\rangle=\left\langle{q\mathbf{u},b\,x^{n}}\right\rangle=0$,
we see that

$\left\langle{\mathbf{u},x^{n}}\right\rangle=\left\langle{\mathbf{u},(a\,p+b\,q)x^{n}}\right\rangle=0.$

The converse is obvious. ∎

###  1.4 Orthogonal polynomial sequences

######  Definition 1.4.

Let ${\bf u}\in\mathcal{P}^{\prime}$ and $\\{P_{n}\\}_{n\geq 0}$ a sequence in
$\mathcal{P}$.

  1. (i)

$\\{P_{n}\\}_{n\geq 0}$ is called an orthogonal polynomial sequence (OP222For
abbreviation, we continue to write OP for orthogonal polynomial sequence.)
with respect to ${\bf u}$ if $\\{P_{n}\\}_{n\geq 0}$ is a simple set (so that
$\deg P_{n}=n$ for all $n$) and there exists a sequence $\\{h_{n}\\}_{n\geq
0}$, with $h_{n}\in\mathbb{C}\setminus\\{0\\}$, such that

$\langle{\bf u},P_{m}P_{n}\rangle=h_{n}\delta_{m,n}\;,\quad
m,n=0,1,2,\ldots\;;$

  2. (ii)

${\bf u}$ is called regular (or quasi-definite) if there exists an OP with
respect to ${\bf u}$.

As usual, denoting by $u_{j}=\langle{\bf u},x^{j}\rangle$,
$j\in\mathbb{N}_{0}$ the moments of ${\bf u}$, we define the associated Hankel
determinant $H_{n}\equiv H_{n}({\bf u})$ as

$H_{-1}=1\;,\quad H_{n}=\det\big{\\{}[u_{i+j}]_{i,j=0}^{n}\big{\\}}\;,\quad
n\in\mathbb{N}_{0}\;.$ (1.16)

It is well known that, given ${\bf u}\in\mathcal{P}^{\prime}$, then ${\bf u}$
is regular if and only if

$H_{n}\neq 0\;,\quad\forall n\in\mathbb{N}_{0}\,.$ (1.17)

One of the most important characterizations of OP relies upon the fact that
any three consecutive polynomials are connected by a very simple relation,
expressed as a three-term recurrence relation (TTRR).

######  Theorem 1.1.

Let $\\{\beta_{n}\\}_{n\geq 0}$ and $\\{\gamma_{n}\\}_{n\geq 0}$ be two
arbitrary sequences of complex numbers, and let $\\{P_{n}\\}_{n\geq 0}$ be a
sequence of (monic) polynomials defined by the three-term recurrence relation

$P_{n+1}(x)=(x-\beta_{n})P_{n}(x)-\gamma_{n}P_{n-1}(x)\;,\quad n=0,1,2,\dots,$
(1.18)

with initial conditions $P_{-1}(x)=0$ and $P_{0}(x)=1$. Then there exists a
unique functional ${\bf u}\in\mathcal{P}^{\prime}$ such that

$\langle{\bf u},1\rangle=u_{0}=\gamma_{0}\;,\qquad\langle{\bf
u},P_{n}P_{m}\rangle=0\quad\mbox{\rm if}\quad n\neq
m\quad(n,m\in\mathbb{N}_{0})\;.$ (1.19)

Moreover, ${\bf u}$ is regular and $\\{P_{n}\\}_{n\geq 0}$ is the
corresponding monic OP if and only if $\gamma_{n}\neq 0$ for each
$n\in\mathbb{N}_{0}$.

######  Remark 1.3.

Note the relations

$\beta_{n}=\frac{\langle{\bf u},xP_{n}^{2}\rangle}{\langle{\bf
u},P_{n}^{2}\rangle}\;,\quad\gamma_{n+1}=\frac{\langle{\bf
u},P_{n+1}^{2}\rangle}{\langle{\bf
u},P_{n}^{2}\rangle}=\frac{H_{n-1}H_{n+1}}{H_{n}^{2}}\;,\quad n=0,1,\ldots\;.$
(1.20)

###  1.5 Orthogonal polynomials and dual basis

Since every OP is a simple set of polynomials, it has an associated dual basis
in $\mathcal{P}^{\prime}$.

######  Theorem 1.2.

Let ${\bf u}\in\mathcal{P}^{\prime}$ be regular, $\\{P_{n}\\}_{n\geq 0}$ the
corresponding monic OP, and $\\{{\bf a}_{n}\\}_{n\geq 0}$ the associated dual
basis. Then:

  * (i)

For each $n\in\mathbb{N}_{0}$, ${\bf a}_{n}$ is explicitly given by

${\bf a}_{n}=\frac{P_{n}}{\langle{\bf u},P_{n}^{2}\rangle}\,{\bf u}\;.$

As a consequence, $\\{P_{n}\\}_{n\geq 0}$ is a monic OP with respect to ${\bf
a}_{0}$, being

${\bf u}=u_{0}\,{\bf a}_{0}\;.$

  * (ii)

Let ${\bf v}\in\mathcal{P}^{\prime}$ and $N\in\mathbb{N}_{0}$ such that

$\langle{\bf v},P_{n}\rangle=0\;\;\mbox{\rm if}\;\;n\geq N+1\;.$

Then,

${\bf v}=\sum_{j=0}^{N}\langle{\bf v},P_{j}\rangle\,{\bf a}_{j}=\phi\,{\bf
u}\;,\quad\phi(x)=\sum_{j=0}^{N}\frac{\langle{\bf v},P_{j}\rangle}{\langle{\bf
u},P_{j}^{2}\rangle}\,P_{j}(x)\;.$

Further, $\deg\phi\leq N$, and $\deg\phi=N$ if and only if $\langle{\bf
v},P_{N}\rangle\neq 0$.

  * (iii)

Let the TTRR fulfilled by $\\{P_{n}\\}_{n\geq 0}$ be $(\ref{TTRRFavard})$.
Then $\\{{\bf a}_{n}\\}_{n\geq 0}$ fulfills

$x\,{\bf a}_{n}={\bf a}_{n-1}+\beta_{n}\,{\bf a}_{n}+\gamma_{n+1}\,{\bf
a}_{n+1}\;,\quad n\in\mathbb{N}_{0}\;,$

with initial conditions ${\bf a}_{-1}={\bf 0}$ and ${\bf
a}_{0}=u_{0}^{-1}\,{\bf u}$.

######  Corollary 1.1.

Let $\\{P_{n}\\}_{n\geq 0}$ be a monic OP (with respect to some functional in
$\mathcal{P}^{\prime}$) and let ${\bf v}\in\mathcal{P}^{\prime}$. Then
$\\{P_{n}\\}_{n\geq 0}$ is a monic OP with respect to ${\bf v}$ if and only if

$\langle{\bf v},1\rangle\neq 0\;,\qquad\langle{\bf v},P_{n}\rangle=0\;,\quad
n=0,1,2\ldots\;\,.$ (1.21)

######  Theorem 1.3.

Under the hypothesis of Proposition 1.4, assume further that
$\\{P_{n}\\}_{n\geq 0}$ is a monic OP with respect to the functional ${\bf
u}\in\mathcal{P}^{\prime}$, and let the TTRR fulfilled by $\\{P_{n}\\}_{n\geq
0}$ be $(\ref{TTRRFavard})$. Then, $\\{Q_{n}\\}_{n\geq 0}$ is a monic OP with
respect to

${\bf v}=\big{(}\bm{h}_{a^{-1}}\circ\bm{\tau}_{-b}\big{)}{\bf u}\;,$ (1.22)

and the TTRR fulfilled by $\\{Q_{n}\\}_{n\geq 0}$ is

$xQ_{n}(x)=Q_{n+1}(x)+\widehat{\beta}_{n}Q_{n}(x)+\widehat{\gamma}_{n}Q_{n-1}(x)\;,\quad
n=0,1,2\ldots\;,$ (1.23)

with initial conditions $Q_{-1}(x)=0$ and $Q_{0}(x)=1$, where

$\widehat{\beta}_{n}=\frac{\beta_{n}-b}{a}\;,\quad\widehat{\gamma}_{n}=\frac{\gamma_{n}}{a^{2}}\;.$
(1.24)

##  2 Distributional differential equation

The distributional differential equation has the form

$D(\phi{\bf u})=\psi{\bf u}\;,$ (2.1)

where $\phi\in\mathcal{P}_{2}$ and $\psi\in\mathcal{P}_{1}$, and ${\bf
u}\in\mathcal{P}^{\prime}$ is the unknown. Notice that we do not require _a
priori_ ${\bf u}$ to be a regular functional. We may write

$\phi(x)=ax^{2}+bx+c\;,\quad\psi(x)=px+q\;,$ (2.2)

being $a,b,c,p,q\in\mathbb{C}$. We also define, for each integer or rational
number $n$,

$\psi_{n}=\psi+n\phi^{\prime}\;,\quad d_{n}=\psi_{n/2}^{\prime}=na+p\;,\quad
e_{n}=\psi_{n}(0)=nb+q\;.$ (2.3)

Notice that $\psi_{n}(x)=d_{2n}x+e_{n}\in\mathcal{P}_{1}$. Finally, for each
${\bf u}\in\mathcal{P}^{\prime}$ and each $n\in\mathbb{N}_{0}$, we set

${\bf u}^{[n]}=\phi^{n}{\bf u}\;.$ (2.4)

We begin with the following elementary result.

######  Lemma 2.1.

Let ${\bf u}\in\mathcal{P}^{\prime}$. Then ${\bf u}$ satisfies the
distributional differential equation $(\ref{Pearson-DEq})$ if and only if the
corresponding sequence of moments, $u_{n}=\langle{\bf u},x^{n}\rangle$,
satisfies the second order linear difference equation

$d_{n}u_{n+1}+e_{n}u_{n}+n\phi(0)u_{n-1}=0\;,\quad n\in\mathbb{N}_{0}\;.$
(2.5)

Moreover, if ${\bf u}$ satisfies $(\ref{Pearson-DEq})$, then ${\bf u}^{[n]}$
satisfies

$D\big{(}\phi{\bf u}^{[n]}\big{)}=\psi_{n}{\bf u}^{[n]}\;,\quad
n\in\mathbb{N}_{0}\;.$ (2.6)

Notice that if both $\phi$ and $\psi$ vanish identically then (2.1) reduces to
a trivial equation, so we will exclude this situation from our study.

######  Lemma 2.2.

Let ${\bf u}\in\mathcal{P}^{\prime}$. Suppose that ${\bf u}$ is regular and
satisfies $(\ref{Pearson-DEq})$, being $\phi\in\mathcal{P}_{2}$ and
$\psi\in\mathcal{P}_{1}$, and assume that at least one of the polynomials
$\phi$ and $\psi$ is nonzero. Then neither $\phi$ nor $\psi$ is the zero
polynomial, and

$\deg\psi=1\;.$ (2.7)

Given a monic polynomial $P_{n}$ of degree $n$ (which needs not to belong to
an OP), we denote by $P_{n}^{[k]}$ the monic polynomial of degree $n$ defined
by

$P_{n}^{[k]}(x)=\frac{{\rm d}^{k}}{{\rm
d}x^{k}}\,\frac{P_{n+k}(x)}{(n+1)_{k}},\quad k,n\in\mathbb{N}_{0}\;,$ (2.8)

where, for a given $\alpha\in\mathbb{C}$, $(\alpha)_{n}$ is the Pochhammer
symbol, defined as

$(\alpha)_{0}=1\;,\qquad(\alpha)_{n}=\alpha(\alpha+1)\cdots(\alpha+n-1)\;,\quad
n\in\mathbb{N}\;.$ (2.9)

Clearly, if $\\{P_{n}\\}_{n\geq 0}$ is a simple set in $\mathcal{P}$, then so
is $\\{P_{n}^{[k]}\\}_{n\geq 0}$. Under such conditions, there is a beautiful
relation between the associated dual basis:

$D^{k}\big{(}\,{\bf a}_{n}^{[k]}\,\big{)}=(-1)^{k}(n+1)_{k}\,{\bf
a}_{n+k}\;,\quad k,n\in\mathbb{N}_{0}\,,$ (2.10)

where $\\{{\bf a}_{n}\\}_{n\geq 0}$ and $\big{\\{}{\bf
a}_{n}^{[k]}\,\big{\\}}_{n\geq 0}$ are the dual basis in
$\mathcal{P}^{\prime}$ associated with $\\{P_{n}\\}_{n\geq 0}$ and
$\\{P_{n}^{[k]}\\}_{n\geq 0}$, respectively.

######  Lemma 2.3.

Let ${\bf u}\in\mathcal{P}^{\prime}$, and suppose that ${\bf u}$ satisfies the
distributional differential equation $(\ref{Pearson-DEq})$, with $\phi$ and
$\psi$ given by $(\ref{Pearson-PhiPsi})$, being at least one of these
polynomials nonzero. Suppose further that ${\bf u}$ is regular. Then

$d_{n}=na+p\neq 0\;,\quad\forall n\in\mathbb{N}_{0}\;.$ (2.11)

Moreover, if $\\{P_{n}\\}_{n\geq 0}$ denotes the monic OP with respect to
${\bf u}$, and $P_{n}^{[k]}$ is defined by $(\ref{Pnk-deriv})$, then ${\bf
u}^{[k]}=\phi^{k}{\bf u}$ is regular and $\\{P_{n}^{[k]}\\}_{n\geq 0}$ is its
monic OP, for each $k\in\mathbb{N}_{0}$.

We may now establish necessary and sufficient conditions ensuring the
regularity of a given functional ${\bf u}\in\mathcal{P}^{\prime}$ satisfying
(2.2). The next results was proved in [5, Theorem 2] (see also [3] in a more
general context).

######  Theorem 2.1.

Let ${\bf u}\in\mathcal{P}^{\prime}\setminus\\{{\bf 0}\\}$, and suppose that
${\bf u}$ satisfies

$D(\phi{\bf u})=\psi{\bf u}\;,$ (2.12)

where $\phi$ and $\psi$ are nonzero polynomials such that
$\phi\in\mathcal{P}_{2}$ and $\psi\in\mathcal{P}_{1}$. Set

$\phi(x)=ax^{2}+bx+c\,,\quad\psi(x)=px+q\,,\quad d_{n}=na+p\,,\quad
e_{n}=nb+q\quad(n\in\mathbb{N}_{0})\,.$

Then, ${\bf u}$ is regular if and only if

$d_{n}\neq 0\,,\quad\phi\Big{(}-\frac{e_{n}}{d_{2n}}\Big{)}\neq
0\,,\quad\forall n\in\mathbb{N}_{0}\;.$ (2.13)

Moreover, under these conditions, the monic OP $\\{P_{n}\\}_{n\geq 0}$ with
respect to ${\bf u}$ is given by the three-term recurrence relation

$P_{n+1}(x)=(x-\beta_{n})P_{n}(x)-\gamma_{n}P_{n-1}(x)\;,\quad
n\in\mathbb{N}_{0}$ (2.14)

with initial conditions $P_{-1}(x)=0$ and $P_{0}(x)=1$, being

$\beta_{n}=\frac{ne_{n-1}}{d_{2n-2}}-\frac{(n+1)e_{n}}{d_{2n}}\,,\quad\gamma_{n+1}=-\frac{(n+1)d_{n-1}}{d_{2n-1}d_{2n+1}}\phi\Big{(}-\frac{e_{n}}{d_{2n}}\Big{)}\;,\quad
n\in\mathbb{N}_{0}\;.$ (2.15)

In addition, for each $n\in\mathbb{N}_{0}$, $P_{n}$ satisfies the
distributional Rodrigues formula

$P_{n}{\bf u}=k_{n}\,D^{n}\big{(}\phi^{n}{\bf u}\big{)}\;,\quad
k_{n}=\prod_{i=0}^{n-1}d_{n+i-1}^{-1}\,.$ (2.16)

##  3 Classical orthogonal polynomials

The _classical functionals_ are the regular solutions (in
$\mathcal{P}^{\prime}$) of the distributional equation (2.1). The
corresponding OP are called _classical orthogonal polynomials_. In this
section we present the most significant results concerning this important
class of OP.

###  3.1 Definition and characterizations

######  Definition 3.1.

Let ${\bf u}\in\mathcal{P}^{\prime}$. ${\bf u}$ is called a classical
functional if the following two conditions hold:

  1. (i)

${\bf u}$ is regular;

  2. (ii)

${\bf u}$ satisfies the distributional differential equation

$D(\phi{\bf u})=\psi{\bf u}\;,$ (3.1)

where $\phi$ and $\psi$ are polynomials fulfilling

$\deg\phi\leq 2\;,\quad\deg\psi=1\;.$ (3.2)

An OP $\\{P_{n}\\}_{n\geq 0}$ with respect to a classical functional is called
a classical OP.

######  Remark 3.1.

According with Lemma 2.2, in the above definition conditions (3.2) may be
replaced by the weaker conditions

$\phi\in\mathcal{P}_{2}\;,\quad\psi\in\mathcal{P}_{1}\;,\quad\\{\phi,\psi\\}\neq\mathcal{P}_{-1}=\\{0\\}\;.$
(3.3)

Theorem 2.1 gives necessary and sufficient conditions for the existence of
solutions of the distributional equation (2.1), characterizing also such
functionals (and, in particular, solving the question of the existence of
classical functionals). Thus, we may state: a functional ${\bf
u}\in\mathcal{P}^{\prime}\setminus\\{{\bf 0}\\}$ is classical if and only if
there exist $\phi\in\mathcal{P}_{2}$ and $\psi\in\mathcal{P}_{1}$ such that
the following conditions hold:

$\begin{array}[]{rl}{\rm(i)}&D(\phi{\bf u})=\psi{\bf u}\,;\\\
{\rm(ii)}&na+p\neq
0\;,\quad\displaystyle\phi\left(-\frac{nb+q}{2na+p}\right)\neq
0\;,\quad\forall n\in\mathbb{N}_{0}\;,\end{array}$ (3.4)

where we have set $\phi(x)=ax^{2}+bx+c$ and $\psi(x)=px+q$.

In the next proposition we state several characterizations of the classical
OP. For convenience, we introduce the concept of admissible pair of
polynomials.

######  Definition 3.2.

$(\phi,\psi)$ is called an admissible pair if

$\phi\in\mathcal{P}_{2}\;,\quad\psi\in\mathcal{P}_{1}\;,\quad
d_{n}=\psi^{\prime}+\mbox{$\frac{n}{2}$}\,\phi^{\prime\prime}\neq
0\;,\;\;\forall n\in\mathbb{N}_{0}.$

Introducing this concept makes sense, since according with conditions (ii) in
(3.4), only admissible pairs may appear in the framework of the theory of
classical OP.

######  Theorem 3.1 (characterizations of the classical OP).

Let ${\bf u}\in\mathcal{P}^{\prime}$, regular, and let $\\{P_{n}\\}_{n\geq 0}$
be its monic OP. Then the following properties are equivalent:

  1. C1.

${\bf u}$ is classical, i.e., there are nonzero polynomials
$\phi\in\mathcal{P}_{2}$ and $\psi\in\mathcal{P}_{1}$ such that ${\bf u}$
satisfies

$D(\phi{\bf u})=\psi{\bf u}\;;$

  2. C1′.

there is an admissible pair $(\phi,\psi)$ such that ${\bf u}$ satisfies

$D(\phi{\bf u})=\psi{\bf u}\;;$

  3. C2.

(Al-Salam $\&$ Chihara) there exist a polynomial $\phi\in\mathcal{P}_{2}$ and,
for each $n\in\mathbb{N}_{0}$, complex parameters $a_{n}$, $b_{n}$ and
$c_{n}$, with $c_{n}\neq 0$ if $n\geq 1$, such that

$\phi(x)P_{n}^{\prime}(x)=a_{n}P_{n+1}(x)+b_{n}P_{n}(x)+c_{n}P_{n-1}(x)\;,\quad
n\geq 0\;;$

  4. C3.

(Hahn) $\Big{\\{}P_{n}^{[k]}=\frac{{\rm d}^{k}}{{\rm
d}x^{k}}\frac{P_{n+k}}{(n+1)_{k}}\Big{\\}}_{n\geq 0}$ is a monic OP for some
$k\in\mathbb{N}\,$;

  5. C3′.

$\big{\\{}P_{n}^{[k]}\big{\\}}_{n\geq 0}$ is a monic OP for each
$k\in\mathbb{N}\,$;

  6. C4.

there exist $k\in\mathbb{N}$ and complex parameters $r_{n}^{[k]}$ and
$s_{n}^{[k]}$ such that

$\displaystyle
P_{n}^{[k-1]}(x)=P_{n}^{[k]}(x)+r_{n}^{[k]}P_{n-1}^{[k]}(x)+s_{n}^{[k]}P_{n-2}^{[k]}(x)\;,\quad
n\geq 2\;;$ (3.5)

  7. C4′. 333It was proved for $k=1$ by Geronimus (see [4, (42)]).

for each $k\in\mathbb{N}$, there exist parameters $r_{n}^{[k]}$ and
$s_{n}^{[k]}$ such that (3.5) holds;

  8. C5.

(Bochner) there exist polynomials $\phi$ and $\psi$ and, for each $n\geq 0$, a
complex parameter $\lambda_{n}$, with $\lambda_{n}\neq 0$ if $n\geq 1$, such
that $y=P_{n}(x)$ is a solution of the second order ordinary differential
equation

$\phi(x)y^{\prime\prime}+\psi(x)y^{\prime}+\lambda_{n}y=0\;,\quad n\geq 0\;;$

  9. C6.

(Maroni) there is an admissible pair $(\phi,\psi)$ so that the formal
Stieltjes series associated with ${\bf u}$, $S_{\bf
u}(z)=-\sum_{n=0}^{\infty}u_{n}/z^{n+1}$, satisfies (formally)

$\phi(z)S_{\bf u}^{\prime}(z)=[\psi(z)-\phi^{\prime}(z)]S_{\bf
u}(z)+(\psi^{\prime}-\mbox{$\frac{1}{2}$}\,\phi^{\prime\prime})u_{0}\;;$

  10. C7.

(McCarthy) there exists an admissible pair $(\phi,\psi)$ and, for each $n\geq
1$, complex parameters $h_{n}$ and $t_{n}$ such that

$\phi(P_{n}P_{n-1})^{\prime}(x)=h_{n}P_{n}^{2}(x)-(\psi-\phi^{\prime})P_{n}P_{n-1}(x)+t_{n}P_{n-1}^{2}(x)\;;$

  11. C8.

(distributional Rodrigues formula) there exist a polynomial
$\phi\in\mathcal{P}_{2}$ and nonzero complex parameters $k_{n}$ such that

$P_{n}(x){\bf u}=k_{n}D^{n}\big{(}\phi^{n}(x){\bf u}\big{)}\;,\quad n\geq
0\;.$

Moreover, the polynomials $\phi$ and $\psi$ may be taken the same in all
properties above where they appear. In addition, let the TTRR fulfilled by the
monic OP $\\{P_{n}\\}_{n\geq 0}$ be

$P_{n+1}(x)=(x-\beta_{n})P_{n}(x)-\gamma_{n}P_{n-1}(x)\;,\quad n\geq 0$

($P_{-1}(x)=0$; $P_{0}(x)=1$). Write $\phi(x)=ax^{2}+bx+c$, $\psi(x)=px+q$,
$d_{n}=na+p$, and $e_{n}=nb+q$. Then

$\beta_{n}=-\frac{d_{-2}q+2bnd_{n-1}}{d_{2n}d_{2n-2}}\;,\quad\gamma_{n}=-\frac{nd_{n-2}}{d_{2n-3}d_{2n-1}}\phi\Big{(}-\frac{e_{n-1}}{d_{2n-2}}\Big{)}\;,$

and the parameters appearing in the above characterizations may be computed
explicitly:

$\displaystyle a_{n}$ $\displaystyle=na,\quad\;\;\qquad
b_{n}=-\mbox{$\frac{1}{2}$}\psi(\beta_{n}),\quad\quad
c_{n}=-d_{n-1}\gamma_{n},$ $\displaystyle r_{n}^{[1]}$
$\displaystyle=\mbox{$\frac{1}{2}$}\frac{\psi(\beta_{n})}{d_{n-1}},\quad
s_{n}^{[1]}=-\frac{(n-1)a}{d_{n-2}}\gamma_{n},$ $\displaystyle\lambda_{n}$
$\displaystyle=-nd_{n-1},\quad h_{n}=d_{2n-3},\quad
t_{n}=-d_{2n-1}\gamma_{n},\quad k_{n}=\prod_{i=0}^{n-1}d_{n+i-1}^{-1}.$

###### Proof.

By Lemma 2.3 and Theorem 2.1, C1$\,\Leftrightarrow\,$C1′,
C1$\,\Rightarrow\,$C3′, and C1′$\,\Leftrightarrow\,$C8. Clearly,
C3′$\,\Rightarrow\,$C3 and C4′$\,\Rightarrow\,$C4. We show that
C3′$\,\Rightarrow\,$C4′ using the same arguments of the proof of
C3$\,\Rightarrow\,$C4 given in bellow. The proof of C1′$\,\Leftrightarrow\,$C6
is left to the reader. Thus, we only need to show that:

C1′$\,\Rightarrow\,$C2$\,\Rightarrow\,$C3$\,\Rightarrow\,$C4$\,\Rightarrow\,$C1$\,,\;\;$
C1$\,\Leftrightarrow\,$C5$\,,\;\;$ C2$\,\Leftrightarrow\,$C7.

(C1′$\,\Rightarrow\,$C2). Assume that C1′ holds. Fix $n\in\mathbb{N}_{0}$.
Since $\deg(\phi P_{n}^{\prime})\leq n+1$, then

$\phi P^{\prime}_{n}=\sum_{j=0}^{n+1}a_{n,j}P_{j}\,,\quad
a_{n,j}=\frac{\langle{\bf u},\phi P_{n}^{\prime}P_{j}\rangle}{\langle{\bf
u},P_{j}^{2}\rangle}\;.$ (3.6)

For each integer number $j$, with $0\leq j\leq n+1$, we deduce

$\begin{array}[]{rl}\langle{\bf u},\phi
P_{n}^{\prime}P_{j}\rangle&=\,\langle\phi{\bf
u},(P_{n}P_{j})^{\prime}-P_{n}P_{j}^{\prime}\rangle=-\langle D(\phi{\bf
u}),P_{n}P_{j}\rangle-\langle\phi{\bf u},P_{n}P_{j}^{\prime}\rangle\\\\[5.0pt]
&=\,-\langle{\bf u},\psi P_{j}P_{n}\rangle-\langle{\bf u},\phi
P_{j}^{\prime}P_{n}\rangle\;.\end{array}$ (3.7)

If $0\leq j\leq n-2$ we obtain $\langle{\bf u},\phi
P_{n}^{\prime}P_{j}\rangle=0$, and so $a_{n,j}=0$. Thus, (3.6) reduces to

$\phi P^{\prime}_{n}=a_{n}P_{n+1}+b_{n}P_{n}+c_{n}P_{n-1}\;,\quad n\geq 0\;,$

where, writing $\phi(x)=ax^{2}+bx+c$ and $\psi(x)=px+q$, $a_{n}=na$ (by
comparison of coefficients), $b_{n}=a_{n,n}$, and $c_{n}=a_{n,n-1}$. Setting
$j=n-1$ in (3.7), we deduce

$\langle{\bf u},\phi P_{n}^{\prime}P_{n-1}\rangle=-\langle{\bf u},(\psi
P_{n-1}+\phi P_{n-1}^{\prime})P_{n}\rangle=-d_{n-1}\langle{\bf
u},P_{n}^{2}\rangle\;,$

hence

$c_{n}=a_{n,n-1}=\frac{\langle{\bf u},\phi
P_{n}^{\prime}P_{n-1}\rangle}{\langle{\bf
u},P_{n-1}^{2}\rangle}=\frac{\langle{\bf u},\phi
P_{n}^{\prime}P_{n-1}\rangle}{\langle{\bf
u},P_{n}^{2}\rangle}\frac{\langle{\bf u},P_{n}^{2}\rangle}{\langle{\bf
u},P_{n-1}^{2}\rangle}=-d_{n-1}\gamma_{n}\;,\quad n\geq 1\;.$

Since, by hypothesis, $(\phi,\psi)$ is an admissible pair, then we may
conclude that $c_{n}\neq 0$ for each $n\geq 1$. Thus C1′$\,\Rightarrow\,$C2.
Notice that taking $j=n$ in (3.7) yields

$\langle{\bf u},\phi
P_{n}^{\prime}P_{n}\rangle=-\mbox{$\frac{1}{2}\,$}\langle{\bf u},\psi
P_{n}^{2}\rangle=-\mbox{$\frac{1}{2}\,$}\big{(}p\langle{\bf
u},xP_{n}^{2}\rangle+q\langle{\bf u},P_{n}^{2}\rangle\big{)}\;,$

hence we deduce the expression for $b_{n}$ given in the statement of the
theorem:

$b_{n}=a_{n,n}=\frac{\langle{\bf u},\phi
P_{n}^{\prime}P_{n}\rangle}{\langle{\bf
u},P_{n}^{2}\rangle}=-\mbox{$\frac{1}{2}\,$}\Big{(}p\frac{\langle{\bf
u},xP_{n}^{2}\rangle}{\langle{\bf
u},P_{n}^{2}\rangle}+q\Big{)}=-\mbox{$\frac{1}{2}$}\psi(\beta_{n})\;.$

(C2$\,\Rightarrow\,$C3). Suppose that C2 holds. We will show that
$\\{P_{n}^{[1]}=P_{n+1}^{\prime}/(n+1)\\}_{n\geq 0}$ is a monic OP with
respect to ${\bf v}=\phi{\bf u}$. Indeed, for each $n\in\mathbb{N}_{0}$ and
$0\leq m\leq n$,

$\displaystyle(n+1)\langle{\bf v},x^{m}P_{n}^{[1]}\rangle$
$\displaystyle=\langle\phi{\bf u},x^{m}P_{n+1}^{\prime}\rangle=\langle{\bf
u},\big{(}\phi P_{n+1}^{\prime}\big{)}x^{m}\rangle$ $\displaystyle=\langle{\bf
u},(a_{n+1}P_{n+2}+b_{n+1}P_{n+1}+c_{n+1}P_{n})x^{m}\rangle$
$\displaystyle=c_{n+1}\langle{\bf u},P_{n}^{2}\rangle\delta_{m,n}\;.$

Therefore, since $c_{n+1}\neq 0$ for each $n\geq 0$, we conclude that
$\\{P_{n}^{[1]}\\}_{n\geq 0}$ is a monic OP (with respect to ${\bf v}=\phi{\bf
u}$).

(C3$\,\Rightarrow\,$C4). By hypothesis, $\\{P_{n}^{[k]}=\frac{{\rm
d}^{k}}{{\rm d}x^{k}}\big{(}\frac{P_{n+k}}{(n+1)_{k}}\big{)}\\}_{n\geq 0}$ is
a monic OP for some (fixed) $k\in\mathbb{N}$. Then there exists
$\beta_{n}^{[k]}\in\mathbb{C}$ and
$\gamma_{n}^{[k]}\in\mathbb{C}\setminus\\{0\\}$ such that

$xP_{n}^{[k]}=P_{n+1}^{[k]}+\beta_{n}^{[k]}P_{n}^{[k]}+\gamma_{n}^{[k]}P_{n-1}^{[k]}\;,\quad
n\in\mathbb{N}_{0}\,.$ (3.8)

Similarly, there exists $\beta_{n}\in\mathbb{C}$ and
$\gamma_{n}\in\mathbb{C}\setminus\\{0\\}$ such that

$xP_{n}=P_{n+1}+\beta_{n}P_{n}+\gamma_{n}P_{n-1}\;,\quad
n\in\mathbb{N}_{0}\,.$ (3.9)

Changing $n$ into $n+k$ in (3.9), then taking the derivative of order $k$ in
both sides of the resulting equation and using Leibnitz rule on the left-hand
side, we find

$xP_{n}^{[k]}+\frac{k}{n+1}P_{n+1}^{[k-1]}=\frac{n+k+1}{n+1}P_{n+1}^{[k]}+\beta_{n+k}P_{n}^{[k]}+\frac{n\gamma_{n+k}}{n+k}P_{n-1}^{[k]}\;,\quad
n\in\mathbb{N}_{0}\,.$

In this equation, replacing $xP_{n}^{[k]}$ by the right-hand side of (3.8),
and then changing $n$ into $n-1$, we obtain (3.5), with

$r_{n}^{[k]}=\frac{n\,\big{(}\beta_{n+k-1}-\beta_{n-1}^{[k]}\big{)}}{k}\;,\quad
s_{n}^{[k]}=\frac{n\,\big{(}(n-1)\gamma_{n+k-1}-(n+k-1)\gamma_{n-1}^{[k]}\big{)}}{k(n+k-1)}\;.$

(C4$\,\Rightarrow\,$C1). By hypothesis (3.5) holds. Let $\\{{\bf
a}_{n}\\}_{n\geq 0}$ and $\\{{\bf a}_{n}^{[k]}\\}_{n\geq 0}$ be the dual basis
for $\\{P_{n}\\}_{n\geq 0}$ and $\\{P_{n}^{[k]}\\}_{n\geq 0}$, respectively.
By Proposition 1.3, $\,{\bf a}_{n}^{[k]}=\sum_{j\geq 0}\langle{\bf
a}_{n}^{[k]},P_{j}^{[k-1]}\rangle{\bf a}_{j}^{[k-1]}$ for each
$n\in\mathbb{N}_{0}$. Using (3.5), we compute

$\displaystyle\langle{\bf a}_{n}^{[k]},P_{j}^{[k-1]}\rangle$
$\displaystyle=\langle{\bf
a}_{n}^{[k]},P_{j}^{[k]}\rangle+r_{j}^{[k]}\langle{\bf
a}_{n}^{[k]},P_{j-1}^{[k]}\rangle+s_{j}^{[k]}\langle{\bf
a}_{n}^{[k]},P_{j-2}^{[k]}\rangle$
$\displaystyle=\left\\{\begin{array}[]{cl}1\;,&\mbox{\rm if}\;j=n\\\\[5.0pt]
r_{n+1}^{[k]}\;,&\mbox{\rm if}\;j=n+1\\\\[5.0pt] s_{n+2}^{[k]}\;,&\mbox{\rm
if}\;j=n+2\\\\[5.0pt] 0\;,&\mbox{\rm otherwise}\,.\end{array}\right.$

Hence

${\bf a}_{n}^{[k]}={\bf a}_{n}^{[k-1]}+r_{n+1}^{[k]}{\bf
a}_{n+1}^{[k-1]}+s_{n+2}^{[k]}{\bf a}_{n+2}^{[k-1]}\;,\quad
n\in\mathbb{N}_{0}\;.$

Taking the (distributional) derivative of order $k$ in both sides of this
equation, and using the relations $D^{j}\big{(}{\bf
a}_{n}^{[j]}\big{)}=(-1)^{j}(n+1)_{j}\,{\bf a}_{n+j}$, we obtain

$D\left(\frac{1}{n+k}\,{\bf a}_{n+k-1}+\frac{r_{n+1}^{[k]}}{n+1}\,{\bf
a}_{n+k}+\frac{(n+k+1)s_{n+2}^{[k]}}{(n+1)(n+2)}\,{\bf a}_{n+k+1}\right)=-{\bf
a}_{n+k}\;,\;n\in\mathbb{N}_{0}\;.$

Therefore, since, by Theorem 1.2, ${\bf a}_{j}=\frac{P_{j}}{\langle{\bf
u},P_{j}^{2}\rangle}\,{\bf u}$ for $j\in\mathbb{N}_{0}$ and, by (1.20),
$\gamma_{j}=\frac{{\langle{\bf u},P_{j}^{2}\rangle}}{{\langle{\bf
u},P_{j-1}^{2}\rangle}}$ for $j\in\mathbb{N}$, being $\gamma_{j}$ the
parameter appearing in (3.9), we deduce

$D\big{(}\Phi_{n+k+1}\,{\bf u}\big{)}=-P_{n+k}\,{\bf u}\;,\quad
n\in\mathbb{N}_{0}\;,$ (3.10)

where $\Phi_{n+k+1}$ is a polynomial of degree at most $n+k+1$, given by

$\displaystyle\Phi_{n+k+1}(x)$
$\displaystyle=\frac{\gamma_{n+k}}{n+k}\,P_{n+k-1}(x)+\frac{r_{n+1}^{[k]}}{n+1}\,P_{n+k}(x)+\frac{(n+k+1)s_{n+2}^{[k]}}{(n+1)(n+2)\gamma_{n+k+1}}\,P_{n+k+1}(x)\,.$

Since $\Phi_{n+k+1}$ is a (finite) linear combination of polynomials of the
simple set $\\{P_{j}\\}_{j\geq 0}$ and $\gamma_{n+k}\neq 0$, then
$\Phi_{n+k+1}$ does not vanishes identically, so
$\Phi_{n+k+1}\in\mathcal{P}_{n+k+1}\setminus\\{0\\}$. Setting $n=0$ and $n=1$
in (3.10) we obtain the two equations

$D\big{(}\Phi_{k+1}\,{\bf u}\big{)}=-P_{k}\,{\bf u}\;,\quad
D\big{(}\Phi_{k+2}\,{\bf u}\big{)}=-P_{k+1}\,{\bf u}\;.$ (3.11)

If $k=1$ it follows immediately from the first of these equations that C1
holds. Henceforth, assume that $k\geq 2$. Setting $n=0$ and $n=1$ in the
definition of $\Phi_{n+k+1}$ and using the TTRR (3.9), we easily deduce

$\left\\{\begin{array}[]{rcl}\Phi_{k+1}(x)&=&E_{0}(x;k)P_{k+1}(x)+F_{1}(x;k)P_{k}(x)\;,\\\\[5.0pt]
\Phi_{k+2}(x)&=&G_{1}(x;k)P_{k+1}(x)+H_{0}(x;k)P_{k}(x)\;,\end{array}\right.$
(3.12)

where $E_{0}(\cdot;k),H_{0}(\cdot;k)\in\mathcal{P}_{0}$ and
$F_{1}(\cdot;k),G_{1}(\cdot;k)\in\mathcal{P}_{1}$, explicitly given by

$\begin{array}[]{l}E_{0}(x;k)=\displaystyle\frac{(k+1)s_{2}^{[k]}}{2\gamma_{k+1}}-\frac{1}{k}\;,\quad
F_{1}(x;k)=\frac{x-\beta_{k}}{k}+r_{1}^{[k]}\;,\\\\[7.5pt]
G_{1}(x;k)=\displaystyle\frac{(k+2)s_{3}^{[k]}(x-\beta_{k+1})}{6\gamma_{k+2}}+\frac{r_{2}^{[k]}}{2}\;,\;H_{0}(x;k)=\frac{\gamma_{k+1}}{k+1}-\frac{(k+2)s_{3}^{[k]}\gamma_{k+1}}{6\gamma_{k+2}}\;.\end{array}$
(3.13)

Let
$\Delta_{2}(x)\equiv\Delta_{2}(x;k)=E_{0}(x;k)H_{0}(x;k)-F_{1}(x;k)G_{1}(x;k)$,
the determinant of the system (3.12). Using (3.11)–(3.13), and taking into
account that ${\bf u}$ is regular, we may prove that
$\Delta_{2}\in\mathcal{P}_{2}\setminus\\{0\\}$ . Solving (3.12) for $P_{k}$
and $P_{k+1}$ we obtain

$\displaystyle\Delta_{2}(x)P_{k+1}(x)=H_{0}(x;k)\Phi_{k+1}(x)-F_{1}(x;k)\Phi_{k+2}(x)\,,$
(3.14)
$\displaystyle\Delta_{2}(x)P_{k}(x)=E_{0}(x;k)\Phi_{k+2}(x)-G_{1}(x;k)\Phi_{k+1}(x)\;.$
(3.15)

Since $P_{k}$ and $P_{k+1}$ cannot share zeros, it follows from (3.14)–(3.15)
that any common zero of $\Phi_{k+1}$ and $\Phi_{k+2}$ (if there is some) must
be a zero of $\Delta_{2}$. Let $\Phi$ be the greatest common divisor of
$\Phi_{k+1}$ and $\Phi_{k+2}$, i.e.,

$\Phi(x)=\mbox{g.c.d.}\,\\{\Phi_{k+1}(x),\Phi_{k+2}(x)\\}\;.$

Any zero of $\Phi$ is also a zero of both $\Phi_{k+1}$ and $\Phi_{k+2}$, and
so it is a zero of $\Delta_{2}$. Therefore,
$\Phi\in\mathcal{P}_{2}\setminus\\{0\\}$. (Notice that indeed $\Phi\not\equiv
0$, since $\Phi_{k+1}\not\equiv 0$ and $\Phi_{k+2}\not\equiv 0$.) Moreover,
there exist polynomials $\Phi_{1,k}$ and $\Phi_{2,k}$, with no common zeros,
such that

$\displaystyle\Phi_{k+1}=\Phi\,\Phi_{1,k}\;,\quad\Phi_{k+2}=\Phi\,\Phi_{2,k}\;,\qquad\qquad\qquad$
(3.16)
$\displaystyle\Phi_{1,k}\in\mathcal{P}_{k+1-\ell}\setminus\\{0\\}\;,\quad\Phi_{2,k}\in\mathcal{P}_{k+2-\ell}\setminus\\{0\\}\;,\quad\ell=\deg\Phi\leq
2\;.$ (3.17)

From (3.11) and (3.16) we deduce

$\Phi_{1,k}D(\Phi{\bf u})=-(P_{k}+\Phi_{1,k}^{\prime}\Phi){\bf
u}\;,\quad\Phi_{2,k}D(\Phi{\bf u})=-(P_{k+1}+\Phi_{2,k}^{\prime}\Phi){\bf
u}\;.$ (3.18)

Combining these two equations yields
$\big{(}\Phi_{1,k}(P_{k+1}+\Phi_{2,k}^{\prime}\Phi)-\Phi_{2,k}(P_{k}+\Phi_{1,k}^{\prime}\Phi)\big{)}{\bf
u}={\bf 0}$, and so, since ${\bf u}$ is regular,
$\Phi_{1,k}(P_{k+1}+\Phi_{2,k}^{\prime}\Phi)=\Phi_{2,k}(P_{k}+\Phi_{1,k}^{\prime}\Phi)$.
Therefore, taking into account that $\Phi_{1,k}$ and $\Phi_{2,k}$ have no
common zeros and (3.17) holds, we may ensure that there exists a polynomial
$\Psi\in\mathcal{P}_{1}$ such that

$P_{k}+\Phi_{1,k}^{\prime}\Phi=-\Psi\Phi_{1,k}\;,\quad
P_{k+1}+\Phi_{2,k}^{\prime}\Phi=-\Psi\Phi_{2,k}\;.$ (3.19)

Combining equations (3.18) and (3.19) we deduce

$\Phi_{1,k}\big{(}D(\Phi{\bf u})-\Psi{\bf
u}\big{)}=\Phi_{2,k}\big{(}D(\Phi{\bf u})-\Psi{\bf u}\big{)}={\bf 0}\;.$

From these equations, and using once again the fact that $\Phi_{1,k}$ and
$\Phi_{2,k}$ have no common zeros, we conclude, by Proposition 1.5, that
$D(\Phi{\bf u})=\Psi{\bf u}$. Thus C4$\,\Rightarrow\,$C1. The formulas for
$r_{n}^{[1]}$ and $s_{n}^{[1]}$ given in the statement of the theorem may be
derived as follows. We have already proved that
C4$\,\Rightarrow\,$C1$\,\Rightarrow\,$C1′$\,\Rightarrow\,$C2$\,\Rightarrow\,$C3$\,\Rightarrow\,$C4,
and we see that the polynomials $\phi$ and $\psi$ appearing in all these
characterizations may be taken the same. As we have seen, the formulas for
$b_{n}$ and $c_{n}$ given in the statement of the theorem hold. We now use
these formulas to obtain the expressions for $r_{n}^{[1]}$ and $s_{n}^{[1]}$.
Set $Q_{n}=P_{n}^{[1]}=P_{n+1}^{\prime}/(n+1)$. By C4,
$P_{n}=Q_{n}+r_{n}^{[1]}Q_{n-1}+s_{n}^{[1]}Q_{n-2}$ if $n\geq 2$. Hence, since
$\\{Q_{n}\\}_{n\geq 0}$ is a monic OP with respect to ${\bf v}=\phi{\bf u}$,
we deduce, for each $n\geq 2$,

$\displaystyle r_{n}^{[1]}$ $\displaystyle=\frac{\langle{\bf u},\phi
P_{n}P_{n}^{\prime}\rangle}{\langle{\bf u},\phi P_{n}^{\prime}P_{n-1}\rangle}$
$\displaystyle=\frac{\langle{\bf u},P_{n-1}^{2}\rangle}{\langle{\bf u},\phi
P_{n}^{\prime}P_{n-1}\rangle}\frac{\langle{\bf u},\phi
P_{n}^{\prime}P_{n}\rangle}{\langle{\bf u},P_{n}^{2}\rangle}\frac{\langle{\bf
u},P_{n}^{2}\rangle}{\langle{\bf
u},P_{n-1}^{2}\rangle}=\frac{1}{c_{n}}b_{n}\gamma_{n}=\mbox{$\frac{1}{2}$}\frac{\psi(\beta_{n})}{d_{n-1}}\,,$

where the third equality holds taking into account C2. Similarly, for each
$n\geq 2$,

$\displaystyle s_{n}^{[1]}$ $\displaystyle=\frac{a\langle{\bf
u},P_{n}^{2}\rangle}{\frac{1}{n-1}\langle{\bf u},\phi
P_{n-1}^{\prime}P_{n-2}\rangle}=\frac{(n-1)a\langle{\bf
u},P_{n}^{2}\rangle}{c_{n-1}\langle{\bf u},P_{n-2}^{2}\rangle}$
$\displaystyle=\frac{(n-1)a}{c_{n-1}}\gamma_{n-1}\gamma_{n}=-\frac{(n-1)a}{d_{n-2}}\gamma_{n}\;.$

(C1$\,\Rightarrow\,$C5). By hypothesis, $D(\phi{\bf u})=\psi{\bf u}$, where
$\phi\in\mathcal{P}_{2}$, $\psi\in\mathcal{P}_{1}$, and $\deg\psi=1$ (cf.
Lemma 2.2). Fix $n\in\mathbb{N}$, and write

$\phi P_{n}^{\prime\prime}+\psi
P_{n}^{\prime}=\sum_{j=0}^{n}\lambda_{n,j}P_{j}\;.$ (3.20)

Then, for each $j$ such that $0\leq j\leq n$,

$\displaystyle\langle{\bf u},P_{j}^{2}\rangle\lambda_{n,j}$
$\displaystyle=\langle{\bf u},(\phi P_{n}^{\prime\prime}+\psi
P_{n}^{\prime})P_{j}\rangle=\langle\phi{\bf
u},P_{n}^{\prime\prime}P_{j}\rangle+\langle\psi{\bf
u},P_{n}^{\prime}P_{j}\rangle$ $\displaystyle=\langle\phi{\bf
u},(P_{n}^{\prime}P_{j})^{\prime}\rangle-\langle\phi{\bf
u},P_{n}^{\prime}P_{j}^{\prime}\rangle+\langle\psi{\bf
u},P_{n}^{\prime}P_{j}\rangle=-\langle\phi{\bf
u},P_{n}^{\prime}P_{j}^{\prime}\rangle\,.$

Since by hypothesis C1 holds, and we have already proved that
C1$\,\Rightarrow\,$C1′$\,\Rightarrow\,$C2$\,\Rightarrow\,$C3, and in the proof
of C2$\,\Rightarrow\,$C3 we have shown that
$\\{Q_{n}=P^{\prime}_{n+1}/(n+1)\\}_{n\geq 0}$ is a monic OP with respect to
${\bf v}=\phi{\bf u}$, then $\langle\phi{\bf
u},P_{n}^{\prime}P_{j}^{\prime}\rangle=0$ if $j\neq n$, hence (3.20) reduces
to

$\phi P_{n}^{\prime\prime}+\psi P_{n}^{\prime}+\lambda_{n}P_{n}=0\;,\quad
n\geq 0\;,$ (3.21)

where $\lambda_{n}=-\lambda_{n,n}$. Comparing leading coefficients in (3.21),
and setting $\phi(x)=ax^{2}+bx+c$ and $\psi(x)=px+q$, we obtain
$\lambda_{n}=-n\big{(}(n-1)a+p\big{)}=-nd_{n-1}$, hence $\lambda_{n}\neq 0$ if
$n\geq 1$ (since C1$\,\Rightarrow\,$C1′, so $(\phi,\psi)$ is an admissible
pair). Thus C1$\,\Rightarrow\,$C5.

(C5$\,\Rightarrow\,$C1). By hypothesis, there extist
$\phi,\psi\in\mathcal{P}$, and $\lambda_{n}\in\mathbb{C}$, with
$\lambda_{n}\neq 0$ if $n\geq 1$, such that $-\phi P_{n+1}^{\prime\prime}=\psi
P_{n+1}^{\prime}+\lambda_{n+1}P_{n+1}$. Taking in this equation $n=0$ and
$n=1$ we deduce
$\psi=-\lambda_{1}P_{1}\in\mathcal{P}_{1}\setminus\mathcal{P}_{0}$ and
$\phi=-(\psi P_{2}^{\prime}+\lambda_{2}P_{2})/2\in\mathcal{P}_{2}$. We will
prove that $D(\phi{\bf u})=\psi{\bf u}$ by showing that the actions of the
functionals $D(\phi{\bf u})$ and $\psi{\bf u}$ coincide on the simple set
$\\{Q_{n}\\}_{n\geq 0}$. Indeed,

$\displaystyle\langle D(\phi{\bf u}),Q_{n}\rangle$
$\displaystyle=\frac{1}{n+1}\langle D(\phi{\bf u}),P^{\prime}_{n+1}\rangle$
$\displaystyle=-\frac{1}{n+1}\langle{\bf u},\phi
P_{n+1}^{\prime\prime}\rangle=\frac{1}{n+1}\langle{\bf u},\psi
P^{\prime}_{n+1}+\lambda_{n+1}P_{n+1}\rangle$ $\displaystyle=\langle{\bf
u},\psi Q_{n}\rangle+\frac{\lambda_{n+1}}{n+1}\langle{\bf
u},P_{n+1}\rangle=\langle\psi{\bf u},Q_{n}\rangle.$

Since at least one of the polynomials $\phi$ and $\psi$ is nonzero (because
$\lambda_{n}\neq 0$), C1 holds.

(C2$\,\Rightarrow\,$C7). Since by hypothesis (C2) holds, we may write

$\displaystyle\phi
P_{n}^{\prime}=a_{n}P_{n+1}+b_{n}P_{n}+c_{n}P_{n-1}\;,\quad$ (3.22)
$\displaystyle\phi
P_{n-1}^{\prime}=a_{n-1}P_{n}+b_{n-1}P_{n-1}+c_{n-1}P_{n-2}\;.$ (3.23)

Multiplying (3.22) by $P_{n-1}$ and (3.23) by $P_{n}$ and adding the resulting
equalities, we find that $\phi(P_{n}P_{n-1})^{\prime}$ is a linear combination
of the polynomials $P_{n}^{2}$, $P_{n}P_{n-1}$, $P_{n-1}^{2}$,
$P_{n+1}P_{n-1}$ and $P_{n}P_{n-2}$. Substituting $P_{n+1}$ and $P_{n-2}$ by
the corresponding expressions given by the TTRR, we deduce

$\phi(P_{n}P_{n-1})^{\prime}=A_{n}P_{n}^{2}+(B_{n}x+C_{n})P_{n}P_{n-1}+D_{n}P_{n-1}^{2}\;,\quad
n\geq 1\;,$ (3.24)

where

$\begin{array}[]{l}A_{n}=a_{n-1}-\frac{c_{n-1}}{\gamma_{n-1}}\;,\quad
B_{n}=a_{n}+\frac{c_{n-1}}{\gamma_{n-1}}\;,\\\\[5.0pt]
C_{n}=-a_{n}\beta_{n}+b_{n}+b_{n-1}-\frac{c_{n-1}}{\gamma_{n-1}}\beta_{n-1}\;,\quad
D_{n}=c_{n}-a_{n}\gamma_{n}\;.\end{array}$

Write $\phi(x)=ax^{2}+bx+c$ and $\psi(x)=px+q$. We have already seen that
C2$\,\Leftrightarrow\,$C1′, and while proving C1′$\,\Rightarrow\,$C2 we have
shown that the coefficients $a_{n}$, $b_{n}$, and $c_{n}$ appearing in (3.22)
are given by $a_{n}=na$, $b_{n}=-\frac{1}{2}\psi(\beta_{n})$, and
$c_{n}=-d_{n-1}\gamma_{n}$. It follows that

$\begin{array}[]{rcl}A_{n}=d_{2n-3}\;,\quad B_{n}=2a-p\;,\quad
D_{n}=-d_{2n-1}\gamma_{n}\;,\\\\[5.0pt]
C_{n}=-\frac{1}{2}\big{(}d_{2n}\beta_{n}-d_{2n-4}\beta_{n-1}\big{)}-q=b-q\;,\end{array}$
(3.25)

where the last equality is easily derived using the expressions for the
$\beta-$parameters given in the statement of the theorem. Therefore,
$B_{n}x+C_{n}=\phi^{\prime}-\psi$ (independent of $n$). Finally, substituting
(3.25) into (3.24) yields the equation appearing in C7, being
$h_{n}=A_{n}=d_{2n-3}$ and $t_{n}=D_{n}=-d_{2n-1}\gamma_{n}$ for each $n\geq
1$. Thus C2$\,\Rightarrow\,$C7.

(C7$\,\Rightarrow\,$C2). Fix an integer $n\geq 1$. For this $n$, rewrite the
equation in (C7) as

$\big{(}\phi P_{n}^{\prime}+\psi P_{n}-t_{n}P_{n-1}\big{)}P_{n-1}=\big{(}-\phi
P_{n-1}^{\prime}+\phi^{\prime}P_{n-1}+h_{n}P_{n}\big{)}P_{n}\;.$

Therefore, since $P_{n}$ and $P_{n-1}$ have no common zeros, there is
$\pi_{1,n}\in\mathcal{P}_{1}$ such that

$\displaystyle\phi P_{n}^{\prime}+\psi P_{n}-t_{n}P_{n-1}=\pi_{1,n}P_{n}\;,$
(3.26) $\displaystyle-\phi
P_{n-1}^{\prime}+\phi^{\prime}P_{n-1}+h_{n}P_{n}=\pi_{1,n}P_{n-1}\;.$ (3.27)

By comparing the leading coefficients on both sides of equation (3.26) we
deduce $\pi_{1,n}(x)=d_{n}x+z_{n}$ for some $z_{n}\in\mathbb{C}$ (and
$d_{n}=na+p$). By hypothesis, $(\phi,\psi)$ is an admissible pair, hence
$d_{n}\neq 0$ and so $\deg\pi_{1,n}=1$. Moreover, by the TTRR for
$\\{P_{n}\\}_{n\geq 0}$, $xP_{n}=P_{n+1}+\beta_{n}P_{n}+\gamma_{n}P_{n-1}$.
Therefore, (3.26) may be rewritten as

$\phi P_{n}^{\prime}=a_{n}P_{n+1}+b_{n}P_{n}+c_{n}P_{n-1}\;,$

where $a_{n}=na$, $b_{n}=na\beta_{n}+z_{n}-q$, and $c_{n}=na\gamma_{n}+t_{n}$.
To conclude the proof we need to show that $c_{n}\neq 0$ for all $n\geq 1$.
Indeed, changing $n$ into $n+1$ in (3.27) and adding the resulting equation
with (3.26), we obtain

$(\psi+\phi^{\prime})P_{n}-t_{n}P_{n-1}+h_{n+1}P_{n+1}=\big{(}(d_{n}+d_{n+1})x+(z_{n}+z_{n+1})\big{)}P_{n}\;.$

Since $\psi+\phi^{\prime}=(2a+p)x+q+b$ and taking into account once again the
TTRR for $\\{P_{n}\\}_{n\geq 0}$, the last equation may be rewritten as a
trivial linear combination of the three polynomials $P_{n+1}$, $P_{n}$, and
$P_{n-1}$. Thus, we deduce

$h_{n+1}=d_{2n-1}\;,\quad z_{n+1}=-z_{n}-d_{2n-1}\beta_{n}+q+b\;,\quad
t_{n}=-d_{2n-1}\gamma_{n}\;.$

Therefore, $c_{n}=na\gamma_{n}+t_{n}=-d_{n-1}\gamma_{n}\neq 0$ (since $n\geq
1$). This completes the proof. ∎

######  Remark 3.2.

The parameters $\beta_{n}$ and $\gamma_{n}$ appearing in Theorem 3.1 may be
written explicitly in terms of the coefficients of $\phi$ and $\psi$ as
follows:

$\begin{array}[]{c}\beta_{n}=\displaystyle-\frac{(-2a+p)q+2bn[(n-1)a+p]}{(2na+p)[(2n-2)a+p]}\;,\\\\[12.50002pt]
\gamma_{n+1}=\displaystyle\frac{-(n+1)[(n-1)a+p][a(nb+q)^{2}-b(nb+q)(2na+p)+c(2na+p)^{2}]}{[(2n-1)a+p](2na+p)^{2}[2(n+1)a+p]}\;.\end{array}$

###  3.2 Classification and canonical representatives

We all always hear say: up to constant factors and affine changes of
variables, there are only four (parametric) families of classical OP, namely,
Hermite, Laguerre, Jacobi, and Bessel polynomials. But, what is the rigorous
meaning of this statement? The corresponding regular functionals will be
denoted by ${\bf u}_{H}$, ${\bf u}_{L}^{(\alpha)}$, ${\bf
u}_{J}^{(\alpha,\beta)}$, and ${\bf u}_{B}^{(\alpha)}$ (resp.) and these will
be called the canonical representatives (or canonical forms) of the classical
functionals. Their description is given in Table 1. Each one of these
functionals fulfils $(\ref{EDClassic1})$, being the corresponding pair
$(\phi,\psi)\equiv(\Phi,\Psi)$ given in the table. The regularity conditions
in the table are determined by conditions (ii) appearing in
$(\ref{P-regular1})$.

Class | ${\bf u}$ | $\Phi$ | $\Psi$ | regularity conditions
---|---|---|---|---
Hermite | ${\bf u}_{H}$ | $1$ | $-2x$ | ——
Laguerre | ${\bf u}_{L}^{(\alpha)}$ | $x$ | $-x+\alpha+1$ | $-\alpha\not\in\mathbb{N}$
Jacobi | ${\bf u}_{J}^{(\alpha,\beta)}$ | $1-x^{2}$ | $-(\alpha+\beta+2)x+\beta-\alpha$ | $-\alpha,-\beta,-(\alpha+\beta+1)\not\in\mathbb{N}$
Bessel | ${\bf u}_{B}^{(\alpha)}$ | $x^{2}$ | $(\alpha+2)x+2$ | $-(\alpha+1)\not\in\mathbb{N}$
Table 1: Classification and canonical forms of the classical functionals

Ultimately, denoting by $[{\bf u}]$ the equivalent class determined by a
functional ${\bf u}\in\mathcal{P}^{\prime}$ (see [11, Section 3.1.2.4, pp.
18-19]), and setting $\mathcal{P}^{\prime}_{C}=\\{{\bf
u}\in\mathcal{P}^{\prime}\,|\,\mbox{\rm${\bf u}$ is classical}\\}$, we will
show that

$\mathcal{P}^{\prime}_{C}/_{\sim}=\big{\\{}\,[{\bf u}]\,|\,{\bf
u}\in\mathcal{P}^{\prime}_{C}\big{\\}}=\big{\\{}[{\bf u}_{H}],[{\bf
u}_{L}^{(\alpha)}],[{\bf u}_{J}^{(\alpha,\beta)}],[{\bf
u}_{B}^{(\alpha)}]\,\big{\\}}\;,$

where the parameters $\alpha$ and $\beta$ vary on $\mathbb{C}$ subject to the
regularity conditions in Table 1, and $\sim$ is an equivalence relation in
$\mathcal{P}^{\prime}$ defined by

${\bf u}\sim{\bf v}\qquad\mbox{\rm iff}\qquad\exists
A\in\mathbb{C}\setminus\\{0\\}\;,\;\;\exists B\in\mathbb{C}\;:\;\;{\bf
v}=\big{(}\bm{h}_{A^{-1}}\circ\bm{\tau}_{-B}\big{)}{\bf u}\;.$ (3.28)

We start by proving a proposition that allow us to ensure that this
equivalence relation preserves the classical character of a given classical
functional.

######  Lemma 3.1.

Let ${\bf u},{\bf v}\in\mathcal{P}^{\prime}$ and suppose that ${\bf u}\sim{\bf
v}$, i.e., $(\ref{u-equiv-v1})$ holds. Suppose that there exist two
polynomials $\phi$ and $\psi$ such that

$D(\phi{\bf u})=\psi{\bf u}\;.$

Let $\Phi(x)=K\phi(Ax+B)$ and $\Psi(x)=KA\psi(Ax+B)$, being
$K\in\mathbb{C}\setminus\\{0\\}$. Then

$D(\Phi{\bf v})=\Psi{\bf v}\;.$

Moreover, if ${\bf u}$ is a classical functional, then so is ${\bf v}$.

###### Proof.

Since ${\bf u}$ and ${\bf v}$ fulfill (3.28), then

$\langle{\bf v},x^{n}\rangle=\Big{\langle}{\bf
u},\Big{(}\mbox{$\frac{x-B}{A}$}\Big{)}^{n}\,\Big{\rangle}\;,\quad
n\in\mathbb{N}_{0}\;.$

Therefore, for each $n\in\mathbb{N}_{0}$, we have

$\displaystyle\langle D(\Phi{\bf v}),x^{n}\rangle$
$\displaystyle=-n\langle{\bf v},\Phi(x)x^{n-1}\rangle=-n\big{\langle}{\bf
u},\big{(}\tau_{B}\circ
h_{A^{-1}}\big{)}\big{(}\Phi(x)x^{n-1}\big{)}\big{\rangle}$
$\displaystyle=-n\Big{\langle}{\bf
u},\Phi\Big{(}\mbox{$\frac{x-B}{A}$}\Big{)}\Big{(}\mbox{$\frac{x-B}{A}$}\Big{)}^{n-1}\Big{\rangle}$
$\displaystyle=-\Big{\langle}{\bf u},K\phi(x)\cdot A\frac{{\rm d}}{{\rm
d}x}\Big{\\{}\Big{(}\mbox{$\frac{x-B}{A}$}\Big{)}^{n}\Big{\\}}\Big{\rangle}$
$\displaystyle=KA\Big{\langle}D\big{(}\phi(x){\bf
u}\big{)},\Big{(}\mbox{$\frac{x-B}{A}$}\Big{)}^{n}\Big{\rangle}=KA\Big{\langle}\psi(x){\bf
u},\Big{(}\mbox{$\frac{x-B}{A}$}\Big{)}^{n}\Big{\rangle}$
$\displaystyle=\Big{\langle}{\bf
u},\Psi\Big{(}\mbox{$\frac{x-B}{A}$}\Big{)}\Big{(}\mbox{$\frac{x-B}{A}$}\Big{)}^{n}\Big{\rangle}=\big{\langle}{\bf
u},\big{(}\tau_{B}\circ
h_{A^{-1}}\big{)}\big{(}\Psi(x)x^{n}\big{)}\big{\rangle}$
$\displaystyle=\langle{\bf v},\Psi(x)x^{n}\rangle=\langle\Psi{\bf
v},x^{n}\rangle\;.$

Finally, the last sentence stated in the lemma follows by using Theorem 1.3. ∎

######  Theorem 3.2 (canonical representatives of the classical functionals).

Let ${\bf u}\in\mathcal{P}^{\prime}$ be a classical functional, so that ${\bf
u}$ fulfils

$D(\phi{\bf u})=\psi{\bf u}\;,$ (3.29)

where $\phi(x)=ax^{2}+bx+c$ and $\psi(x)=px+q$, subject to the regularity
conditions

$na+p\neq 0\;,\quad\phi\left(-\frac{nb+q}{2na+p}\right)\neq 0\;,\quad\forall
n\in\mathbb{N}_{0}\;.$ (3.30)

Then, there exists a regular functional ${\bf v}\in\mathcal{P}^{\prime}$ such
that

${\bf u}\sim{\bf v}\;,\quad D(\Phi{\bf v})=\Psi{\bf v}\;,$ (3.31)

where, for each classical functional determined by the pair $(\phi,\psi)$, the
corresponding pair $(\Phi,\Psi)$ is given by Table 1. More precisely, setting

$\Delta=b^{2}-4ac\;;\quad
d=\psi\left(-\mbox{$\frac{b}{2a}$}\right)\;\;\mbox{\rm if}\;\;a\neq 0\;,$

the following holds:

  * 1\. (Hermite) if $a=b=0$, then:

${\bf v}=\big{(}\bm{h}_{\sqrt{-p/(2c)}}\circ\bm{\tau}_{q/p}\big{)}{\bf u}={\bf
u}_{{}_{H}}\;;$

  * 2\. (Laguerre) if $a=0$ and $b\neq 0$, then:

${\bf v}=\big{(}\bm{h}_{-p/b}\circ\bm{\tau}_{c/b}\big{)}{\bf u}={\bf
u}_{{}_{L}}^{(\alpha)}\;,\quad\alpha=-1+(qb-pc)/b^{2}\;;$

  * 3\. (Bessel) if $a\neq 0$ and $\Delta=0$, then:

${\bf v}=\big{(}\bm{h}_{2a/d}\circ\bm{\tau}_{b/(2a)}\big{)}{\bf u}={\bf
u}_{{}_{B}}^{(\alpha)}\;,\quad\alpha=-2+p/a\;;$

  * 4\. (Jacobi) if $a\neq 0$ and $\Delta\neq 0$, then:

${\bf v}=\big{(}\bm{h}_{-2a/\sqrt{\Delta}}\circ\bm{\tau}_{b/(2a)}\big{)}{\bf
u}={\bf u}_{{}_{J}}^{(\alpha,\beta)}\;,$
$\alpha=-1+p/(2a)-d/\sqrt{\Delta}\;,\quad\beta=-1+p/(2a)+d/\sqrt{\Delta}\;.$

###### Proof.

Taking into account Lemma 3.1, the theorem will be proved if we are able to
show that, for each given pair $(\phi,\psi)$, and for each corresponding pair
$(\Phi,\Psi)$ given by Table 1—where the “corresponding pair” $(\Phi,\Psi)$ is
the one in the table such that $\phi$ and $\Phi$ have the same degree and
their zeros the same multiplicity—, there exist
$A,K\in\mathbb{C}\setminus\\{0\\}$ and $B\in\mathbb{C}$ such that the
relations

$\Phi(x)=K\phi(Ax+B)\;,\quad\Psi(x)=KA\psi(Ax+B)=KA^{2}px+KA(Bp+q)$ (3.32)

hold, for appropriate choices of the parameters $\alpha$ and $\beta$ appearing
in Table 1 for the Laguerre, Bessel, and Jacobi cases. Indeed, considering the
four possible cases determined by the polynomial $\phi$, we have:

1\. Assume $a=b=0$, i.e., $\phi(x)=c$. The regularity conditions (3.30) ensure
that $p\neq 0$ and $c\neq 0$. Therefore, since in this case we require
$(\Phi,\Psi)=(1,-2x)$, from (3.32) we obtain the equations

$1=Kc\;,\quad-2=KA^{2}p\;,\quad 0=Bp+q\;.$

A solution of this system of equations is

$K=1/c\;,\quad A=\sqrt{-2c/p}\;,\quad B=-q/p\;,$

which gives the desired result for the Hermite case, by Lemma 3.1.

2\. Assume $a=0$ and $b\neq 0$, so that $\phi(x)=bx+c$. Since in this case we
require $(\Phi,\Psi)=(x,-x+\alpha+1)$, from (3.32) we obtain

$1=KAb\;,\quad 0=bB+c\;,\quad-1=KA^{2}p\;,\quad\alpha+1=KA(Bp+q)\;.$

Solving this system we find

$K=-p/b^{2}\;,\quad B=-c/b\;,\quad A=-b/p\;,\quad\alpha=-1+(qb-pc)/b^{2}\;.$

Notice that, in this case,

$d_{n}=p\;,\quad\phi\left(-\frac{nb+q}{2na+p}\right)=-\frac{b^{2}}{p}\big{(}n+\alpha+1\big{)}\;,$

hence the regularity conditions (3.30) ensure that $p\neq 0$ (and so $K$ and
$A$ are well defined, being both nonzero complex numbers) and
$-\alpha\not\in\mathbb{N}$.

3\. Assume $a\neq 0$ and $\Delta=0$. Then
$\phi(x)=a\Big{(}x+\frac{b}{2a}\Big{)}^{2}$. In this case we require
$(\Phi,\Psi)=\big{(}x^{2},(\alpha+2)x+2\big{)}$, hence from (3.32) we obtain

$1=KA^{2}a\;,\quad 0=B+b/(2a)\;,\quad\alpha+2=KA^{2}p\;,\quad 2=KA(Bp+q)\;.$

Therefore, taking into account that
$d=\psi\left(-\mbox{$\frac{b}{2a}$}\right)=(2aq-pb)/(2a)$, we deduce

$K=4a/d^{2}\;,\quad B=-b/(2a)\;,\quad A=d/(2a)\;,\quad\alpha=-2+p/a\;.$

In this case we have

$d_{n}=a(n+\alpha+2)\;,\quad\phi\left(-\frac{nb+q}{2na+p}\right)=\frac{d^{2}}{a(2n+\alpha+2)^{2}}\;,$

hence conditions (3.30) ensure that $-(\alpha+1)\not\in\mathbb{N}$ and $d\neq
0$, and so, in particular, $K$ is well defined, being both $K$ and $A$ nonzero
complex numbers.

4\. Finally, assume $a\neq 0$ and $\Delta\neq 0$. Writing
$\phi(x)=a\Big{[}\Big{(}x+\frac{b}{2a}\Big{)}^{2}-\frac{\Delta}{4a^{2}}\Big{]}$,
since in this case we require
$(\Phi,\Psi)=\big{(}1-x^{2},-(\alpha+\beta+2)x+\beta-\alpha\big{)}$, from
(3.32) we obtain

$\begin{array}[]{c}-1=KA^{2}a\;,\quad 0=B+b/(2a)\;,\quad
1=Ka\Big{[}\Big{(}B+\frac{b}{2a}\Big{)}^{2}-\frac{\Delta}{4a^{2}}\Big{]}\;,\\\\[5.0pt]
-(\alpha+\beta+2)=KA^{2}p\;,\quad\beta-\alpha=KA(Bp+q)\;.\end{array}$

A solution of this system of five equations is

$\begin{array}[]{c}K=-4a/\Delta\;,\quad B=-b/(2a)\;,\quad
A=-\sqrt{\Delta}/(2a)\;,\\\\[2.5pt]
\alpha=-1+p/(2a)-d/\sqrt{\Delta}\;,\quad\beta=-1+p/(2a)+d/\sqrt{\Delta}\;.\end{array}$

(We choose $A$ with the minus sign since whenever $(\phi,\psi)=(\Phi,\Psi)$
that choice implies $A=1$ and $B=0$, hence ${\bf u}={\bf v}={\bf
u}_{J}^{(\alpha,\beta)}$, and so it is a more natural choice.) Adding and
subtracting the last two equations for $\alpha$ and $\beta$, we find
$\alpha+\beta+2=p/a$ and $\alpha-\beta=-2d/\sqrt{\Delta}$, hence we deduce

$d_{n}=a(n+\alpha+\beta+2)\;,\quad\phi\left(-\frac{nb+q}{2na+p}\right)=-\frac{\Delta}{a}\frac{(n+\alpha+1)(n+\beta+1)}{(2n+\alpha+\beta+2)^{2}}\;,$

Therefore, conditions (3.30) ensure that $-(\alpha+\beta+1)\not\in\mathbb{N}$,
$-\alpha\not\in\mathbb{N}$, and $-\beta\not\in\mathbb{N}$. This completes the
proof. ∎

######  Remark 3.3.

It follows from the proof of Theorem 3.2 that the parameters $\alpha$ and
$\beta$ defined in the statement of this theorem (in cases 2, 3, and 4) fulfil
the regularity conditions appearing in Table 1.

The preceding theorem allows us to classify each classical functional
according with the degree of the polynomial $\phi$ appearing in equation
$(\ref{EDClassic1})$.

######  Corollary 3.1.

Let ${\bf u}$ be a classical functional, fulfilling
$(\ref{EDClassic1})$–$(\ref{grauPhiPsi})$.

  1. (i)

if $\deg\phi=0$ (hence $\phi$ is a nonzero constant), then ${\bf u}\sim{\bf
u}_{H}\,$;

  2. (ii)

if $\deg\phi=1$, then ${\bf u}\sim{\bf u}_{L}^{(\alpha)}\,$ for some $\alpha$;

  3. (iii)

if $\deg\phi=2$ and $\phi$ has simple zeros, then ${\bf u}\sim{\bf
u}_{J}^{(\alpha,\beta)}\,$ for some pair $(\alpha,\beta)$;

  4. (iv)

if $\deg\phi=2$ and $\phi$ has a double zero, then ${\bf u}\sim{\bf
u}_{B}^{(\alpha)}\,$ for some $\alpha$.

The monic OP with respect to the canonical representatives ${\bf u}_{H}$,
${\bf u}_{L}^{(\alpha)}$, ${\bf u}_{J}^{(\alpha,\beta)}$, and ${\bf
u}_{B}^{(\alpha)}$ will be denoted by $\\{\widehat{H}_{n}\\}$,
$\\{\widehat{L}_{n}^{(\alpha)}\\}$, $\\{\widehat{P}_{n}^{(\alpha,\beta)}\\}$,
and $\\{\widehat{B}_{n}^{(\alpha)}\\}$ (resp.), and they will be called the
(monic) Hermite, Laguerre, Jacobi, and Bessel polynomials. Table 2 summarizes
the corresponding parameters appearing in all characterizations presented in
Theorem 3.1. In view of Theorem 3.2 and Theorem 1.3, we may now justify a
sentence made at the beginning of the section regarding Hermite, Laguerre,
Jacobi, and Bessel polynomials. As Maroni said in an interview when asked
about Bessel polynomials: “comme dans le roman d’Alexandre Dumas, les trois
mousquetaires étaient quatre en réalité”.

| $\widehat{H}_{n}$ | $\widehat{L}_{n}^{(\alpha)}$ | $\widehat{P}_{n}^{(\alpha,\beta)}$ | $\widehat{B}_{n}^{(\alpha)}$
---|---|---|---|---
$\lambda_{n}$ | ${\scriptstyle{2n}}$ | ${\scriptstyle{n}}$ | ${\scriptstyle{n(n+\alpha+\beta+1)}}$ | ${\scriptstyle{-n(n+\alpha+1)}}$
$\beta_{n}$ | ${\scriptstyle{0}}$ | ${\scriptstyle{2n+\alpha+1}}$ | $\frac{\beta^{2}-\alpha^{2}}{(2n+\alpha+\beta)(2n+2+\alpha+\beta)}$ | $\frac{-2\alpha}{(2n+\alpha)(2n+2+\alpha)}$
$\gamma_{n}$ | $\frac{n}{2}$ | ${\scriptstyle{n(n+\alpha)}}$ | $\frac{4n(n+\alpha)(n+\beta)(n+\alpha+\beta)}{(2n+\alpha+\beta-1)(2n+\alpha+\beta)^{2}(2n+\alpha+\beta+1)}$ | $\frac{-4n(n+\alpha)}{(2n+\alpha-1)(2n+\alpha)^{2}(2n+\alpha+1)}$
$a_{n}$ | ${\scriptstyle{0}}$ | ${\scriptstyle{0}}$ | ${\scriptstyle{-n}}$ | ${\scriptstyle{n}}$
$b_{n}$ | ${\scriptstyle{0}}$ | ${\scriptstyle{n}}$ | $\frac{2(\alpha-\beta)n(n+\alpha+\beta+1)}{(2n+\alpha+\beta)(2n+2+\alpha+\beta)}$ | $\frac{-4n(n+\alpha+1)}{(2n+\alpha)(2n+2+\alpha)}$
$c_{n}$ | ${\scriptstyle{n}}$ | ${\scriptstyle{n(n+\alpha)}}$ | $\frac{4n(n+\alpha)(n+\beta)(n+\alpha+\beta)(n+\alpha+\beta+1)}{(2n+\alpha+\beta-1)(2n+\alpha+\beta)^{2}(2n+\alpha+\beta+1)}$ | $\frac{4n(n+\alpha)(n+\alpha+1)}{(2n+\alpha-1)(2n+\alpha)^{2}(2n+\alpha+1)}$
$r_{n}^{[1]}$ | ${\scriptstyle{0}}$ | ${\scriptstyle{n}}$ | $\frac{2(\alpha-\beta)n}{(2n+\alpha+\beta)(2n+2+\alpha+\beta)}$ | $\frac{4n}{(2n+\alpha)(2n+2+\alpha)}$
$s_{n}^{[1]}$ | ${\scriptstyle{0}}$ | ${\scriptstyle{0}}$ | $\frac{-4(n-1)n(n+\alpha)(n+\beta)}{(2n+\alpha+\beta-1)(2n+\alpha+\beta)^{2}(2n+\alpha+\beta+1)}$ | $\frac{4(n-1)n}{(2n+\alpha-1)(2n+\alpha)^{2}(2n+\alpha+1)}$
$h_{n}$ | ${\scriptstyle{-2}}$ | ${\scriptstyle{-1}}$ | ${\scriptstyle{-(2n+\alpha+\beta-1)}}$ | ${\scriptstyle{2n+\alpha-1}}$
$t_{n}$ | ${\scriptstyle{n}}$ | ${\scriptstyle{n(n+\alpha)}}$ | $\frac{4n(n+\alpha)(n+\beta)(n+\alpha+\beta)}{(2n+\alpha+\beta-1)(2n+\alpha+\beta)^{2}}$ | $\frac{4n(n+\alpha)}{(2n+\alpha-1)(2n+\alpha)^{2}}$
$k_{n}$ | $\frac{(-1)^{n}}{2^{n}}$ | ${\scriptstyle{(-1)^{n}}}$ | $\frac{(-1)^{n}}{(n+\alpha+\beta+1)_{n}}$ | $\frac{1}{(n+\alpha+1)_{n}}$
Table 2: Parameters for the classical monic OP appearing in Theorem 3.1 with
respect to the canonical forms given in Table 1.

## Acknowledgements

This work is supported by the Centre for Mathematics of the University of
Coimbra-UIDB/00324/2020, funded by the Portuguese Government through FCT/
MCTES.

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