#### Abstract

In this paper, we apply to (almost) all the “named”
polynomials of the Askey scheme, as defined by their standard
three-term recursion relations, the machinery developed in
previous papers. For each of these polynomials we identify at
least one additional recursion relation involving a shift in some
of the parameters they feature, and for several of these
polynomials characterized by special values of their parameters,
factorizations are identified yielding some or all of their
zeros—generally given by simple expressions in terms of
*integers* (*Diophantine*
relations). The factorization findings generally are applicable
for values of the Askey polynomials that extend beyond those for
which the standard orthogonality relations hold. Most of these
results are not (yet) reported in the standard compilations.

#### 1. Introduction

Recently *Diophantine* findings and conjectures
concerning the eigenvalues of certain tridiagonal matrices, and correspondingly
the zeros of the polynomials associated with their secular equations, were arrived at via the
study of the behavior of certain *isochronous* many-body problems of Toda
type in the neighborhood of their equilibria [1, 2] (for a review of these and
other analogous results, see [3, Appendix C]). To prove (some of) these conjectures a theoretical
framework was then developed [4–6], involving polynomials defined by three-term recursion
relations—hence being, at least for appropriate ranges of the parameters they
feature, orthogonal. (This result is generally referred to as “Favard
theorem,” on the basis of [7]; however, as noted by Ismail, a more
appropriate name is “spectral theorem for orthogonal polynomials”
[8]). Specific
conditions were identified—to be satisfied by the coefficients, featuring a
parameter of these recursion relations—sufficient to
guarantee that the corresponding polynomials also satisfy a *second* three-term recursion relation involving shifts in that parameter ;
and via this *second* recursion relation, *Diophantine* results of
the kind indicated above were obtained [5]. In Section 2, in order to make this paper essentially
self-contained, these developments are tersely reviewed—and also marginally
extended, with the corresponding proofs relegated to an appendix to avoid
interrupting the flow of the presentation. We then apply, in Section 3, this
theoretical machinery to the “named” polynomials of the Askey scheme
[9], as defined by
the basic three-term recursion relation they satisfy: this entails the identification
of the parameter —which can often be done in more than one
way, especially for the named polynomials involving several parameters—and
yields the identification of additional recursion relations satisfied by (most
of) these polynomials. Presumably such results (especially *after* they
have been discovered) could also be obtained by other routes—for instance, by
exploiting the relations of these polynomials with hypergeometric functions: we
did not find them (except in some very classical cases) in the standard
compilations [9–13], where they in our opinion deserve to be eventually
recorded. Moreover, our machinery yields factorizations of certain of these
polynomials entailing the identification of some or all of their zeros, as well
as factorizations relating some of these polynomials (with different
parameters) to each other. Again, most of these results seem *new* and
deserving to be eventually recorded in the standard
compilations although they generally require that the parameters of the named
polynomials do *not* satisfy the standard restrictions required for the
orthogonality property. To clarify this restriction let us remark that an
elementary example of such factorizations—which might be considered the *prototype* of formulas reported below for many of the polynomials of the Askey
scheme—reads as follows:
where is the standard (generalized) Laguerre
polynomial of order ,
for whose orthogonality,it is, however, generally
required that This formula, (1.1a), is well known and it is
indeed displayed in some of the standard compilations reporting results for
classical orthogonal polynomials (see, e.g., page 109 of the classical book by
Magnus and Oberhettinger [14] or [11, Equation 8.973.4]). And this remark applies as well to the following
neat generalization of this formula, reading
which qualifies as
well as the *prototype* of formulas reported
below for many of the polynomials of the Askey scheme. (Note,
incidentally, that this formula can be inserted without difficulty in the
standard orthogonality relation for generalized Laguerre polynomials, (1.1b),
reproducing the standard relation: the singularity of the weight function gets
indeed neatly compensated by the term appearing in the right-hand side of (1.1c).
Presumably, this property—and the analogous version for Jacobi
polynomials—is well known to most experts on orthogonal polynomials; e.g., a referee of this paper wrote “Although I have known of (1.1c) for
a long time, I have neither written it down nor saw it stated explicitly. It is
clear from reading [15, Paragraph 6.72] that Szëgo was aware of (1.1c) and the more general case
of Jacobi polynomials.”) Most of the formulas (analogous to (1.1c) and (1.1a))
for the named polynomials of the Askey scheme that are reported below
are instead, to the best of our knowledge, *new*: they do not appear in
the standard compilations where we suggest they should be eventually recorded,
in view of their neatness and their *Diophantine* character. They could of
course be as well obtained by other routes than those we followed to identify
and prove them (it is indeed generally the case that formulas involving special
functions, *after* they have been discovered, are easily proven via
several different routes). Let us however emphasize that although the results
reported below have been obtained by a rather systematic application
of our approach to all the polynomials of the Askey scheme, we do not claim
that the results reported exhaust all those of this kind featured by these
polynomials. And let us also note that, as it is generally done in the standard
treatments of “named” polynomials [9–13], we have treated *separately* each of the
differently “named” classes of these polynomials, even though
“in principle” it would be sufficient to only treat the most general
class of them—Wilson polynomials—that encompasses all the other classes
via appropriate assignments (including limiting ones) of the 4 parameters it
features. Section 4 mentions tersely
possible future developments.

#### 2. Preliminaries and Notation

In this section we report tersely the key points of our approach, mainly in order to make this paper self-contained—as indicated above—and also to establish its notation: previously known results are of course reported without their proofs, except for an extension of these findings whose proof is relegated to Appendix A.

Hereafter we consider classes of monic polynomials , of degree in their argument and depending on a parameter , defined by the three-term recursion relation: with the “initial” assignmentsclearly entailing and so on. (In some cases the left-hand side of the first (2.1b) might preferably be replaced by , to take account of possible indeterminacies of .)

*Notation. *Here and hereafter the index is a *nonnegative integer* (but some of
the formulas written below might make little sense for ,
requiring a—generally quite obvious—special interpretation), and are functions of this index and of the parameter .
They might—indeed they often do—also depend on other parameters besides (see below); but this parameter plays a crucial role, indeed the results
reported below emerge from the identification of special values of it
(generally simply related to the index ).

Let us recall that the theorem which guarantees that
these polynomials, being defined by the three-term recursion relation
(2.1), are
orthogonal (with a positive definite, albeit a priori unknown, weight
function), requires that the coefficients and be *real* and that the latter be *negative*, (see, e.g., [16]).

##### 2.1. Additional Recursion Relation

Proposition 2.1. * If the quantities and satisfy the nonlinear recursion relation
** with the boundary condition **
(where, without significant loss
of generality, this constant is set to zero rather than to an arbitrary -independent value :
see [5, Equation (4a)]; and we also replaced, for notational convenience, the quantity previously used [5] with ), and if the coefficients and are defined in terms of these quantities by the following formulas:
**
then the polynomials identified by the recursion relation
(2.1) satisfy the following additional recursion relation (involving a shift
both in the order of the polynomials and in the parameter ):
*

*with*

This proposition corresponds to [5, Proposition 2.3]. (As suggested by a referee, let us also mention that recursions in a parameter—albeit of a very special type and different from that reported above—were also presented long ago in a paper by Dickinson et al. [17].)

Alternative conditions sufficient for the validity of Proposition 2.1 and characterizing directly the coefficients , and read as follows (see [5, Appendix B]): withand the “initial” conditionentailing via (2.5c) (with ) and via (2.5a) (with )

Proposition 2.2. *Assume that the class of (monic, orthogonal) polynomials defined by the recursion (2.1) satisfies Proposition
2.1, hence that they also obey the (“second”) recursion
relation (2.4). Then, there also holds the relations:
**
in addition to
**
as well as
*

These findings correspond to [6, Proposition 1].

##### 2.2. Factorizations

In the following we introduce a second parameter but for notational simplicity we do *not* emphasize *explicitly* the dependence of the various quantities on this
parameter.

Proposition 2.3. * If the (monic, orthogonal) polynomials are defined by the recursion relation
(2.1) and
the coefficients satisfy the relation **entailing that for the recursion relation (2.1a)
reads **then there holds the factorization*

*with the “complementary” polynomials (of course of degree ) defined by the following three-term recursion relation analogous (but not identical) to (2.1):*

*entailing*

*with*

*and so on.*

This is a slight generalization (proven below, in Appendix A) of [5, Proposition 2.4]. Note incidentally that also the complementary polynomials being defined by three-terms recursion relations, see (2.12a), may belong to orthogonal families, hence they should have to be eventually investigated in such a context, perhaps applying also to them the kind of findings reported in this paper.

The following two results are immediate consequences of Proposition 2.3.

Corollary 2.4. * If (2.9) holds—entailing (2.10)
and (2.11) with (2.12)—the polynomial has the zero **and the polynomial has the two zeros (see (2.12e)),
**
The first of these results is a
trivial consequence of (2.10); the second is evident from (2.11) and (2.12d). Note,
moreover, that from the factorization formula (2.11), one can likewise find explicitly 3 zeros of and 4 zeros of by evaluation from (2.12) and and by taking advantage of the explicit solvability of algebraic equations of degrees 3 and 4.*

These findings often have a *Diophantine* connotation, due to the neat expressions of the zeros and in terms of *integers*.

Corollary 2.5. * If (2.9) holds—entailing (2.10) and (2.11) with (2.12)—and moreover the
quantities and satisfy the properties*

*then clearly*

*entailing that the*

*factorization*(2.11) takes the neat form*Note that—for future convenience, see below—one has emphasized explicitly the possibility that the polynomials depend on additional parameters (indicated with the vector variables , resp., ; these additional parameters must of course be independent of , but they might depend on ).*

The following remark is relevant when both Propositions 2.1 and 2.2 hold.

*Remark 2.6. * As implied by (2.3b), the condition (2.9) can be enforced via the
assignmententailing that the nonlinear recursion
relation (2.3a) reads

Corollaries 2.4 and 2.5 and Remark 2.6 are analogous to [5, Corollaries 2.5 and 2.6 and Remark 2.7].

##### 2.3. Complete Factorizations and Diophantine Findings

The *Diophantine* character of the findings
reported below is due to the generally neat expressions of the
following zeros in terms of *integers* (see in particular the examples in
Section 3).

Proposition 2.7. * If the (monic, orthogonal) polynomials are defined by the three-term recursion
relations (2.1) with coefficients and satisfying the requirements sufficient for the
validity of both Propositions 2.1 and 2.2 (namely (2.3), with (2.2) and (2.9), or just with (2.18)), then
**with the expressions (2.6b) of the
zeros and the standard convention according to which
a product equals unity when its lower limit exceeds its upper limit. Note that
these zeros are -independent (except for their number). In particular,
**
and so on.*

These findings correspond to [6, Proposition 2.2 (first part)].

The following results are immediate consequences of Proposition 2.7 and of Corollary 2.4.

Corollary 2.8. * If Proposition 2.7 holds, then also the polynomials and (in addition to ,
see (2.19)) can be written in the following completely factorized form (see (2.6b)
and (2.12e)):
**Analogously, complete factorizations can clearly be
written for the polynomials and ,
see the last part of Corollary 2.4. **And of course the factorization (2.11) together
with (2.19a) entails the (generally Diophantine) finding that the
polynomial with features the zeros , see (2.6b): *

Proposition 2.9. * Assume that, for the class of polynomials ,
there hold the preceding Proposition 2.1, and moreover that, for some
value of the parameter (and of course for all nonnegative integer values of ), the coefficients vanish (see (2.7a) and (2.7c)),
*

*then the polynomials factorize as follows:*

*entailing*

*and so on.*

*Likewise, if for all*

*nonnegative**integer*values of the following*two*properties hold (see (2.8a), (2.8c), and (2.8d)):*then the polynomials factorize as follows:*

*entailing*

*and so on.*

*Here of course the -independent!) zeros , respectively, are defined by (2.7b), respectively, (2.8b).*

These findings correspond to [6, Proposition 2].

#### 3. Results for the Polynomials of the Askey Scheme

In this section, we apply to the polynomials of the
Askey scheme [9] the
results reviewed in the previous section. This class of polynomials (including
the classical polynomials) may be introduced in various manners: via generating
functions, Rodriguez-type formulas, their connections with hypergeometric
formulas, and so forth. In order to apply our machinery, as outlined in the
preceding section, we introduce them via the three-term recursion relation they
satisfy:
with the “initial”
assignments
clearly entailing
and so on. Here the components
of the vector denote the *additional parameters* generally featured by these polynomials.

Let us emphasize that in this manner we introduced the *monic* (or “normalized” [9]) version of these polynomials; below we
generally also report the relation of this version to the more standard version
[9].

To apply our machinery we must identify, among the parameters characterizing these polynomials, the single parameter playing a special role in our approach. This can be generally done in several ways (even for the same class of polynomials, see below). Once this identification (i.e., the assignment ) has been made, the recursion relations (3.1) coincide with the relations (2.1) via the self-evident notational identification:

Before proceeding with the report of our results, let
us also emphasize that when the polynomials considered below feature symmetries
regarding the dependence on their parameters—for instance, they are invariant
under exchanges of some of them—obviously *all* the properties of these
polynomials reported below can be duplicated via such symmetry
properties; but it would be a waste of space to report explicitly the
corresponding formulas, hence such duplications are hereafter omitted (except
that sometimes results arrived at by different routes
can be recognized as trivially related via such symmetries: when this happens
this fact is explicitly noted). We will use systematically the notation of
[9]—up to obvious
changes made whenever necessary in order to avoid interferences with our
previous notation. When we obtain a result that we deem interesting but is not reported
in the standard compilations [9–13], we identify it as *new* (although given the
very large literature on orthogonal polynomials, we cannot be certain that such
a result has not been already published; indeed we will be grateful to any
reader who were to discover that this is indeed the case and will let us know).
And let us reiterate that even though we performed an extensive search
for such results, this investigation cannot be considered
“exhaustive”: additional results might perhaps be discovered via
assignments of the -dependence different from those considered below.

##### 3.1. Wilson

The monic Wilson polynomials (see [9], and note the notational replacement of the 4 parameters used there with ) are defined by the three-term recursion relations (3.1) with where

The standard version of these polynomials reads (see [9]):

Let us also recall that these polynomials are invariant under any permutation of the 4 parameters .

As for the identification of the parameter (see (3.2)), two possibilities are listed in the following subsections.

###### 3.1.1. First Assignment

With this assignment, one can set, consistently with
our previous treatment,
implying, via (2.2), (2.3), that
the polynomials defined by the three-term recurrence relations
(2.1) coincide with the normalized Wilson polynomials (3.3):Hence, with this identification, Proposition
2.1 becomes applicable, entailing (*new* finding) that these
normalized Wilson polynomials satisfy the second recursion relation (2.4a)
withNote that this finding is
obtained without requiring any limitation on the 4 parameters of the Wilson
polynomials .

It is, moreover, plain that with the assignmentthe factorizations implied by Proposition
2.3, and the properties implied by Corollary 2.4, become applicable
with These are *new* findings. As for the
additional findings entailed by Corollary 2.5, they are reported in
Section 3.1.3. And Proposition 2.7 becomes as well applicable,
entailing (*new* finding) the *Diophantine* factorizationwhile Corollary 2.8 entails even more general properties, such as (*new* finding)

*Remark 3.1. * A look at the formulas (3.3) suggests other possible assignments of the parameter satisfying (2.9), such as ,
namely, .
However, these assignments actually fail to satisfy (2.9) for *all* values
of ,
because for this to happen, it is *not* sufficient that the numerator in
the expression of vanish, it is, moreover, required that the
denominator in that expression *never* vanish. In the following, we will
consider only assignments of the parameter in terms of that satisfy these requirements.

###### 3.1.2. Second Assignment

With this assignment, one can set, consistently with
our previous treatment,
implying, via (2.2), (2.3), that
the polynomials defined by the three-term recurrence relations
(2.1) coincide with the normalized Wilson polynomials (3.3):Hence, with this identification, Proposition
2.1 becomes applicable, entailing (*new* finding) that these
normalized Wilson polynomials satisfy the second recursion relation (2.4a)
with Note that this assignment entails
now the (single) restriction on the 4 parameters of the Wilson polynomials .

It is, moreover, plain that with the assignments
respectively,
the factorizations implied by Proposition
2.3 and the properties implied by Corollary 2.4 become applicable
with ,
respectively, .
These are *new* findings. As for the additional findings entailed by Corollary
2.5, they are reported in Section 3.1.3. And Proposition 2.7 becomes as well applicable, entailing the *Diophantine* factorizations
respectively,
(A referee pointed out that (3.17a) is not new, as one can evaluate
explicitly when which is indeed the case in (3.17a); and,
moreover, that the two formulas (3.17b) and (3.17c) coincide, since their left-hand
sides are identical as a consequence of the symmetry property of Wilson
polynomials under the transformation .)

And Corollary 2.8 entails even more
general properties, such as (*new* finding)
respectively,

Moreover, with the assignments
respectively,
Proposition 2.9 becomes
applicable, entailing (*new* findings) the *Diophantine* factorizations
respectively,
obviously implying the relation

###### 3.1.3. Factorizations

The following *new* relations among monic Wilson
polynomials are implied by Proposition 2.3 with Corollary 2.5:
Note that the polynomials
appearing as second factors in the right-hand side of these formulas are
completely factorizable, see (3.10) and (3.17b) (we will not repeat this remark in
the case of analogous formulas below).

##### 3.2. Racah

The monic Racah polynomials (see [9])
are defined by the three-term
recursion relations (3.1) with
where
The standard version of these
polynomials reads (see [9])
Note, however, that in the
following we do *not* require the parameters of these polynomials to
satisfy one of the restrictions ,
or with a positive integer and whose validity is instead required for the
standard Racah polynomials [9].

Let us recall that these polynomials are invariant under various reshufflings of their parameters:

Let us now identify the parameter as follows (see (3.2)):

With this assignment, one can set, consistently with
our previous treatment,
implying, via (2.2), (2.3), that
the polynomials defined by the three-term recurrence relations
(2.1) coincide with the normalized Racah polynomials (3.22):Hence, with this identification, Proposition
2.1 becomes applicable, entailing (*new* finding) that these
normalized Racah polynomials satisfy the second recursion relation (2.4a)
withNote that this finding is
obtained without requiring any limitation on the 4 parameters of the Racah
polynomials .

It is, moreover, plain that with the assignments
respectively,
the factorizations implied by Proposition
2.3 and the properties implied by Corollary 2.4 become applicable
with respectively, .
These are *new* findings. As for the additional findings entailed by Corollary
2.5, they are reported in Section 3.2.1. And Proposition 2.7 becomes as well applicable, entailing (*new* findings) the *Diophantine* factorizations
respectively,
And Corollary 2.8 entails even more general properties, such as (*new* findings)
respectively,

###### 3.2.1. Factorizations

The following *new* relations among Racah
polynomials are implied by Proposition 2.3 with Corollary 2.5:

##### 3.3. Continuous Dual Hahn (CDH)

In this section (some results of which were already reported in [5]) we focus on the monic continuous dual Hahn (CDH) polynomials (see [9], and note the notational replacement of the 3 parameters used there with ), defined by the three-term recursion relations (3.1) with The standard version of these polynomials reads (see [9])

Let us recall that these polynomials are invariant under any permutation of the three parameters .

Let us now proceed and provide two identifications of the parameter see (3.2).

###### 3.3.1. First Assignment

With this assignment, one can set, consistently with
our previous treatment,
implying, via (2.2), (2.3), that
the polynomials defined by the three-term recurrence relations
(2.1) coincide with the normalized CDH polynomials (3.32):Hence, with this identification, Proposition
2.1 becomes applicable, entailing (*new* finding) that these
normalized CDH polynomials satisfy the second recursion relation (2.4a)
withNote that this finding is
obtained without requiring any limitation on the 3 parameters of the CDH
polynomials .

It is, moreover, plain that with the
assignmentthe factorizations implied by Proposition
2.3 and the properties implied by Corollary 2.4 become applicable
with .
These are *new* findings. As for the additional findings entailed by Corollary
2.5, they are reported in Section 3.3.3. And Proposition 2.7 becomes as well applicable, entailing (*new* findings) the *Diophantine* factorizationAnd Corollary 2.8 entails even more general properties, such as (*new* finding)

Likewise, with the assignment Proposition 2.9 becomes
applicable, entailing (*new* finding) the *Diophantine* factorization

###### 3.3.2. Second Assignment

where is an a priori arbitrary parameter.

With this assignment, one can set, consistently with
our previous treatment,
implying, via (2.2), (2.3), that
the polynomials defined by the three-term recurrence relations
(2.1) coincide with the normalized CDH polynomials (3.32):Hence, with this identification, Proposition
2.1 becomes applicable, entailing (*new* finding) that these
normalized CDH polynomials satisfy the second recursion relation (2.4a)
withNote that this assignment entails
the (single) limitation on the parameters of the CDH polynomials.

It is, moreover, plain that with the
assignmentthe factorizations implied by Proposition
2.3 and the properties implied by Corollary 2.4 become applicable
with .
These are *new* findings. As for the additional findings entailed by Corollary
2.5, they are reported in Section 3.3.3. And Proposition 2.7 becomes as well applicable, entailing (*new* findings) the *Diophantine* factorizationAnd Corollary 2.8 entails even more general properties, such as (*new* finding)

Likewise with the assignments
respectively,
Proposition 2.9 becomes
applicable, entailing (*new* findings) the *Diophantine* factorizations
respectively,

Note that the right-hand sides of the last two
formulas coincide; this implies (*new* finding) that the left-hand sides
coincide as well.

###### 3.3.3. Factorizations

The following *new* relations among continuous
dual Hahn polynomials are implied by Proposition 2.3 with Corollary
2.5:

##### 3.4. Continuous Hahn (CH)

The monic continuous Hahn (CDH) polynomials (see [9], and note the notational replacement of the 4 parameters used there with ), are defined by the three-term recursion relations (3.1) with where The standard version of these polynomials reads (see [9])

Let us recall that these polynomials are symmetrical under the exchange of the first two and last two parameters:

Let us now proceed and provide two identifications of the parameter see (3.2).

###### 3.4.1. First Assignment

With this assignment, one can set, consistently with
our previous treatment,
implying, via (2.2), (2.3), that
the polynomials defined by the three-term recurrence relations
(2.1) coincide with the normalized CH polynomials (3.53):Hence, with this identification, Proposition
2.1 becomes applicable, entailing (*new* finding) that these
normalized CH polynomials satisfy the second recursion relation (2.4a)
withNote that this assignment entails
no restriction on the 4 parameters of the CH polynomials .

It is, moreover, plain that with the
assignmentthe factorizations implied by Proposition
2.3 and the properties implied by Corollary 2.4 become applicable
with .
These are *new* findings. And Proposition 2.7 becomes as well
applicable, entailing (*new* findings) the *Diophantine* factorizationAnd Corollary 2.8 entails even more general properties, such as (*new* findings)

###### 3.4.2. Second Assignment

Analogous results also obtain from the assignment

With this assignment, one can set, consistently with
our previous treatment,
implying, via (2.2), (2.3), that
the polynomials defined by the three-term recurrence relations
(2.1) coincide with the normalized CH polynomials (3.53):Hence, with this identification, Proposition
2.1 becomes applicable, entailing (*new* finding) that these
normalized CH polynomials satisfy the second recursion relation (2.4a)
withNote that this assignment entails
no restriction on the 4 parameters of the CH polynomials .

It is, moreover, plain that with the
assignmentthe factorizations implied by Proposition
2.3, and the properties implied by Corollary 2.4, become applicable
with .
These are *new* findings. And Proposition 2.7 becomes as well
applicable, entailing (*new* finding) the *Diophantine* factorizationAnd Corollary 2.8 entails even more general properties, such as (*new* finding)

##### 3.5. Hahn

In this subsection, we introduce a somewhat
generalized version of the standard (monic) Hahn polynomials. These
(generalized) monic Hahn polynomials (see [9], and note the replacement of the integer parameter with the arbitrary parameter :
hence the standard Hahn polynomials are only obtained for with a *positive integer* and ),
are defined by the three-term
recursion relations (3.1) with
where
The standard version of these
polynomials reads (see [9])

Let us now proceed and provide three identifications of the parameter see (3.2).

###### 3.5.1. First Assignment

With this assignment, one can set, consistently with
our previous treatment,
implying, via (2.2), (2.3), that
the polynomials defined by the three-term recurrence relations
(2.1) coincide with the normalized Hahn polynomials (3.69):Hence, with this identification, Proposition
2.1 becomes applicable, entailing (*new* finding) that these
normalized Hahn polynomials satisfy the second recursion relation (2.4a)
withNote that this assignment entails
no restriction on the 3 parameters of the Hahn polynomials .

It is, moreover, plain that with the assignments
respectively,
the factorizations implied by Proposition
2.3 and the properties implied by Corollary 2.4 become applicable
with .
These are *new* findings. And Proposition 2.7 becomes as
well applicable, entailing (*new* findings) the *Diophantine* factorizations
respectively,
And Corollary 2.8 entails even more general properties, such as (*new* findings)
respectively,

###### 3.5.2. Second Assignment

where is an arbitrary parameter.

With this assignment, one can set, consistently with
our previous treatment,
implying, via (2.2), (2.3), that
the polynomials defined by the three-term recurrence relations
(2.1) coincide with the normalized Hahn polynomials (3.69):Hence, with this identification, Proposition
2.1 becomes applicable, entailing (*new* finding) that these
normalized Hahn polynomials satisfy the second recursion relation (2.4a)
withNote that this assignment entails
no restriction on the 3 parameters of the Hahn polynomials .

It is, moreover, plain that with the assignments
respectively,
the factorizations implied by Proposition
2.3 and the properties implied by Corollary 2.4 become applicable
with ,
respectively, .
These are *new* findings. And Proposition 2.7 becomes as well
applicable, entailing (*new* findings) the *Diophantine* factorizations
respectively,
And Corollary 2.8 entails even more general properties, such as (*new* findings)
respectively,

###### 3.5.3. Third Assignment

where is an arbitrary parameter.

With this assignment, one can set, consistently with
our previous treatment,
implying, via (2.2), (2.3), that
the polynomials defined by the three-term recurrence relations
(2.1) coincide with the normalized Hahn polynomials (3.69):Hence, with this identification, Proposition
2.1 becomes applicable, entailing (*new* finding) that these
normalized Hahn polynomials satisfy the second recursion relation (2.4a)
withNote that this assignment entails
no restriction on the 4 parameters of the Hahn polynomials .

It is, moreover, plain that with the
assignmentthe factorizations implied by Proposition
2.3 and the properties implied by Corollary 2.4 become applicable
with .
These are *new* findings. And Proposition 2.7 becomes as well
applicable, entailing (*new* finding) the *Diophantine* factorizationAnd Corollary 2.8 entails even more general properties, such as (*new* finding)

##### 3.6. Dual Hahn

In this subsection, we introduce a somewhat
generalized version of the standard (monic) dual Hahn polynomials. These (generalized)
monic dual Hahn polynomials (see [9], and note the replacement of the integer parameter with the arbitrary parameter :
hence the standard Hahn polynomials are only obtained for with a *positive integer* and ),
are defined by the three-term
recursion relations (3.1) with
where
The standard version of these
polynomials reads (see [9])

Let us now proceed and provide two identifications of the parameter .

###### 3.6.1. First Assignment

With this assignment, one can set, consistently with
our previous treatment,
implying, via (2.2), (2.3), that
the polynomials defined by the three-term recurrence relations
(2.1) coincide with the normalized dual Hahn polynomials (3.92):Hence, with this identification, Proposition
2.1 becomes applicable, entailing (*new* finding) that these
normalized dual Hahn polynomials satisfy the second recursion relation (2.4a) withNote that this assignment entails
no restriction on the 3 parameters of the dual Hahn polynomials .

It is, moreover, plain that with the assignments
(which is, however, incompatible
with the requirement characterizing the *standard* dual Hahn polynomials: with a *positive integer* and ), respectively,
the factorizations implied by Proposition
2.3 and the properties implied by Corollary 2.4 become applicable
with ,
respectively, .
These are *new* findings. As for the additional findings entailed by Corollary
2.5, they are reported in Section 3.6.3. And Proposition 2.7 becomes as well applicable, entailing (*new* findings) the *Diophantine* factorizations
respectively,
And Corollary 2.8 entails even more general properties, such as (*new* findings)
respectively,

While for
respectively,
Proposition 2.9 becomes
applicable, entailing (*new* findings) the *Diophantine* factorizations
respectively,

###### 3.6.2. Second Assignment

With this assignment, one can set, consistently with
our previous treatment,
implying, via (2.2), (2.3), that
the polynomials defined by the three-term recurrence relations
(2.1) coincide with the normalized Dual Hahn polynomials (3.92):Hence, with this identification, Proposition
2.1 becomes applicable, entailing (*new* finding) that these
normalized dual Hahn polynomials satisfy the second recursion relation (2.4a) withNote that this assignment
entails no restriction on the 3 parameters of the dual Hahn polynomials .

It is, moreover, plain that with the assignments
respectively,
the factorizations implied by Proposition
2.3 and the properties implied by Corollary 2.4 become applicable
with ,
respectively, .
These are *new* findings. As for the additional findings entailed by Corollary
2.5, they are reported in Section 3.6.3. And Proposition 2.7 becomes as well applicable, entailing (*new* findings) the *Diophantine* factorizations
respectively,
And Corollary 2.8 entails even more general properties, such as (*new* findings)
respectively,

While for
respectively,
Proposition 2.9 becomes
applicable, entailing (*new* findings) the *Diophantine* factorizations
respectively,

###### 3.6.3. Factorizations

The following *new* relations among dual Hahn
polynomials are implied by Proposition 2.3 with Corollary 2.5:

##### 3.7. Shifted Meixner-Pollaczek (SMP)

In this subsection, we introduce and treat a modified version of the standard (monic) Meixner-Pollaczek polynomials. The standard (monic) Meixner-Pollaczek (MP) polynomials (see [9]), are defined by the three-term recursion relations (3.1) with The standard version of these polynomials reads (see [9])

However, we have not found any assignment of the
parameters and in terms of allowing the application of our machinery. We,
therefore, consider the (monic) “shifted Meixner-Pollaczek” (sMP) polynomialsdefined of course via the
three-term recursion relation
(3.1) withThen, with the assignment(entailing no restriction on the parameters , in as much as the two parameters and are arbitrary), one can set, consistently with
our previous treatment,implying, via (2.2), (2.3), that
the polynomials defined by the three-term recurrence relations
(2.1) coincide with the normalized shifted Meixner-Pollaczek
polynomials:Hence, with this identification, Proposition
2.1 becomes applicable, entailing (*new* finding) that these
(normalized) shifted Meixner-Pollaczek polynomials satisfy the second recursion
relation (2.4a) with

It is, moreover, plain that with the
assignmentthe factorizations implied by Proposition
2.3, and the properties implied by Corollary 2.4, become applicable
with .
And Proposition 2.7 becomes as well applicable, entailing (*new* finding)
the *Diophantine* factorizationAnd Corollary 2.8 entails even more general properties, such as (*new* finding)

##### 3.8. Meixner

In this section (some results of which were already reported in [5]), we focus on the monic Meixner polynomials (see [9]), defined by the three-term recursion relations (3.1) with The standard version of these polynomials reads (see [9]):

We now identify the parameter via the assignmentOne can then set, consistently
with our previous treatment,implying, via (2.2), (2.3), that
the polynomials defined by the three-term recurrence relations
(2.1) coincide with the normalized Meixner polynomials (3.25):Hence, with this identification, Proposition
2.1 becomes applicable, entailing (*new* finding) that these
normalized Meixner polynomials satisfy the second recursion relation (2.4a)
withNote that this assignment entails
no restriction on the 2 parameters of the Meixner polynomials .

It is, moreover, plain that with the
assignmentthe factorizations implied by Proposition
2.3, and the properties implied by Corollary 2.4, become applicable
with .
These are *new* findings. And Proposition 2.7 becomes as well
applicable, entailing (*new* finding) the *Diophantine* factorizationAnd Corollary 2.8 entails even more general properties, such as (*new* finding)

Likewise for Proposition 2.9 becomes
applicable, entailing (*new* finding) the *Diophantine* factorization

##### 3.9. Krawtchouk

The monic Krawtchouk polynomials (see [9]: and note the notational replacement of the
parameters and used there with the parameters and used here, implying that only when and with a *positive integer* these polynomials coincide with the standard Krawtchouk
polynomials),
are defined by the three-term
recursion relations (3.1) with
The standard version of these
polynomials reads (see [9])

We now identify the parameter via the assignmentOne can then set, consistently
with our previous treatment,implying, via (2.2), (2.3), that
the polynomials defined by the three-term recurrence relations
(2.1) coincide with the normalized Krawtchouk polynomials (3.135):Hence, with this identification, Proposition
2.1 becomes applicable, entailing (*new* finding) that these
normalized Krawtchouk polynomials satisfy the second recursion relation (2.4a)
withNote that this assignment entails
no restriction on the 2 parameters of the Krawtchouk polynomials .

It is, moreover, plain that with the
assignment(which is, however, incompatible
with the definition of the standard Krawtchouk polynomials: and with a *positive integer*), the factorizations
implied by Proposition 2.3, and the properties implied by Corollary
2.4, become applicable with .
These are *new* findings. And Proposition 2.7 becomes as well
applicable, entailing (*new* finding) the *Diophantine* factorizationAnd Corollary 2.8 entails even more general properties, such as (*new* findings)

Likewise for(which is also incompatible with
the definition of the standard Krawtchouk polynomials: and with a *positive integer*), Proposition 2.9 becomes applicable, entailing (*new* finding) the *Diophantine* factorization

##### 3.10. Jacobi

In this section (most results of which were already reported in [5]), we focus on the monic Jacobi polynomials (see [9]), defined by the three-term recursion relations (3.1) with The standard version of these polynomials reads (see [9])

Let us recall that for the Jacobi polynomials there holds the symmetry relation

We now identify the parameter as follows:With this assignment one can
set, consistently with our previous treatment,implying, via (2.2), (2.3), that
the polynomials defined by the three-term recurrence relations
(2.1) coincide with the normalized Jacobi polynomials (3.146):Hence, with this identification, Proposition
2.1 becomes applicable, entailing (*well-known* result) that these
normalized Jacobi polynomials satisfy the second recursion relation (2.4a)
with

It is, moreover, plain that with the
assignmentthe factorizations implied by Proposition
2.3, and the properties implied by Corollary 2.4, become applicable
with .
These seem *new* findings. As for the additional findings entailed by Corollary
2.5, they are reported in Section 3.10.1. And Proposition 2.7 becomes as well applicable, entailing (*well-known* result) the *Diophantine* factorizationAnd Corollary 2.8 entails even more general properties, such as the fact that the Jacobi polynomials ,
feature as a zero of order

###### 3.10.1. Factorizations

The following (*not new*) relations among Jacobi
polynomials are implied by Proposition 2.3 with Corollary 2.5 (and
see (3.153), of which the following formula is a generalization, just as
(1.1c) is a generalization of (1.1a)):

##### 3.11. Laguerre

In this section (most results of which were already reported in [5]), we focus on the monic Laguerre polynomials (see [9]), defined by the three-term recursion relations (3.1) with The standard version of these polynomials reads (see [9])

We now identify the parameter as follows:With this assignment, one can
set, consistently with our previous treatment,implying, via (2.2), (2.3), that
the polynomials defined by the three-term recurrence relations
(2.1) coincide with the normalized Laguerre polynomials (3.155):Hence, with this identification, Proposition
2.1 becomes applicable, entailing (*well-known* result) that the
normalized Laguerre polynomials satisfy the second recursion relation (2.4a)
with

It is, moreover, plain that with the
assignmentthe factorizations implied by Proposition
2.3, and the properties implied by Corollary 2.4, become applicable
with .
These seem *new* findings. As for the additional findings entailed by Corollary
2.5, they are reported in Section 3.11.1. And Proposition 2.7 becomes as well applicable, entailing (*well-known* result) the formula
(see (1.1a))And Corollary 2.8 entails even more general properties, such as the fact that the Laguerre polynomials ,
feature as a zero of order ,
see (1.1c) or, equivalently, the next formula.

###### 3.11.1. Factorizations

The following (*not new*) relations among
Laguerre polynomials are implied by Proposition 2.3 with Corollary
2.5 (and see (3.162), of which the following formula—already reported
above, see (1.1c)—is a generalization):

##### 3.12. Modified Charlier

In this subsection, we introduce and treat a modified version of the standard (monic) Charlier polynomials. The standard (monic) Charlier polynomials (see [9]), are defined by the three-term recursion relations (3.1) with

The standard version of these polynomials reads (see [9])

However, we have not found any assignment of the parameters in terms of allowing the application of our machinery. To
nevertheless proceed, we introduce the class of (monic) “modified
Charlier” polynomials characterized by the three-term recursion relation
(3.1) withthat obviously reduce to the
monic Charlier polynomials for .
Assigning insteadone can set, consistently with
our previous treatment,implying, via (2.2), (2.3), that
the polynomials defined by the three-term recurrence relations
(2.1) coincide with these (monic) modified Charlier polynomials:Hence, with this identification, Proposition
2.1 becomes applicable, entailing (*new* finding) that these (monic)
modified Charlier polynomials satisfy the second recursion relation (2.4a)
with

There does not seem to be any interesting results for the zeros of these polynomials.

#### 4. Outlook

Other classes of orthogonal polynomials to which our
machinery is applicable, partly overlapping with those reported in this paper,
have been identified by finding explicit classes of coefficients and (defining these classes of orthogonal polynomials
via the three-term recursion relations (2.1)) that do satisfy the nonlinear
relations entailing the validity of the various propositions reported
above. Hence, for these classes of orthogonal polynomials analogous results to
those reported above hold, namely an additional three-term recursion relation
involving shifts in the parameter ,
and possibly as well factorizations identifying *Diophantine* zeros. These
findings will be reported, hopefully soon, in a subsequent paper, where we also
elucidate and exploit the connection about the machinery reported above and the
wealth of known results on *discrete integrable* systems [18]. Other developments
connected with the machinery reported above are as well under investigation,
including *inter alia* other types of *additional* recursion
relations satisfied by the classes of orthogonal polynomials defined by the
three-term recursion relations (2.1) (for appropriate choices of the coefficients and ) and the investigation of other properties of
such polynomials—possibly including the identification of ODEs satisfied by
them.

#### Appendix

#### A. A proof

In this Appendix, for completeness, we provide a proof of the factorization (2.11) with (2.12a) (see Proposition 2.3) although this proof is actually quite analogous to that provided (for the special case with ) in [5, Section 4]. We proceed by induction, assuming that (2.11) holds up to and then showing that it holds for . Indeed, by using it in the right-hand side of the relation (2.1a) with , we get and clearly by using the recursion relation (2.12a) the square bracket in the right-hand side of this equation can be replaced by yieldingNote that for , this formula is an identity, since see (2.12b); likewise, this formula clearly also holds for , provided that (2.9) holds, see (2.1a) with and (2.12c).

But this is just the formula (2.11) with replaced by . Q. E. D.

*Remark A .1. *The hypothesis (2.9) has been used above, in this proof of Proposition 2.3, only to prove the validity of the final
formula, (A.2), for .
Hence one might wonder whether this hypothesis, (2.9), was redundant, since the
validity of the final formula (A.2) for seems to be implied by (A.1) with (2.12c) and
(2.12b), without the need to invoke (2.9). But in fact, by setting in the basic recurrence relation (2.1a), it is clear that (2.12c) and (2.12b) only hold provided (2.9) also holds.

#### Acknowledgments

It is a pleasure to acknowledge useful discussions with H. W. Capel, Frank Nijhoff, Peter van der Kampf and, last but not least, Paul Edward Spicer who provided us with a copy of his Ph.D. thesis entitled “On Orthogonal Polynomials and Related Discrete Integrable Systems.” These interactions occurred mainly in July 2007 during the XIV Workshop on Nonlinear Evolution Equations and Dynamical Systems (NEEDS 2007), the organizers of which—David Gómez Ullate, Andy Hone, Sara Lombardo, and Joachim Puig—we also like to thank for the excellent organization and the pleasant working atmosphere of that meeting.