^{1,2}

^{1,2}

^{3}

^{1}

^{2}

^{3}

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

Recently

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

Here and hereafter the index

Let us recall that the theorem which guarantees that
these polynomials, being defined by the three-term recursion relation
(

If the quantities

This proposition corresponds to [

Alternative conditions sufficient for the validity of Proposition

Assume that the class of (monic, orthogonal) polynomials

These findings correspond to [

In the following we introduce a second parameter

If the (monic, orthogonal) polynomials

This is a slight generalization (proven below, in Appendix A) of [

The following two results are immediate consequences
of Proposition

If (

These findings often have a

If (

The following remark is relevant when both Propositions

As implied by (

Corollaries

The

If the (monic, orthogonal) polynomials

These findings correspond to [

The following results are immediate consequences of Proposition

If Proposition

Analogously, complete factorizations can clearly be
written for the polynomials

And of course the factorization (

Assume that, for the class of polynomials

Likewise, if for all

Here of course the

These findings correspond to
[

In this section, we apply to the polynomials of the
Askey scheme [

Let us emphasize that in this manner we introduced the

To apply our machinery we must identify, among the
parameters characterizing these polynomials, the single parameter

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

The monic Wilson polynomials (see [

The standard version of these polynomials reads (see
[

Let us also recall that these polynomials

As for the identification of the parameter

With this assignment, one can set, consistently with
our previous treatment,

It is, moreover, plain that with the assignment

A look at the formulas (

With this assignment, one can set, consistently with
our previous treatment,

It is, moreover, plain that with the assignments

And Corollary

Moreover, with the assignments

The following

The monic Racah polynomials (see [

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

Let us now identify the parameter

With this assignment, one can set, consistently with
our previous treatment,

It is, moreover, plain that with the assignments

The following

In this section (some results of which were already
reported in [

Let us recall that these polynomials

Let us now proceed and provide two identifications of
the parameter

With this assignment, one can set, consistently with
our previous treatment,

It is, moreover, plain that with the
assignment

Likewise, with the assignment

With this assignment, one can set, consistently with
our previous treatment,

It is, moreover, plain that with the
assignment

Likewise with the assignments

Note that the right-hand sides of the last two
formulas coincide; this implies (

The following

The monic continuous Hahn (CDH) polynomials

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

With this assignment, one can set, consistently with
our previous treatment,

It is, moreover, plain that with the
assignment

Analogous results also obtain from the
assignment

With this assignment, one can set, consistently with
our previous treatment,

It is, moreover, plain that with the
assignment

In this subsection, we introduce a somewhat
generalized version of the standard (monic) Hahn polynomials. These
(generalized) monic Hahn polynomials

Let us now proceed and provide three identifications
of the parameter

With this assignment, one can set, consistently with
our previous treatment,

It is, moreover, plain that with the assignments

With this assignment, one can set, consistently with
our previous treatment,

It is, moreover, plain that with the assignments

With this assignment, one can set, consistently with
our previous treatment,

It is, moreover, plain that with the
assignment

In this subsection, we introduce a somewhat
generalized version of the standard (monic) dual Hahn polynomials. These (generalized)
monic dual Hahn polynomials

Let us now proceed and provide two identifications of
the parameter

With this assignment, one can set, consistently with
our previous treatment,

It is, moreover, plain that with the assignments

While for

With this assignment, one can set, consistently with
our previous treatment,

It is, moreover, plain that with the assignments

While for

The following

In this subsection, we introduce and treat a modified
version of the standard (monic) Meixner-Pollaczek polynomials. The standard
(monic) Meixner-Pollaczek (MP) polynomials

However, we have not found any assignment of the
parameters

It is, moreover, plain that with the
assignment

In this section (some results of which were already
reported in [

We now identify the parameter

It is, moreover, plain that with the
assignment

Likewise for

The monic Krawtchouk polynomials

We now identify the parameter

It is, moreover, plain that with the
assignment

Likewise for

In this section (most results of which were already
reported in [

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

We now identify the parameter

It is, moreover, plain that with the
assignment

The following (

In this section (most results of which were already
reported in [

We now identify the parameter

It is, moreover, plain that with the
assignment

The following (

In this subsection, we introduce and treat a modified
version of the standard (monic) Charlier polynomials. The standard (monic)
Charlier polynomials

The standard version of these polynomials reads (see
[

However, we have not found any assignment of the parameters

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

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

In this Appendix, for completeness, we provide a proof
of the factorization (

But this is just the formula (

The hypothesis (

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.