with the Polynomials of the Askey Scheme

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Volume 2009, Article ID 268134,43pages doi:10.1155/2009/268134

Research Article

Additional Recursion Relations, Factorizations, and Diophantine Properties Associated

with the Polynomials of the Askey Scheme

M. Bruschi,

^{1, 2}

F. Calogero,

^{1, 2}

and R. Droghei

³

1Dipartimento di Fisica, Universit`a di Roma “La Sapienza”, 00185 Roma, Italy

2Istituto Nazionale di Fisica Nucleare, Sezione di Roma, 00185 Roma, Italy

3Dipartimento di Fisica, Universit`a Roma Tre, 00146 Roma, Italy

Correspondence should be addressed to F. Calogero,francesco.calogero@roma1.infn.it Received 29 July 2008; Accepted 1 December 2008

Recommended by M. Lakshmanan

In this paper, we apply toalmostall 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 integersDiophantine 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 notyetreported in the standard compilations.

Copyrightq2009 M. Bruschi et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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 equilibria1,2 for a review of these and other analogous results, see3, Appendix C. To provesome ofthese conjectures a theoretical framework was then developed4–6, 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

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coeﬃcients, featuring a parameterν,of these recursion relations—suﬃcient to guarantee that the corresponding polynomials also satisfy a second three-term recursion relation involving shifts in that parameterν; and via this second recursion relation, Diophantine results of the kind indicated above were obtained5. InSection 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 scheme9, as defined by the basic three-term recursion relation they satisfy: this entails the identification of the parameterν—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 resultsespecially after they have been discoveredcould also be obtained by other routes—for instance, by exploiting the relations of these polynomials with hypergeometric functions: we did not find themexcept 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 polynomialswith diﬀerent parametersto 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:

L⁻ⁿ_n x −xⁿ

n! , n0,1,2, . . . , 1.1a

where L^α_n x is the standard generalized Laguerre polynomial of order n, for whose orthogonality,

_∞

0

dx x^αexp−xL^αn xL^αm x δ_nmΓnα1

n! , 1.1b

it is, however, generally required that Reα > −1. 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 Oberhettinger14or11, Equation 8.973.4. And this remark applies as well to the following neat generalization of this formula, reading

L^−m_n x −1^mn−m!

n! x^mL^m_n−mx, m0,1, . . . , n, n0,1,2, . . . , 1.1c which qualifies as well as the prototype of formulas reported below for many of the poly- nomials 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

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compensated by the term x^m 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 1.1cfor 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 themit 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” polynomials9–13, 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 assignmentsincluding limiting onesof 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 p_n^νx, of degree n in their argumentxand depending on a parameterν, defined by the three-term recursion relation:

p^ν_n1x

xa^ν_n

p^ν_n x b^ν_n p^ν_n−1x 2.1a

with the “initial” assignments

p^ν₋₁x 0, p^ν₀ x 1, 2.1b

clearly entailing

p^ν₁ x xa^ν₀ , p₂^νx

xa^ν₁

xa^ν₀

b₁^ν, 2.1c

and so on.In some cases the left-hand side of the first2.1bmight preferably be replaced by b^ν₀ p^ν₋₁x, to take account of possible indeterminacies ofb^ν₀ .

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Notation. Here and hereafter the indexnis a nonnegative integerbut some of the formulas written below might make little sense for n 0, requiring a—generally quite obvious—

special interpretation, anda^ν_n , b^ν_n are functions of this indexn and of the parameter ν.

They might—indeed they often do—also depend on other parameters besidesνsee below;

but this parameterνplays a crucial role, indeed the results reported below emerge from the identification of special values of itgenerally simply related to the indexn.

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 coeﬃcientsa^ν_n andb^ν_n be real and that the latter be negative,b^ν_n <0see, e.g.,16.

2.1. Additional Recursion Relation

Proposition 2.1. If the quantitiesA^ν_n andω^νsatisfy the nonlinear recursion relation A^ν_n−1−A^ν−1_n−1

A^ν_n −A^ν−1_n−1 ω^ν

A^ν−1_n−1 −A^ν−2_n−1

A^ν−1_n−1 −A^ν−2_n−2 ω^ν−1

2.2a with the boundary condition

A^ν₀ 0 2.2b

(where, without significant loss of generality, this constant is set to zero rather than to an arbitrary ν-independent valueA: see [5, Equation (4a)]; and we also replaced, for notational convenience, the quantityα^νpreviously used [5] withω^ν), and if the coeﬃcientsa^ν_n andb^ν_n are defined in terms of these quantities by the following formulas:

a^ν_n A^ν_n1−A^ν_n , 2.3a

b_n^ν

A^ν_n −A^ν−1_n

A^ν_n −A^ν−1_n−1 ω^ν

, 2.3b

then the polynomialsp_n^νxidentified by the recursion relation (2.1) satisfy the following additional recursion relation (involving a shift both in the ordernof the polynomials and in the parameterν):

p^ν_n x p^ν−1_n x g_n^νp^ν−1_n−1 x 2.4a

with

g_n^νA^ν_n −A^ν−1_n . 2.4b

This proposition corresponds to5, 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.

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Alternative conditions suﬃcient for the validity ofProposition 2.1and characterizing directly the coeﬃcientsa^ν_n , b^ν_n , andg_n^νread as followssee5, Appendix B:

a^ν_n −a^ν−1_n g_n1^ν −g_n^ν, 2.5a b^ν−1_n−1 g_n^ν−b_n^νg_n−1^ν 0, 2.5b

with

g_n^ν−b_n^ν−b^ν−1_n

a^ν_n −a^ν−1_n−1 , 2.5c

and the “initial” condition

g₁^νa^ν₀ −a^ν−1₀ , 2.5d

entailing via2.5c withn1

b^ν₁ −b₁^ν−1

a^ν₀ −a^ν−1₀

a^ν₁ −a^ν−1₀

0 2.5e

and via2.5a withn0

g₀^ν0. 2.5f

Proposition 2.2. Assume that the class of (monic, orthogonal) polynomialsp_n^νx 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:

p^ν_n x

x−x^1,ν_n

p^ν−1_n−1 x b^ν−1_n−1 p_n−2^ν−1x, 2.6a x^1,ν_n −

a^ν−1_n−1 g_n^ν

, 2.6b

in addition to

p^ν_n x

x−x^2,ν_n

p^ν−2_n−1 x c^ν_n p^ν−2_n−2 x, 2.7a x^2,ν_n −

a^ν−2_n−1 g_n^νg_n^ν−1

, 2.7b

c_n^νb_n−1^ν−2g_n^νg_n−1^ν−1, 2.7c

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as well as

p^νn x

x−xn^3,ν

p^ν−3_n−1 x d^νn p^ν−3_n−2 x e^νn p^ν−3_n−3 x, 2.8a

x^3,ν_n −

a^ν−3_n−1 g_n^νg_n^ν−1g_n^ν−2

, 2.8b

d^ν_n b^ν−3_n−1 g_n^νg_n−1^ν−2g_n^ν−1g_n−1^ν−2g_n^νg_n−1^ν−1, 2.8c e^ν_n g_n^νg^ν−1_n−1 g_n−2^ν−2. 2.8d

These findings correspond to6, Proposition 1.

2.2. Factorizations

In the following we introduce a second parameterμ,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) polynomialsp^ν_n xare defined by the recursion relation (2.1) and the coeﬃcientsb^ν_n satisfy the relation

b_n^nμ 0, 2.9

entailing that forνnμ,the recursion relation2.1areads

p^nμ_n1 x

xa^nμ_n

p^nμ_n x, 2.10

then there holds the factorization

p_n^mμx p^−m_n−mxpm^mμx, m0,1, . . . , n, 2.11

with the “complementary” polynomialsp^−m_n x(of course of degreen) defined by the following three- term recursion relation analogous (but not identical) to (2.1):

p^−m_n1 x

xa^mμ_nm

p^−m_n x b^mμ_nm p^−m_n−1 x, 2.12a

p^−m₋₁ x 0, p^−m₀ x 1, 2.12b

entailing

p^−m₁ x xa^mμ_m , 2.12c

p^−m₂ x

xa^mμ_m1

xa^mμ_m

b^mμ_m1

x−x_m

x−x_m⁻

2.12d

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with

x^±_m 1 2

−a^mμ_m −a^mμ_m1 ±

a^mμ_m −a^mμ_m1 ²

−4b^mμ_m1 ^1/2

, 2.12e

and so on.

This is a slight generalizationproven below, in Appendix Aof5, Proposition 2.4.

Note incidentally that also the complementary polynomialsp^−m_n x,being defined by three- terms recursion relations, see2.12a, 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 ofProposition 2.3.

Corollary 2.4. If2.9holds—entailing2.10and2.11with (2.12)—the polynomialpⁿ⁻¹_n xhas the zero−aⁿ⁻¹_n−1 ,

p^n−1μ_n

−a^n−1μ_n−1

0, 2.13a

and the polynomialp^n−2μ_n xhas the two zerosx^±_n−2(see2.12e),

p^n−2μ_n x^±_n−2

0. 2.13b

The first of these results is a trivial consequence of 2.10; the second is evident from2.11and 2.12d. Note, moreover, that from the factorization formula2.11, one can likewise find explicitly 3 zeros ofp^n−3μn xand 4 zeros ofpn^n−4μx,by evaluation from (2.12)p^−m₃ xandp^−m₄ xand 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−a^n−1μ_n−1 andx^±_n−2in terms of integers.

Corollary 2.5. If2.9holds—entailing2.10and2.11with (2.12)—and moreover the quantities a^mn andb^mn satisfy the properties

a^−mμ_n−m ρ

a^m_n ^μ ρ

, b_n−m^−mμ ρ

b_n^m^μ ρ

, 2.14

then clearly

p^m_n

x;ρ

p_n^m^μ x;ρ

, 2.15

entailing that the factorization2.11takes the neat form p_n^mμ

x;ρ

p^−m_n−m^μ x;ρ

p^mμ_m x;ρ

, m0,1, . . . , n. 2.16

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Note that—for future convenience, see below—one has emphasized explicitly the possibility that the polynomials depend on additional parameters (indicated with the vector variablesρ, resp.,ρ; these additional parameters must of course be independent ofn, but they might depend onm).

The following remark is relevant when both Propositions2.1and2.2hold.

Remark 2.6. As implied by2.3b, the condition2.9can be enforced via the assignment

ω^ν A^ν−1μ_ν−1 −A^νμ_ν , 2.17

entailing that the nonlinear recursion relation2.3areads A^ν_n−1−A^ν−1_n−1

A^ν_n −A^ν−1_n−1 A^ν−1μ_ν−1 −A^νμ_ν

A^ν−1_n−1 −A^ν−2_n−1

A^ν−1_n−1 −A^ν−2_n−2 A^ν−2μ_ν−2 −A^ν−1μ_ν−1 .

2.18

Corollaries2.4and2.5andRemark 2.6are analogous to5, 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 p^ν_n x are defined by the three-term recursion relations (2.1) with coeﬃcientsa^ν_n andb^ν_n satisfying the requirements suﬃcient for the validity of both Propositions2.1and2.2(namely (2.3), with (2.2) and2.9, or just with2.18), then

p^nμ_n x ⁿ

m1

x−x_m^1,mμ

, 2.19a

with the expressions 2.6b of the zeros x_m^1,ν and the standard convention according to which a product equals unity when its lower limit exceeds its upper limit. Note that these nzeros are n- independent (except for their number). In particular,

p^μ₀ x 1, p₁^1μx x−x^1,1μ₁ , p₂^2μx

x−x₁^1,2μ

x−x₂^1,2μ

, 2.19b

and so on.

These findings correspond to6, Proposition 2.2first part.

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

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Corollary 2.8. IfProposition 2.7 holds, then also the polynomialsp^n−1μ_n xand p^n−2μ_n x(in addition top_n^nμx, see (2.19)) can be written in the following completely factorized form (see2.6b and2.12e):

p_n^n−1μx

xaⁿ⁻¹_n−1 ⁿ⁻¹

m1

x−x^1,mμ_m

, 2.20a

p^n−2μ_n x

x−x_m

x−x_m⁻ⁿ⁻²

m1

x−x^1,mμ_m

. 2.20b

Analogously, complete factorizations can clearly be written for the polynomialsp^n−3μ_n xand p^n−4μn x, see the last part ofCorollary 2.4.

And of course the factorization2.11together with2.19aentails the (generally Diophantine) finding that the polynomialp^mμ_n xwithm1, . . . , nfeatures themzerosx^1,μ , 1, . . . , m, see2.6b:

p_n^mμ

x^1,μ

0, 1, . . . , m, m1, . . . , n. 2.21

Proposition 2.9. Assume that, for the class of polynomials pn^νx, there hold the preceding Proposition 2.1, and moreover that, for some value of the parameterμ(and of course for all nonnegative integer values ofn), the coeﬃcientsc^2nμ_n vanish (see2.7aand2.7c),

c^2nμ_n b^2nμ−2_n−1 g_n^2nμg_n−1^2nμ−1 0, 2.22a

then the polynomialsp^2nμn xfactorize as follows:

p_n^2nμx ⁿ

m1

x−x_m^2,2mμ

, 2.22b

entailing

p^μ₀ x 1, p^2μ₁ x x−x₁^2,2μ, p^4μ₂ x

x−x^2,2μ₁

x−x^2,4μ₂

, 2.22c

and so on.

Likewise, if for all nonnegative integer values ofn,the following two properties hold (see2.8a, 2.8c, and2.8d):

d_n^3nμb^3nμ−3_n−1 g_n^3nμg^3nμ−2_n−1 g_n^3nμ−1g_n−1^3nμ−2g_n^3nμg_n−1^3nμ−10, 2.23a e^3nμ_n 0, that is, g_n^3nμ0 or g_n−1^3nμ−10 or g_n−2^3nμ−20, 2.23b

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then the polynomialsp^3nμ_n xfactorize as follows:

pn^3nμx ⁿ

m1

x−xm^3,3mμ

, 2.23c

entailing

p^μ₀ x 1, p^3μ₁ x x−x^3,3μ₁ , p^6μ₂ x

x−x^3,3μ₁

x−x^3,6μ₂

, 2.23d

and so on.

Here of course the n (n-independent!) zeros x^2,2mμ_m , respectively, x_m^3,3mμ are defined by 2.7b, respectively,2.8b.

These findings correspond to6, Proposition 2.

3. Results for the Polynomials of the Askey Scheme

In this section, we apply to the polynomials of the Askey scheme9the results reviewed in the previous section. This class of polynomialsincluding the classical polynomialsmay 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:

p_n1 x;η

xa_n

η p_n

x;η b_n

η p_n−1

x;η

3.1a

with the “initial” assignments

p₋₁ x;η

0, p₀ x;η

1, 3.1b

clearly entailing

p1

x;η

xa0

η , p2

x;η

xa1

η xa0

η b1

η

, 3.1c

and so on. Here the components of the vector η denote the additional parameters generally featured by these polynomials.

Let us emphasize that in this manner we introduced the monicor “normalized”9 version of these polynomials; below we generally also report the relation of this version to the more standard version9.

To apply our machinery we must identify, among the parameters characterizing these polynomials, the single parameter ν playing a special role in our approach. This can be generally done in several wayseven for the same class of polynomials, see below. Once this

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identification i.e., the assignment η ≡ ην has been made, the recursion relations 3.1 coincide with the relations2.1via the self-evident notational identification:

p^ν_n x≡pn

x;ην

, a^ν_n ≡an

ην

, b^ν_n ≡bn

ην

. 3.2

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 ν-dependence ην diﬀerent from those considered below.

3.1. Wilson

The monic Wilson polynomials see 9, and note the notational replacement of the 4 parametersa, b, c, dused there withα, β, γ, δ

pnx;α, β, γ, δ≡pn

x;η

3.3a

are defined by the three-term recursion relations3.1with

an

η

α²−An−Cn, bn

η

−A_n−1Cn, 3.3b

where

An nαβnαγnαδn−1ασ

2n−1ασ2nασ , 3.3c

C_n nn−1βγn−1βδn−1γδ

2n−2ασ2n−1ασ , 3.3d

σ≡βγδ, ρ≡βγβδγδ, τ ≡βγδ. 3.3e

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The standard version of these polynomials readssee9:

W_nx;α, β, γ, δ −1ⁿn−1αβγδ_np_nx;α, β, γ, δ. 3.4

Let us also recall that these polynomials pnx;α, β, γ, δ are invariant under any permutation of the 4 parametersα, β, γ, δ.

As for the identification of the parameterνsee3.2, two possibilities are listed in the following subsections.

3.1.1. First Assignment

α−ν. 3.5

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

A^ν_n

62n−2−νσ₋₁ n

4−5σ6ρ−6τ 5−6σ6ρν

−109σ−6ρ −96σν n 8−4σ4νn²−2n³

,

3.6a

ω^ν−ν², 3.6b

implying, via2.2,2.3, that the polynomialsp^ν_n xdefined by the three-term recurrence relations2.1coincide with the normalized Wilson polynomials3.3:

p^ν_n x p_nx;−ν, β, γ, δ. 3.7

Hence, with this identification, Proposition 2.1becomes applicable, entailingnew finding that these normalized Wilson polynomials satisfy the second recursion relation2.4awith

g_n^ν nn−1βγn−1βδn−1γδ

2n−2−νσ2n−1−νσ . 3.8

Note that this finding is obtained without requiring any limitation on the 4 parameters of the Wilson polynomialsp_nx;α, β, γ, δ.

It is, moreover, plain that with the assignment

νn−1β, namely, α−n1−β, 3.9

the factorizations implied by Proposition 2.3, and the properties implied by Corollary 2.4, become applicable with μ β−1. These are new findings. As for the additional findings

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entailed byCorollary 2.5, they are reported inSection 3.1.3. AndProposition 2.7becomes as well applicable, entailingnew findingthe Diophantine factorization

pnx;−n1−β, β, γ, δ ⁿ

m1

x m−1β²

, 3.10

whileCorollary 2.8entails even more general properties, such asnew finding p_n

−−1β²;−m1−β, β, γ, δ

0, 1, . . . , m, m1, . . . , n. 3.11

Remark 3.1. A look at the formulas3.3suggests other possible assignments of the parameter ν satisfying 2.9, such as ν n − 2 σ, namely, α 2 − n − σ. However, these assignments actually fail to satisfy2.9for all values ofn, because for this to happen, it is not suﬃcient that the numerator in the expression ofb_n^νμ vanish, it is, moreover, required that the denominator in that expression never vanish. In the following, we will consider only assignments of the parameterνin terms ofnthat satisfy these requirements.

3.1.2. Second Assignment

α−ν

2, β 1−ν

2 . 3.12

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

64n−3−2ν2γ2δ₋₁ n

3−4γ−4δ6γδ 7−9γ−9δ12γδν31−γ−δν²

−

11−12γ−12δ12γδ35−4γ−4δν3ν² n 432ν−2γ−2δn²−4n³

,

3.13a ω^ν−ν²

4, 3.13b implying, via2.2,2.3, that the polynomialsp^ν_n xdefined by the three-term recurrence relations2.1coincide with the normalized Wilson polynomials3.3:

p_n^νx pn

x;−ν

2,1−ν 2 , γ, δ

. 3.14

Hence, with this identification, Proposition 2.1becomes applicable, entailingnew finding that these normalized Wilson polynomials satisfy the second recursion relation2.4awith

g_n^ν nn−1γδ2n−1−ν2γ2n−1−ν2δ

4n−3−2ν2γ2δ4n−1−2ν2γ2δ . 3.15

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Note that this assignment entails now thesinglerestrictionβα1/2 on the 4 parameters of the Wilson polynomialsp_nx;α, β, γ, δ.

It is, moreover, plain that with the assignments

νn−1

2, henceα−n 2 1

4, β−n 2 3

4, 3.16a νn−22δ, γδ−1

2, α−n

2 1−δ, β−n 2 3

2−δ, 3.16b

respectively,

νn−12δ, γδ1

2, α−n 2 1

2−δ, β−n

2 1−δ, 3.16c

the factorizations implied by Proposition 2.3 and the properties implied by Corollary 2.4 become applicable with μ −1/2, μ −2 2δ, respectively, μ −1 2δ. 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

p_n

x;−n 2 1

4,−n 2 3

4, γ, δ

ⁿ

m1

x

2m−1 4

₂

, 3.17a

p_n

x;−n

2 1−δ,−n 2 3

2 −δ, δ− 1 2, δ

ⁿ

m1

x

m−22δ 2

2

, 3.17b

respectively,

p_n

x;−n 2 1

2−δ,−n

2 1−δ, δ1 2, δ

ⁿ

m1

x

m−12δ 2

₂

. 3.17c

A referee pointed out that3.17ais not new, as one can evaluate explicitlypnx;α, β, γ, δ whennαβ10,which is indeed the case in3.17a; and, moreover, that the two formulas 3.17band3.17ccoincide, since their left-hand sides are identical as a consequence of the symmetry property of Wilson polynomials under the transformationδ⇒δ1/2.

AndCorollary 2.8entails even more general properties, such asnew finding

p_n

−

2−1 4

₂

;−m 2 1

4,−m 2 3

4, γ, δ

0, 1, . . . , m, m1, . . . , n, 3.18a

p_n

−

−22δ 2

2

;−m

2 1−δ,−m 2 3

2 −δ, δ−1 2, δ

0, 1, . . . , m, m1, . . . , n, 3.18b

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respectively,

p_n

−

−12δ 2

2

;−m 2 1

2−δ,−m

2 1−δ, δ1 2, δ

0, 1, . . . , m, m1, . . . , n.

3.18c

Moreover, with the assignments

ν2n−22δ, α−n1−δ, β−n 3

2−δ, 3.19a respectively,

ν2n−12δ, α−n1

2 −δ, β−n1−δ, 3.19b Proposition 2.9becomes applicable, entailingnew findingsthe Diophantine factorizations

pn

x;−n1−δ,−n3

2−δ, γ, δ

ⁿ

m1

x m−1δ²

, 3.20a

respectively,

p_n

x;−n1

2 −δ,−n1−δ, γ, δ

ⁿ

m1

x m−1δ²

, 3.20b

obviously implying the relation

p_n

x;−n1−δ,−n3

2 −δ, γ, δ

p_n

x;−n1

2 −δ,−n1−δ, γ, δ

. 3.20c

3.1.3. Factorizations

The following new relations among monic Wilson polynomials are implied byProposition 2.3 withCorollary 2.5:

p_nx;−m1−β, β, γ, δ

p_n−mx;mβ, γ,1−β, δpmx;−m1−β, β, γ, δ, m0,1, . . . , n, 3.21a

p_n

x;−m

2 1−δ,−m 2 3

2−δ, δ−1 2, δ

p_n−m

x;m 2 − 1

2δ,m

2 δ,1−δ,−δ 3 2

pm

x;−m

2 1−δ,−m 2 3

2−δ, δ−1 2, δ

, m0,1, . . . , n.

3.21b

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Note that the polynomials appearing as second factors in the right-hand side of these formulas are completely factorizable, see3.10and3.17b we will not repeat this remark in the case of analogous formulas below.

3.2. Racah

The monic Racah polynomialssee9

p_nx;α, β, γ, δ≡p_n x;η

3.22a

a_n η

A_nC_n, b_n η

−A_n−1C_n, 3.22b

where

An n1αn1αβn1βδn1γ

2n1αβ2n2αβ , 3.22c C_n nnαβ−γnα−δnβ

2nαβ2n1αβ . 3.22d

The standard version of these polynomials readssee9

Rnx;α, β, γ, δ nαβ1_n

α1_nβδ1_nγ1_npnx;α, β, γ, δ. 3.23a

Note, however, that in the following we do not require the parameters of these polynomials to satisfy one of the restrictionsα−N, βδ−N, orγ −N,withNa positive integer and n0,1, . . . , N,whose validity is instead required for the standard Racah polynomials9.

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

pnx;α, β, γ, δ pnx;α, β, βδ, γ−β p_nx;βδ, α−δ, γ, δ

p_nx;γ, αβ−γ, α,−αγδ.

3.23b

Let us now identify the parameterνas followssee3.2:

α−ν. 3.24

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A^ν_n 62n−νβ⁻¹n

β23γ3δ−23γδ 6γβδν 46γδ3βγ−βδ2γδ−32βγδνn 4−νβn²2n³

,

3.25a

ω^ν ν−1νγδ, 3.25b

implying, via2.2,2.3, that the polynomialsp^ν_n xdefined by the three-term recurrence relations2.1coincide with the normalized Racah polynomials3.22:

p^ν_n x p_nx;−ν, β, γ, δ. 3.26

Hence, with this identification, Proposition 2.1becomes applicable, entailingnew finding that these normalized Racah polynomials satisfy the second recursion relation2.4awith

g_n^ν− nnβnβδnγ

2n−νβ2n1−νβ. 3.27

Note that this finding is obtained without requiring any limitation on the 4 parameters of the Racah polynomialspnx;α, β, γ, δ.

It is, moreover, plain that with the assignments

νn, hence α−n, 3.28a

νn−δ, hence α−nδ, 3.28b

respectively,

νnβ−γ, hence α−n−βγ, 3.28c

the factorizations implied by Proposition 2.3 and the properties implied by Corollary 2.4 become applicable with μ 0, μ −δ, respectively, μ β −γ. 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

p_nx;−n, β, γ, δ ⁿ

m1

x−m−1mγδ

, 3.29a

pnx;−nδ, β, γ, δ ⁿ

m1

x−mγm−δ−1

, 3.29b

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respectively,

p_nx;−n−βγ, β, γ, δ ⁿ

m1

x−m−1β−γmβδ

. 3.29c

AndCorollary 2.8entails even more general properties, such asnew findings p_n

−1γδ;−m, β, γ, δ

0, 1, . . . , m, m1, . . . , n, 3.30a p_n

γ−δ−1;−mδ, β, γ, δ

0, 1, . . . , m, m1, . . . , n, 3.30b

respectively, pn

−1β−γβδ;−m−βγ, β, γ, δ

0, 1, . . . , m, m1, . . . , n. 3.30c

The following new relations among Racah polynomials are implied byProposition 2.3with Corollary 2.5:

p_nx;−m, β,−1,1 p_n−mx;m, β,−1,1pmx;−m, β,−1,1, m0,1, . . . , n, 3.31a pnx;−mδ, β,−δ, δp_n−mx;m−δ,2δβ, δ,−δpmx;−mδ, β,−δ, δ, m0,1, . . . , n,

3.31b p_nx;−m−βγ, β, γ, c−γ

p_n−mx;mβ−γc,−β2γ−c, γ, c−γpmx;v−m−βγ, β, γ, c−γ, m0,1, . . . , n, 3.31c p_nx;α,−m, γ, δ p_n−mx;α, m, δ, γpmx;α,−m, γ, δ, m0,1, . . . , n, 3.31d p_nx, α,−m−αη, η, δpn−mx, η, m, ηδ−α, αpmx, α,−m−αη, η, δ, m0,1, . . . , n.

3.31e

3.3. Continuous Dual Hahn (CDH)

In this sectionsome results of which were already reported in5we focus on the monic continuous dual Hahn CDH polynomials pnx;α, β, γ see 9, and note the notational replacement of the 3 parametersa, b, cused there withα, β, γ,

p_nx;α, β, γ≡p_n x;η

, 3.32a

defined by the three-term recursion relations3.1with an

η

α²−nαβnαγ−nn−1βγ, 3.32b b_n

η

−nn−1αβn−1αγn−1βγ. 3.32c

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Snx;α, β, γ −1ⁿpnx;α, β, γ. 3.33

Let us recall that these polynomialspnx;α, β, γare invariant under any permutation of the three parametersα, β, γ.

Let us now proceed and provide two identifications of the parameterν,see3.2.

α−ν. 3.34

A^ν_n n

−5

6βγ−βγ βγ−1ν 3

2−β−γν

n−2 3n²

, 3.35a ω^ν−ν², 3.35b

implying, via2.2,2.3, that the polynomialsp^ν_n xdefined by the three-term recurrence relations2.1coincide with the normalized CDH polynomials3.32:

p_n^νx pnx;−ν, β, γ. 3.36

Hence, with this identification, Proposition 2.1becomes applicable, entailingnew finding that these normalized CDH polynomials satisfy the second recursion relation2.4awith

g_n^νnn−1βγ. 3.37

Note that this finding is obtained without requiring any limitation on the 3 parameters of the CDH polynomialsp_nx;α, β, γ.

νn−1β, hence α−n1−β, 3.38

the factorizations implied by Proposition 2.3 and the properties implied by Corollary 2.4 become applicable withμ −1β. These are new findings. As for the additional findings entailed byCorollary 2.5, they are reported inSection 3.3.3. AndProposition 2.7becomes as well applicable, entailingnew findingsthe Diophantine factorization

pnx;−n1−β, β, γ n m1

x m−1β²

. 3.39

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AndCorollary 2.8entails even more general properties, such asnew finding p_n

−−1β²;−m1−β, β, γ

0, 1, . . . , m, m1, . . . , n. 3.40

Likewise, with the assignment

ν2nβ, α−2n−β, γ 1

2, 3.41

Proposition 2.9becomes applicable, entailingnew findingthe Diophantine factorization

p_n

x;−2n−β, β,1 2

ⁿ

m1

x 2m−1β²

. 3.42

α−1

2νc, β−1

2ν1 c 3.43

wherecis an a priori arbitrary parameter.

A^ν_n n

−4 3 3

2γ5

2c−c²−2γc

−5 4 γc

ν−1

4ν² 2−γ−2cνn−2 3n²

, 3.44a ω^ν−1

41−2cν², 3.44b

implying, via2.2,2.3, that the polynomialsp^ν_n xdefined by the three-term recurrence relations2.1coincide with the normalized CDH polynomials3.32:

p^ν_n x pn

x;c−ν

2, c−ν 2 −1

2, γ

. 3.45

Hence, with this identification, Proposition 2.1becomes applicable, entailingnew finding that these normalized CDH polynomials satisfy the second recursion relation2.4awith

g_n^νn

n−1−ν 2 γc

. 3.46

Note that this assignment entails thesinglelimitationβα−1/2 on the parameters of the CDH polynomials.

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νn2c− 3

2, henceα−n 2 3

4, β−n 2 1

4, 3.47

the factorizations implied by Proposition 2.3 and the properties implied by Corollary 2.4 become applicable withμ2c−3/2. These are new findings. As for the additional findings entailed byCorollary 2.5, they are reported inSection 3.3.3. AndProposition 2.7becomes as well applicable, entailingnew findingsthe Diophantine factorization

p_n

x;−n 2 3

4,−n 2 1

4, γ

ⁿ

m1

x

2m−1 4

₂

. 3.48

AndCorollary 2.8entails even more general properties, such asnew finding

p_n

−

2−1 4

₂

;−m 2 3

4,−m 2 1

4, γ

0, 1, . . . , m, m1, . . . , n. 3.49

Likewise with the assignments

ν2n−1cγ, henceα−n1−γ, β−n1

2 −γ, 3.50a

respectively,

ν2

n−3 2 cγ

, hence α−n3

2 −γ, β−n1−γ, 3.50b

Proposition 2.9becomes applicable, entailingnew findingsthe Diophantine factorizations

pn

x;−n1−γ,−n 1 2−γ, γ

ⁿ

m1

x m−1γ²

, 3.51a

respectively,

p_n

x;−n3

2−γ,−n1−γ, γ

ⁿ

m1

x m−1γ²

. 3.51b

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

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

pnx;−m1−β, β, γ p_n−mx;mβ,1−β, γpmx;−m1−β, β, γ, m0,1, . . . , n.

3.52

3.4. Continuous Hahn (CH)

The monic continuous Hahn CDH polynomials p_nx;α, β, γ, δ see 9, and note the notational replacement of the 4 parametersa, b, c, dused there withα, β, γ, δ,

p_nx;α, β, γ, δ≡p_n x;η

, 3.53a

a_n η

−i

αA_nC_n

, b_n η

A_n−1C_n, 3.53b

where

An−n−1αβγδnαγnαδ

2n−1αβγδ2nαβγδ, 3.53c Cn nn−1βγn−1βδ

2nαβγδ−12nαβγδ−2. 3.53d

S_nx;α, β, γ, δ −1ⁿp_nx;α, β, γ, δ. 3.54a

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

p_nx;α, β, γ, δ p_nx;β, α, γ, δ p_nx;α, β, δ, γ p_nx;β, α, δ, γ. 3.54b

Let us now proceed and provide two identifications of the parameterν,see3.2.

α−ν. 3.55

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A^ν_n in−βγδ−2γδ 1−2βν β−γ−δ−νn

22−β−γ−δν−2n , 3.56a

ω^ν−i ν, 3.56b

implying, via2.2,2.3, that the polynomialsp^ν_n xdefined by the three-term recurrence relations2.1coincide with the normalized CH polynomials3.53:

p^ν_n x pnx;−ν, β, γ, δ. 3.57

Hence, with this identification, Proposition 2.1becomes applicable, entailingnew finding that these normalized CH polynomials satisfy the second recursion relation2.4awith

g_n^ν inn−1βγn−1βδ

2n−2−νβγδ2n−1−νβγδ. 3.58

Note that this assignment entails no restriction on the 4 parameters of the CH polynomials p_nx;α, β, γ, δ.

νn−1γ, hence α−n1−γ, 3.59

the factorizations implied by Proposition 2.3 and the properties implied by Corollary 2.4 become applicable withμ −1γ. These are new findings. And Proposition 2.7 becomes as well applicable, entailingnew findingsthe Diophantine factorization

pnx;−n1−γ, β, γ, δ ⁿ

m1

xim−1γ

. 3.60

AndCorollary 2.8entails even more general properties, such asnew findings

pn

−i−1γ;−m1−γ, β, γ, δ

0, 1, . . . , m, m1, . . . , n. 3.61

Analogous results also obtain from the assignment

γ−ν. 3.62