Additional Recursion Relations , Factorizations , and Diophantine Properties Associated with the Polynomials of the Askey Scheme

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.


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 manybody 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 threeterm 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 Advances in Mathematical Physics coefficients, featuring a parameter ν, of these recursion relations-sufficient to guarantee that the corresponding polynomials also satisfy a second three-term recursion relation involving shifts in that parameter ν; 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 ν-which can often be done in more than one way, especially for the named polynomials involving several parametersand 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 schemereads as follows: , n 0, 1, 2, . . ., 1.1a where L α n x is the standard generalized Laguerre polynomial of order n, for whose orthogonality, it is, however, generally required that Re α > −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 Oberhettinger 14 or 11, Equation 8.973.4 .And this remark applies as well to the following neat generalization of this formula, reading x , m 0, 1, . . ., n, n 0, 1, 2, . . ., 1.1c 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 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 1.1c 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.

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 ν n x , of degree n in their argument x and depending on a parameter ν, defined by the three-term recursion relation: with the "initial" assignments and so on.In some cases the left-hand side of the first 2.1b might preferably be replaced by b

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Notation.Here and hereafter the index n is a nonnegative integer but some of the formulas written below might make little sense for n 0, requiring a-generally quite obviousspecial interpretation , and a ν n , b ν n are functions of this index n and of the parameter ν.They might-indeed they often do-also depend on other parameters besides ν see below ; but this parameter ν plays a crucial role, indeed the results reported below emerge from the identification of special values of it generally simply related to the index n .
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 a ν n and b ν n be real and that the latter be negative, b ν n < 0 see, e.g., 16 .

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

2.4b
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 and the "initial" condition and via 2.5a with n 0 g ν 0 0.

2.5f
Proposition 2.2.Assume that the class of (monic, orthogonal) polynomials p ν n x 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: Advances in Mathematical Physics as well as

Factorizations
In the following we introduce a second parameter μ, but for notational simplicity we do not emphasize explicitly the dependence of the various quantities on this parameter.x (of course of degree n) defined by the following threeterm recursion relation analogous (but not identical) to (2.1): Advances in Mathematical Physics 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 p −m n x , being defined by threeterms 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.

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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 variables ρ, resp., ρ; these additional parameters must of course be independent of n, but they might depend on m).
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 assignment entailing that the nonlinear recursion relation 2.3a reads

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 .(2.3), with (2.2) and 2.9 , or just with 2.18 ), then with the expressions 2.6b of the zeros x 1,ν m and the standard convention according to which a product equals unity when its lower limit exceeds its upper limit.Note that these n zeros are nindependent (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 p n−1 μ n x and p n−2 μ n x (in addition to p n μ n
x , see (2.19)) can be written in the following completely factorized form (see 2.6b and 2.12e ):

2.21
Proposition 2.9.Assume that, for the class of polynomials p ν n x , there hold the preceding Proposition 2.1, and moreover that, for some value of the parameter μ (and of course for all nonnegative integer values of n), the coefficients c 2n μ n vanish (see 2.7a and 2.7c ), then the polynomials p 2n μ n x factorize as follows: and so on.Likewise, if for all nonnegative integer values of n, the following two properties hold (see 2.8a , 2.8c , and 2.8d ): Advances in Mathematical Physics then the polynomials p 3n μ n x factorize as follows: and so on.

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 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 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 ν 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 η ≡ η ν 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 ν-dependence η ν different from those considered below.

Wilson
The monic Wilson polynomials see 9 , and note the notational replacement of the 4 parameters a, b, c, d used there with α, β, γ, δ p n x; α, β, γ, δ ≡ p n x; η 3.3a are defined by the three-term recursion relations 3.1 with where

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The standard version of these polynomials reads see 9 : Let us also recall that these polynomials p n x; α, β, γ, δ are invariant under any permutation of the 4 parameters α, β, γ, δ.
As for the identification of the parameter ν see 3.2 , two possibilities are listed in the following subsections.

First Assignment
With this assignment, one can set, consistently with our previous treatment, implying, via 2.2 , 2.3 , that the polynomials p ν n x 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 finding is obtained without requiring any limitation on the 4 parameters of the Wilson polynomials p n x; α, β, γ, δ .It is, moreover, plain that with the assignment However, these assignments actually fail to satisfy 2.9 for all values of n, because for this to happen, it is not sufficient that the numerator in the expression of b ν μ n vanish, it is, moreover, required that the denominator in that expression never vanish.In the following, we will consider only assignments of the parameter ν in terms of n that satisfy these requirements.

3.12
With this assignment, one can set, consistently with our previous treatment, implying, via 2.2 , 2.3 , that the polynomials p ν n x 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

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Note that this assignment entails now the single restriction β α 1/2 on the 4 parameters of the Wilson polynomials p n x; α, β, γ, δ .It is, moreover, plain that with the assignments

3.17c
A referee pointed out that 3.17a is not new, as one can evaluate explicitly p n x; α, β, γ, δ when n α β 1 0, 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 δ ⇒ δ 1/2.And Corollary 2.8 entails even more general properties, such as new finding , δ 0, 1, . . ., m, m 1, . . ., n.

3.18c
Moreover, with the assignments Proposition 2.9 becomes applicable, entailing new findings the Diophantine factorizations obviously implying the relation

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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 .

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

3.23a
Note, however, that in the following we do not require the parameters of these polynomials to satisfy one of the restrictions α −N, β δ −N, or γ −N, with N a positive integer and n 0, 1, . . ., N, 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:

3.23b
Let us now identify the parameter ν as follows see 3.2 : α −ν.

3.24
With this assignment, one can set, consistently with our previous treatment,

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

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 p n x; α, β, γ see 9 , and note the notational replacement of the 3 parameters a, b, c used there with α, β, γ , p n x; α, β, γ ≡ p n x; η , 3.32a defined by the three-term recursion relations 3.1 with

3.33
Let us recall that these polynomials p n x; α, β, γ are invariant under any permutation of the three parameters α, β, γ.
Let us now proceed and provide two identifications of the parameter ν, see 3.2 .

3.34
With this assignment, one can set, consistently with our previous treatment, 3.42

Second Assignment
where c 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 p ν n x defined by the three-term recurrence relations 2.1 coincide with the normalized CDH polynomials 3.32 :

3.49
Likewise with the assignments Proposition 2.9 becomes applicable, entailing new findings the Diophantine factorizations

3.51b
Note that the right-hand sides of the last two formulas coincide; this implies new finding that the left-hand sides coincide as well.

Factorizations
The following new relations among continuous dual Hahn polynomials are implied by Proposition 2.3 with Corollary 2.5: 3.52

3.54b
Let us now proceed and provide two identifications of the parameter ν, see 3.2 .

3.55
With this assignment, one can set, consistently with our previous treatment,  x i m − 1 γ .

3.60
And Corollary 2.8 entails even more general properties, such as new findings 3.61

Second Assignment
Analogous results also obtain from the assignment γ −ν.

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With this assignment, one can set, consistently with our previous treatment, x − i m − 1 α .

3.67
And Corollary 2.8 entails even more general properties, such as new finding 3.68

Hahn
In this subsection, we introduce a somewhat generalized version of the standard monic Hahn polynomials.These generalized monic Hahn polynomials p n x; α, β, γ see 9 , and note the replacement of the integer parameter N with the arbitrary parameter γ: hence the standard Hahn polynomials are only obtained for γ N with N a positive integer and n 1, 2, . . ., N , p n x; α, β, γ ≡ p n x; η , 3.69a are defined by the three-term recursion relations 3.1 with where The standard version of these polynomials reads see 9

3.70
Let us now proceed and provide three identifications of the parameter ν, see 3.2 .

3.71
With this assignment, one can set, consistently with our previous treatment, implying, via 2.2 , 2.3 , that the polynomials p ν n x defined by the three-term recurrence relations 2.1 coincide with the normalized Hahn polynomials 3.69 :

3.73
Hence, with this identification, Proposition 2.1 becomes applicable, entailing new finding that these normalized Hahn polynomials satisfy the second recursion relation 2.4a with Note that this assignment entails no restriction on the 3 parameters of the Hahn polynomials p n x; α, β, γ .

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It is, moreover, plain that with the assignments x − m 1 , 3.76a respectively,

Second Assignment
where c is an arbitrary parameter.
With this assignment, one can set, consistently with our previous treatment, implying, via 2.2 , 2.3 , that the polynomials p ν n x defined by the three-term recurrence relations 2.1 coincide with the normalized Hahn polynomials 3.69 : p n x; α, −ν γ c, γ .

3.80
Hence, with this identification, Proposition 2.1 becomes applicable, entailing new finding that these normalized Hahn polynomials satisfy the second recursion relation 2.4a with Note that this assignment entails no restriction on the 3 parameters of the Hahn polynomials p n x; α, β, γ .It is, moreover, plain that with the assignments the factorizations implied by Proposition 2.3 and the properties implied by Corollary 2.4 become applicable with μ γ c, respectively, μ 1 α 2γ c.These are new findings.And Proposition 2.7 becomes as well applicable, entailing new findings the Diophantine factorizations x m α .

3.83b
And Corollary 2.8 entails even more general properties, such as new findings

3.84b
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Third Assignment
where c is an arbitrary parameter.
With this assignment, one can set, consistently with our previous treatment, implying, via 2.2 , 2.3 , that the polynomials p ν n x defined by the three-term recurrence relations 2.1 coincide with the normalized Hahn polynomials 3.69 :

3.87
Hence, with this identification, Proposition 2.1 becomes applicable, entailing new finding that these normalized Hahn polynomials satisfy the second recursion relation 2.4a with

Dual Hahn
In this subsection, we introduce a somewhat generalized version of the standard monic dual Hahn polynomials.These generalized monic dual Hahn polynomials p n x; γ, δ, η see 9 , and note the replacement of the integer parameter N with the arbitrary parameter η: hence the standard Hahn polynomials are only obtained for η N with N a positive integer and n 1, 2, . . ., N , p n x; γ, δ, η ≡ p n x; η , 3.92a are defined by the three-term recursion relations 3.1 with where

3.92c
The standard version of these polynomials reads see 9

3.93
Let us now proceed and provide two identifications of the parameter ν.

3.94
With this assignment, one can set, consistently with our previous treatment, x − m − 1 m c , 3.108a respectively,  Proposition 2.9 becomes applicable, entailing new findings the Diophantine factorizations

Factorizations
The following new relations among dual Hahn polynomials are implied by Proposition 2.3 with Corollary 2.5: 3.112d

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 p n x; α, λ see 9 , p n x; α, λ ≡ p n x; η , 3.113a 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 α and λ in terms of ν allowing the application of our machinery.We, therefore, consider the monic "shifted Meixner-Pollaczek" sMP polynomials Then, with the assignment entailing no restriction on the parameters α, λ, β, in as much as the two parameters c and C are arbitrary , one can set, consistently with our previous treatment,

3.120
It is, moreover, plain that with the assignment

3.122
And Corollary 2.8 entails even more general properties, such as new finding 3.123

Meixner
In this section some results of which were already reported in 5 , we focus on the monic Meixner polynomials p n x; β, c see 9 , p n x; β, c ≡ p n x; η , 3.124a defined by the three-term recursion relations 3.1 with

3.124b
The standard version of these polynomials reads see 9 :

3.125
We now identify the parameter ν via the assignment β −ν.

3.126
One can then set, consistently with our previous treatment, x − 2m 1 . 3.134

Krawtchouk
The monic Krawtchouk polynomials p n x; α, β see 9 : and note the notational replacement of the parameters p and N used there with the parameters α and β used here, implying that only when β N and n 1, 2, . . ., N with N a positive integer these polynomials p n x; α, β coincide with the standard Krawtchouk polynomials ,

3.135b
The standard version of these polynomials reads see 9  β x − 1 n .

3.153
And Corollary 2.8 entails even more general properties, such as the fact that the m Jacobi polynomials p n x; −m, β , m 1, . . ., n, feature x 1 as a zero of order m.

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 :

Laguerre
In this section most results of which were already reported in 5 , we focus on the monic Laguerre polynomials p n x; α see 9 ,

3.155b
The standard version of these polynomials reads see 9 −1 n n! p n x; α .3.156

Modified Charlier
In this subsection, we introduce and treat a modified version of the standard monic Charlier polynomials.The standard monic Charlier polynomials p n x; α see 9 ,

3.164b
The standard version of these polynomials reads see 9 C n x; α −α −n p n x; α .

3.165
However, we have not found any assignment of the parameters α in terms of ν allowing the application of our machinery.To nevertheless proceed, we introduce the class of monic "modified Charlier" polynomials p n x; α, β, γ characterized by the three-term recursion relation 3.

3.169
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 g ν n −n.

3.170
There does not seem to be any interesting results for the zeros of these polynomials.

− 1 x
, to take account of possible indeterminacies of b ν 0 .

in terms of integers. Corollary 2.5. If 2.9 holds-entailing 2.10 and 2.11 with (2.12)-and moreover the quantities a
following two results are immediate consequences of Proposition 2.3.
, complete factorizations can clearly be written for the polynomials p , 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 p x with m 1, . . ., n features the m zeros x 1, μ , 1, . . ., m, see 2.6b : 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 factorization 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.
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 by Proposition 2.3 and the properties implied by Corollary 2.4 become applicable with μ 1 β γ.These are new findings.And Proposition 2.7 becomes as well applicable, entailing new findings the Diophantine factorizations 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 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 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 with Note that this assignment entails no restriction on the 2 parameters of the Krawtchouk polynomials p n x; α, β .It is, moreover, plain that with the assignment ν n − 1, hence β n − 1 3.141 which is, however, incompatible with the definition of the standard Krawtchouk polynomials: β N and n 1, 2, . . ., N with N a positive integer , the factorizations implied by Proposition 2.3, and the properties implied by Corollary 2.4, become applicable with μ −1.These are new findings.And Proposition 2.7 becomes as well applicable, entailing new finding the Diophantine factorization p n x; α, n − 1 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 by Proposition 2.3, and the properties implied by Corollary 2.4, become applicable with μ 0. 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 factorization p n x; −n,