Terminating Basic Hypergeometric Representations and Transformations for the Askey–Wilson Polynomials

In this survey paper, we exhaustively explore the terminating basic hypergeometric representations of the Askey–Wilson polynomials and the corresponding terminating basic hypergeometric transformations that these polynomials satisfy.


Introduction
This paper is a study in q-calculus (typically taken with |q| < 1) directly connected to properties of the Askey-Wilson polynomials p n (x; a|q) [1, §14.1] which are at the very top of the q-Askey scheme (see e.g., [1,Chapter 14]) and whose definition can be found in Theorem 3.1 below.The Askey-Wilson polynomials are symmetric with respect to its four free parameters, that is, they remain unchanged upon interchange of any two of the four free parameters.
The Askey-Wilson polynomials can be defined in terms of terminating basic hypergeometric orthogonal polynomials (see, e.g., (2.11)), which in turn are defined in terms of a sum of products of q-Pochhammer symbols.Using the properties of q-Pochhammer symbols, it is straightforward to replace q → 1/q in the complex plane in order to obtain an extension of these polynomials with |q| > 1.One often refers to polynomials obtained as such as q −1 or 1/q polynomials.Since these algebraic factors are difficult to search on in the literature, we refer to this extension specifically as the q-inverse Askey-Wilson polynomials.
It should however be noted that while the Askey-Wilson polynomials represent an infinite-family of orthogonal polynomials (n ∈ N 0 ), orthogonal with respect to a weight function on [−1, 1] [1, (14. 1.2)] (which gives restrictions on the values of the free parameters), the q-inverse Askey-Wilson polynomials represent a finite-family of basic hypergeometric orthogonal polynomials (n ∈ {0, . . ., N}, N ∈ N 0 ) (see e.g., [2]).Not only that, but one also knows that the q-inverse Askey-Wilson polynomials are simply a scaled version of the Askey-Wilson polynomials with their parameters replaced by their reciprocals (see Remark 3.12, below).
In the sequel, from the basic hypergeometric representations of the Askey-Wilson polynomials, we derive basic hypergeometric representations for the q-inverse Askey-Wilson polynomials.From these two families of basic hypergeometric polynomials, one can easily derive transformation formulae for the terminating basic hypergeometric functions which appear as representations of these polynomials.
The main focus of this survey paper will be to exhaustively describe the transformation identities for the terminating basic hypergeometric functions which appear as representations for these polynomials.Some of these transformation identities are well-known in the literature, but we also give the transformation identities for these basic hypergeometric functions which are obtained by the symmetry of the polynomials under parameter interchange, and under the map θ → −θ, for x = cos θ.

Preliminaries
We adopt the following set notations: N 0 := {0} ∪ N = {0, 1, 2, 3, . ..}, and we use the sets Z, R, C which represent the integers, real numbers and complex numbers respectively, C * := C \ {0}.We also adopt the following notation and conventions.Let a 13 := a 1 a 3 , a 23 := a 2 a 3 , a 123 := a 1 a 2 a 3 , a 1234 := a 1 a 2 a 3 a 4 , etc. Consider a sequence of complex numbers {a k }, k ∈ N 0 .We adopt the following sum and product notations Furthermore, let s, r ∈ N 0 , with s < r.Then we assume that the the empty sum vanishes and the empty product is unity, namely

Definition 2.1. Throughout this paper we adopt the following conventions for succinctly writing elements of lists.
To indicate sequential positive and negative elements, we write ±a := {a, −a}.
In the same vein, consider a finite sequence f s ∈ C with s ∈ S ⊂ N.Then, the notation { f s } represents the sequence of all complex numbers f s such that s ∈ S. Furthermore, consider some p ∈ S, then the notation { f s } s =p represents the sequence of all complex numbers f s such that s ∈ S \{p}.Also, for the empty list, n = 0, we take {a 1 , . . ., a n } := ∅.
In order to obtain our derived identities, we rely on properties of the q-Pochhammer symbol (q-shifted factorial).For any n ∈ N 0 , a, q ∈ C, the Pochhammer symbol, and q-Pochhammer symbols are defined as (2.1) One may also define where |q| < 1.Furthermore, define where aq b ∈ Ω q .We will also use the common notational product conventions The following properties for the q-Pochhammer symbol can be found in (a; q) n+k = (a; q) k (aq k ; q) n = (a; q) n (aq n ; q) k , (2.4) (a 2 ; q 2 ) n = (±a; q) n , ( (a; q) 2n = (a, aq; q 2 ) n = (± √ a, ± √ qa; q) n . (2.8) Observe that by using (2.1) and (2.7), one obtains (2.9) The basic hypergeometric series, which we will often use, is defined for q, z ∈ C * such that |q|, |z| < 1, s, r ∈ N 0 , b j ∈ Ω q , j = 1, ..., s, as [1, (1.10.1)] Note that we refer to a basic hypergeometric series as ℓ-balanced A basic hypergeometric series r+1 φ r is well-poised if the parameters satisfy the relations It is very-well poised if in addition, {a 2 , a 3 } = ±q √ a 1 .
Similarly for terminating basic hypergeometric series which appear in basic hypergeometric orthogonal polynomials, one has , . . ., qb a r+1 ∈ Ω q .When the very-well poised basic hypergeometric series is terminating then one has r+1 W r b; q −n , a 5 , . . . ,a r ; q, z = r+1 φ r b, ±q √ b, q −n , a 5 , . . ., a r ± √ b, q n+1 b,  Some classical transformations for basic hypergeometric series which we will use include Watson's q-analog of Whipple's theorem which relates a terminating balanced 4 φ 3 to a terminating very-well poised 8 W 7 (cf.[4, (17.9.15)]) 4 φ 3 q −n , a, b, c d, e, f ; q, q = de ab , de ac ; q de a , de abc ; q where qabc = de f .
In [3, Exercise 1.4ii], one finds the inversion formula for terminating basic hypergeometric series. (2.17) Proof.Take r = s, in (2.16) and using the definition (2.10) completes the proof.
Note that in Corollary 2.3 if the terminating basic hypergeometric series on the left-hand side is balanced then the argument of the terminating basic hypergeometric series on the right-hand side is q 2 /z.
Applying Corollary 2.3 to the definition of r+1 W r , we obtain the following result for a terminating very-well poised basic hypergeometric series r+1 W r .
∈ Ω n q , k = 4, . . ., r.Then one has the following transformation formula for a very-well poised terminating basic hypergeometric series: Proof.Use Corollary 2.3 and (2.13).
An interesting and useful consequence of this formula is the r = 7 special case, (2.18) Note that in the case when the one obtains an 8 W 7 from a balanced 4 φ 3 using (2.15), then Another equality we can use is the following connecting relation between basic hypergeometric series on q, and on q −1 : In order to understand the procedure for obtaining the q-inverse analogues of the basic hypergeometric orthogonal polynomials studied in this manuscript, let's consider a special case in detail.Let f r (q) := f r (q; z(q); a(q), b(q)) be defined as f r (q) := g r (q) r+1 φ r q −n , a(q) b(q) ; q, z(q) , (2.20) where a(q) := {a 1 (q), . . ., a r (q)} b(q) := {b 1 (q), . . ., b r (q)}    , which will suffice for instance, for the study of the terminating basic hypergeometric representations for the Askey-Wilson polynomials.In order to obtain the corresponding q-inverse hypergeometric representations of f r (q), one only needs to consider the corresponding q-inverted function: Define a multiplier function g r (q) := g r (q; z(q); a(q); b(q)) which is not of basic hypergeometric type (some multiplicative combination of powers and q-Pochhammer symbols), and z(q) := z(q; a(q); b(q)).Then, define f r (q) as in (2.20), and one has 3) repeatedly with the definition (2.10), in (2.21), obtains the q-inverted terminating representation (2.22) which corresponds to the original terminating basic hypergeometric representation (2.20).This completes the proof.
We will obtain new transformations for basic hypergeometric orthogonal polynomials by taking advantage of the following remark.
Remark 2.6.Since x = cos θ is an even function of θ, all polynomials in cos θ will be invariant under the map θ → −θ.
Remark 2.7.Observe in the following discussion we will often be referring to a collection of constants a, b, c, d, e, f .
In such cases, which will be clear from context, then the constant e should not be confused with Euler's number, the base of the natural logarithm, i.e., log e = 1.

The Askey-Wilson polynomials
Define the sets 4 One may obtain alternative nonterminating representations of the Askey-Wilson polynomials using [3, (2.10.7)],namely 4 φ 3 q −n , q n−1 a 1234 , a p e ±iθ {a ps } s =p ; q, q = q 1−n e 2iθ , q 1−n a tu , q 2−n e iθ a prt , q 2−n e iθ a pru ; q ∞ q 1−n e iθ a t , q 1−n e iθ a u , q 2−n e 2iθ a pr , However, these nonterminating representations will not be further discussed in this paper.
Different series representations are useful for obtaining different properties and formulae for these polynomials.So it is very useful to have at hand an exhaustive list.The discussion contained in this section is an attempt to summarize, in an in-depth manner, an exhaustive description of the representation and transformation properties of the terminating 4 φ 3 and 8 W 7 basic hypergeometric representations of the Askey-Wilson polynomials.

The Askey-Wilson polynomial representations
Theorem 3.1.Let n ∈ N 0 , p, s, r, t, u ∈ 4, p, r, t, u distinct and fixed, q ∈ C * such that |q| = 1.Then, the Askey-Wilson polynomials have the following terminating basic hypergeometric series representations given by: p n (x; a|q):= a −n p {a ps } s =p ; q n 4 φ 3 q −n , q n−1 a 1234 , a p e ±iθ {a ps } s =p ; q, q (3.3) a 1234 , = e inθ a pr , a t e −iθ , a u e −iθ ; q n 4 φ 3   q −n , a p e iθ , a r e iθ , q 1−n a tu a pr , = e inθ a 1234 q ; q 2n a s e −iθ s =p , a 1234 e −iθ qa p ; q n a 1234 q ; q n a 1234 e −iθ qa p ; = e inθ a p e −iθ , { a 1234 a ps } s =p ; q n a 1234 e iθ a p ; q n 8 W 7 a 1234 e iθ qa p ; q −n , {a s e iθ } s =p , q n−1 a 1234 ; q, qe −iθ a p ( = a −n p a pt , a pu , a r e ±iθ ; q n a r a p ; q n 8 W 7 q −n a p a r ; q −n , q 1−n a rt , q 1−n a ru , a p e ±iθ ; q, q n a tu (3.8) = e inθ {a s e −iθ }; q n e −2iθ ; q n 8 W 7 q −n e 2iθ ; q −n , {a s e iθ }; q, ) to (3.5), (3.6), (3.7), (3.9) simply takes θ → −θ, and applying it to (3.8) interchanges a p and a r .Mapping θ → −θ may give additional representations, however those are omitted.
The Askey-Wilson polynomials are symmetric in its four parameters, the 8 W 7 representation in which this symmetry is evident demonstrates this symmetry.On the other hand, the polynomial nature of the Askey-Wilson polynomials is not clearly evident from the 8 W 7 representation.In the first 4 φ 3 representation, the polynomial nature of evident.

Terminating 4-parameter symmetric interchange transformations
The evidence that the first (and second) 8 W 7 in Corollary 3.3 is symmetric in the variables c, d, e, f is clear.Therefore, all of the formulas in this corollary are invariant under the interchange of any two of those variables.This is true whether the symmetry between those variables is evident in the corresponding mathematical expression or not.Perhaps, the most famous parameter interchange transformation of this sort is Sears' balanced 4 φ 3 transformations [4, (17.9.14)] which demonstrate the invariance (and provide specific transformation formulas) of the Askey-Wilson polynomials under parameter interchange.Other interesting parameter interchange transformations of this type can be obtained, such as by (3.12) with c ↔ d (preserves the argument), c ↔ e, c ↔ f , d ↔ e, d ↔ f interchanged (the invariance under the interchange e ↔ f is evident).Furthermore, when the symmetry within a set of variables is evident in the transformation corollaries presented below, then due to this symmetry, non-trivial transformation formulas can be obtained by equating the two expressions with certain variables interchanged.
In this subsection we present the entirety of all of the parameter interchange transformations for terminating basic hypergeometric transformations which arise from the Askey-Wilson polynomials.
Corollary 3.5.Let n ∈ N 0 , b, c, d, e, f , q ∈ C * , such that |q| = 1.Then, one has the following parameter interchange transformations for a terminating 8 W 7 : Corollary 3.8.Let n ∈ N 0 , b, c, d, e, f , q ∈ C * , such that |q| = 1.Then, one has the following parameter interchange transformations for a terminating 4 φ 3 : Corollary 3.10.Let n ∈ N 0 , b, c, d, e, f , q ∈ C * , such that |q| = 1.Then, one has the following parameter interchange transformations for a terminating 4 φ 3 : The q-inverse Askey-Wilson polynomials are simply a scaled version of the Askey-Wilson polynomials with the free parameters a k replaced by their reciprocals a −1 k .We demonstrate this in the following remark.
Remark 3.12.Let p n (θ; a 1 , a 2 , a 3 , a 4 |q) := p n (x; a|q), where x = cos θ, be any representation of the Askey-Wilson polynomials.Then the q-inverse Askey-Wilson polynomials p n (x; a|q −1 ) are given by p n (θ; a 1 , a 2 , a 3 , a 4 |q −1 ) = q −3( n 2 ) (−a 1234 ) n p n −θ; a −1 1 , a −1 2 , a −1 3 , a −1 where the second equality follows from Remark 2.6.So aside from a specific normalization, as is well-known, the q-inverse Askey-Wilson polynomials are the Askey-Wilson polynomials with the parameters taken to be their reciprocals.Note that this will not be the case for the symmetric subfamilies of the Askey-Wilson polynomials.
Nonetheless, we give in the following corollary the terminating basic hypergeometric representations of these polynomials.
and x = cos θ ∈ [−1, 1].The Askey-Wilson polynomials p n (x; a|q) are a family of polynomials symmetric in four free parameters.These polynomials have a long and in-depth history and their properties have been studied in detail.The basic hypergeometric series representation of the Askey-Wilson polynomials fall into four main categories: (1) terminating 4 φ 3 representations; (2) terminating 8 W 7 representations; (3) nonterminating 8 W 7 representations; and (4) nonterminating 4 φ 3 representations.

Remark 3 . 4 .
Notice that in Corollary 3.3, our order of the representations begins with the principal 8 W 7 representation in which the symmetry in the parameters c, d, e, f is evident and ends with the representation corresponding to the classical 4 φ 3 basic hypergeometric representation of the Askey-Wilson polynomials (3.3).On the other hand, in Theorem 3.1, we have reversed the order of the corresponding representations.The reason why we have used the ordering as such is because the 4 φ 3 representation of the Askey-Wilson polynomials (3.3) is Applying Theorem 2.5 to the basic hypergeometric representations of the Askey-Wilson polynomials (3.3)-(3.9)produces the inverted basic hypergeometric representations (3.45)-(3.51).This completes the proof.Author Contributions: H.S.C., R.S.C.-S.and L.G. conceived the mathematics; H.S.C., R.S.C.-S.and L.G. wrote the paper.