Generalizations of generating functions for basic hypergeometric orthogonal polynomials

We derive generalized generating functions for basic hypergeometric orthogonal polynomials by applying connection relations with one free parameter to them. In particular, we generalize generating functions for the Askey-Wilson, continuous $q$-ultraspherical/Rogers, little $q$-Laguerre/Wall, and $q$-Laguerre polynomials. Depending on what type of orthogonality these polynomials satisfy, we derive corresponding definite integrals, infinite series, bilateral infinite series, and $q$-integrals.


Introduction
In this paper we work with generating functions and connection problems associated with different families of q-hypergeometric orthogonal polynomials.
On one hand a generating function for a sequence of functions (f n ) is a series of the form for some suitable multipliers (λ n ).Moreover, it is usual that G(x, t) is a hypergeometric function in one or several variables, often written as a special case.
Of the many uses of generating functions for orthogonal polynomials, the one that is most commonly applied is Darboux's method [16,Chapter 9].This is used to find asymptotic properties of orthogonal polynomials.These asymptotic properties are in turn, essential in determining orthogonal polynomial orthogonality measures.
An example of an universal application of generating functions is how to obtain the well known Shannon entropy H n (P ) [20] of some distribution P using information theory.Define the information generating function by where (p i ) is a complete probability distribution, i ∈ N , N being a discrete sample space, and t ∈ R or t ∈ C. The first derivative of I at a point t = 1, gives the negative Shannon entropy of the corresponding probability distribution, namely − dI(t) dt t=1 = − i∈N p i log p i =: H n (P ).
Observe that this technique works equally well for discrete and continuous distributions.Moreover, the information generating function summarizes those aspects of the distribution which are invariant under a measure preserving the rearrangements of the probability space.
On the other hand, obtaining connection relations for arbitrary orthogonal polynomials in terms of a series of hypergometric orthogonal polynomials with fixed parameters, is a matter of great interest.Problems such as these have only been solved in relatively few cases [2,3,4,5,15]; although the most general connection problem for Askey-Wilson polynomials with four free parameters is known in closed form [8]. Connection problems are also relevant to the deeper problem of linearization [1], namely to obtain orthogonal polynomial expansion coefficients for a product of two orthogonal polynomials.Usually, the determination of the expansion coefficients in these particular cases require a deep knowledge of the special functions involved, and at times, ingenious induction arguments based on three-term recurrence relations which the orthogonal polynomials satisfy [1].
One of the reasons for the increased interest in connection and linearization problems are applications of such problems in mathematics and physics.For example, Gasper in his paper [15] motivated the connection and linearization problem in the framework of positivity.In fact, the Bieberbach conjecture (|a n | ≤ n) for analytic and univalent functions of the form f (z) = z + n≥2 a n z n in |z| < 1, was solved by Louis de Branges [14] in 1985 using an inequality proved in Askey & Gasper (1976) [7] using the framework of positivity.In fact, they proved that where (a) n is the Pochhammer symbol, 3 F 2 is a generalized hypergeometric function, and P (α,β) n denotes the classical Jacobi polynomial.Indeed, the computation of connection coefficients is often a challenge.In this manuscript we describe a nice way to obtain such coefficients by using Rodrigues-type formulas associated with some q-polynomial families.Note that even though connection coefficients for Askey-Wilson polynomials with one free parameter are well-known, and as well, the limiting relations with other families are also known, obtaining simple expressions for the resulting connection coefficients is often non-trivial.
As far as we are aware, the connection made in this paper between basic hypergeometric orthogonal polynomial generating functions and connection relations for these polynomials is new.In the context of generalized hypergeometric orthgononal polynomials, Cohl (2013) [11] (see (2.1) therein), developed a series rearrangement technique which produces a generalization of the generating function for Gegenbauer polynomials.We have since demonstrated that this technique is valid for a larger class of generalized hypergeometric orthogonal polynomials.For instance, in Cohl (2013) [10], we applied this same technique to Jacobi polynomials and in Cohl, MacKenzie & Volkmer (2013) [13], we extended this technique to many generating functions for Jacobi, Gegenbauer, Laguerre, and Wilson polynomials.
The series rearrangement technique combines a connection relation with a generating function, resulting in a series with multiple sums.The order of summations are then rearranged and the result often simplifies to produce a generalized (or basic) generating function whose coefficients are given in terms of generalized hypergeometric functions.This technique is especially productive when using connection relations with one free parameter, since the relation is most often a product of gamma functions.One important class of hypergeometric orthogonal polynomial generating functions which does not seem amenable to our series rearrangement technique are bilinear generating functions.The existence of an extra orthogonal polynomial in the generating function, produces multiple summation expressions via the introduction of connection relations for one or both of the polynomials with the sums being formidable to evaluate in closed form.
In this paper, we apply this technique to generalize generating functions for basic hypergeometric orthogonal polynomials in the q-analog of the Askey scheme [18,Chapter 14].These are the Askey-Wilson (Section 3), Rogers/continuous q-ultraspherical polynomials (Section 4), little q-Laguerre polynomials (Setion 5), and q-Laguerre (Section 6) polynomials.In Section 7, we have also computed new definite integrals, infinite series, and Jackson integrals (hereafter q-integrals) corresponding to our generalized generating function expansions using orthogonality for the studied basic hypergeometric orthogonal polynomials.
The first identity follows using (1) and ( 8), and the second identity follows using (1), replacing a → a 2 , q → q 2 , and splitting every term into complex conjugate pairs.Note that a similar result is available for all roots of unity.We have also taken advantage of the q-binomial theorem [18, (1.11.1)] where we have used (2).The basic hypergeometric series, which we will often use, is defined as [18, (1.10.1)] Let us prove some inequalities that we will later use.
Throughout this section and the next section on Rogers/continuous q-ultraspherical polynomials, x = cos θ.We derive generalizations of generating functions for the Askey-Wilson polynomials [18, (14.1.13-15)]using its connection relation with one free parameter [16, (7.6.8-9)] where Due to the symmetry in a, b, c, d of the Askey-Wilson polynomials, the generating functions [18, (14.1.13-15)]are all equivalent.Therefore, we will only consider [18, (14.1.13)].
By starting with generating functions for q-Laguerre polynomials [18, (14.21.14-16)], we derive generalizations of these generating functions using the connection relation for q-Laguerre polynomials (41).Note however that the generating function for q-Laguerre polynomials [18, (14.21.13)] remains unchanged when one applies the connection relation (41).
Proof .We start with the generating function for q-Laguerre polynomials found in Koekoek et al. (2010) [18, (14.21.14)] Using the connection relation (41) in (43), reversing the orders of the summations, shifting the n index by j, and using (3) through (12), obtains the desired result since n (x; q) (q α+1 ; q) n 1 φ 1 q α−β q α+n+1 ; q, tq n .(45) Proof .We start with the generating function for q-Laguerre polynomials found in Koekoek et al. (2010) [18, (14.21.15) Using the connection relation ( 41) in (46), reversing the orders of the summations, shifting the n index by j, and using (3) through (12), obtains the desired result since, again, (γ; q) n (tq α−β ) n (q α+1 ; q) n L (β) n (x; q) 2 φ 1 q α−β , γq n q α+n+1 ; q, t . (47) Proof .We start with the generating function for q-Laguerre polynomials found in Koekoek et al. (2010) [18, (14.21.15)] Using the connection relation ( 41) in (48), reversing the orders of the summations, shifting the n index by j, and using (3) through (12), obtains the desired result since 7 Definite integrals, infinite series, and q-integrals Consider a sequence of orthogonal polynomials (p k (x; α)) (over a domain A, with positive weight w(x; α)) associated with a linear functional u, where α is a set of fixed parameters.Define s k , k ∈ N 0 by In order to justify interchange between a generalized generating function via connection relation and an orthogonality relation for p k , we show that the double sum/integral converges in the L 2 -sense with respect to the weight w(x; α).This requires where Here, a n is the coefficient multiplying the orthogonal polynomial in the original generating function, and c k,n is the connection coefficient for p k (with appropriate set of parameters).Proof .Let n ∈ N 0 , then The result follows.
Given |p k (x; α)| ≤ K(k+1) σ γ k , with K, σ and γ constants independent of k, an orthogonality relation for p k , and |t| < 1/γ, one has Therefore one has confirmed (49), indicating that we are justified in reversing the order of our generalized sums and the orthogonality relations under the above assumptions, All polynomial families used throughout this paper fulfill such assumptions.See for instance (20), ( 24), (40), (44).Such inequalities depend entirely on the representation of the linear functional.In this section one has integral representations, infinite series, and representations in terms of the q-integral.In all the cases Lemma 16 can be applied and we are justified in interchanging the linear form and the infinite sum.
The property of orthogonality for Rogers/continuous q-ultraspherical polynomials found in Koekoek et al. (2010) [18, (3.10.16)] is given by where w : (−1, 1) → [0, ∞) is the weight function defined by We will use this orthogonality relation for proofs of the following definite integrals.
j;a,b p j (x; b|q), where c n,j;a,b = u β , p n (x; a|q)p j (x; b|q) u b , p j (x; b|q)p j (x; b|q) , with u b , p(x) := ∞ k=0

Lemma 16 .
Let u be a classical linear functional and let (p n (x)), n ∈ N 0 be the sequence of orthogonal polynomials associated with u.If |p n (x)| ≤ K(n + 1) σ γ n , with K, σ and γ constants independent of n, then |s n | ≤ K(n + 1) σ γ n |s 0 |.