Askey--Wilson Integral and its Generalizations

We expand the Askey--Wilson (AW) density in a series of products of continuous $q-$Hermite polynomials times the density that makes these polynomials orthogonal. As a by-product we obtain the value of the AW integral as well as the values of integrals of $q-$Hermite polynomial times the AW density ($q-$Hermite moments of AW density). Our approach uses nice, old formulae of Carlitz and is general enough to venture a generalization. We prove that it is possible and pave the way how to do it. As a result we obtain system of recurrences that if solved successfully gives a sequence of generalized AW densities with more and more parameters.


Introduction and Preliminaries
1.1. Introduction. We consider sequence of nonnegative, integrable functions: g n : [−1, 1] −→ R + defined by the formula: where a (n) = (a 1 , . . . , a n ), functions f h and ϕ h defined by (1.16) and (1.14) denote in fact respectively the density of measure that makes the so called continuous q−Hermite polynomials orthogonal and the characteristic function of these polynomials calculated at points a j , j = 1, . . . , n. Naturally functions g n are symmetric with respect to vectors a (n) .
Our elementary but crucial for this paper observation is that examples of such functions are the densities of measures that make orthogonal respectively the so called continuous q−Hermite (q-Hermite, n = 0), big q−Hermite (bqH, n = 1), Al-Salam-Chihara (ASC, n = 2), continuous dual Hahn (C2H, n = 3), Askey-Wilson (AW, n = 4) polynomials. This observation makes functions g n important and what is more exciting allows possible generalization of both AW integral as well as AW polynomials, i.e. go beyond n = 4.
Similar observations were in fact made in [10] when commenting on formula 10.11.19. Hence one can say that we are developing certain idea of [10].
On the other hand by the observation that these functions are symmetric in variables a (n) we enter the fascinating world of symmetric functions.
The paper is organized as follows. Next Subsection 1.2 presents used notation and basic families of orthogonal polynomials that will appear in the sequel. We also present here important properties of these polynomials. Section 2 is devoted to expanding functions g n in the series of the form: where {h n } denote q-Hermite polynomials, T and symbol (q) j is explained at the beginning of next Subsection. We do this effectively for n = 0, . . . 4, obtaining known results in a new way. In Section 3 we show that defined above sequences do exist and present the way how to obtain them recursively. We are unable however to present nice compact forms of these sequences resembling those obtained for n ≤ 4, thus posing several open questions (see Subsection 3.2) and leaving the field to younger and more talented researchers.
For q = 0, the case important for the rapidly developing so called 'free probability', we give simple, compact form for 1 −1 g 5 x|a (5) , 0 dx (see Theorem 2, ii)) paving the way to conjecture the compact form of (3.3).
Tedious, uninteresting proofs are shifted to Section 4.
We will use the so called q−Pochhammer symbol for n ≥ 1 : with (a; q) 0 = 1. Often (a; q) n as well as (a 1 , a 2 , . . . , a k ; q) n will be abbreviated to (a) n and (a 1 , a 2 , . . . , a k ) n , if it will not cause misunderstanding.
It is easy to notice that (q) n = (1 − q) n [n] q ! and that We will need the following sets of orthogonal polynomials The Rogers-Szegö polynomials that are defined by the equality: for n ≥ 0 and w −1 (x|q) = 0. They will be playing an auxiliary rôle in the sequel.
In particular one shows (see e.g. [8]) that the polynomials defined by: where x = cos θ, satisfy the following 3-term recurrence: These polynomials are called continuous q−Hermite polynomials. A lot is known about their properties. For good reference see [8]. In particular we know that sup |x|≤1 |h n (x|q)| ≤ w n (1|q) .

Remark 2.
Notice that h n (x|0) equals to n − th Chebyshev polynomial of the second kind. More about these polynomials one can find in e.g. [10]. To analyze the case q = 1 let us consider rescaled polynomials h n i.e. H n (x|q) = h n x √ 1 − q/2|q / (1 − q) n/2 . Then equation (1.4) takes a form: which shows that H n (x|q) = H n (x), where {H n } denote the so called 'probabilistic' Hermite polynomials i.e. polynomials orthogonal with respect to the measure with density equal to exp −x 2 /2 / √ 2π. This observation suggests that although the case q = 1 lies within our interest it requires special approach. In fact it will be solved completely in Section 3. For now we will assume that |q| < 1.
In the sequel the following identities discovered by Carlitz (see Exercise 12.3(b) and 12.3(c) of [8]), true for |q| , |t| < 1 : will enable to show convergence of many series considered in the sequel. We have also the following so called 'linearization formula' ( [8], 13.1.25) which can be dated back in fact to Rogers and Carlitz (see [10], 10.11.10 with β = 0 or [16] for Rogers-Szegö polynomials): that will be our basic tool.
It is elementary to prove the following two properties of the polynomials µ n , hence we present them without the proof. Proposition 1.
To perform our calculations we will need also the following two functions. The generating function of the q−Hermite polynomials that is given by the formula below (see [7], 3.26.11): where v (x|t) = 1 − 2tx + t 2 . Notice that v (x|t) ≥ 0 for |x| ≤ 1 and that from (1.5) it follows that series in (1.3) converges for |t| < 1. Notice also that from (1.5) it follows that: The density of the measure with respect to which polynomials h n are orthogonal is given in e.g. [7], (3.26.2). Following it we have where δ mn denotes Kronecker's delta, and Remark 3. We have for |x| , |a| < 1.
After proper rescaling and normalization similar to the one performed in Remark 2, the case q = 1 leads to: for x, a ∈ R, as respectively the density of orthogonalizing measure and the characteristic function. For details see [11] or [4].

Main results
Since in our approach symmetric polynomials will appear let us introduce the following set of symmetric polynomials of k variables: Proof. Obvious since ii) for |t| < 1 and ∀j = 1, . . . , k n−m (a j+1 , . . . , a k |q) , iii) . Now the assertion is easy. ii) follows either direct calculation or i) and the properties of characteristic functions. iii) We use (2.1).
Recall (i.e. [8] or [7]) that there exist sets of orthogonal polynomials forming a part of the so called 'AW scheme' that are orthogonal with respect to measures with densities mentioned below. Although our main interest is in providing simple proof of the so called AW integral we will list related densities for better exposition and for indicating the ways of possible generalization of AW integrals and polynomials.
So let us mention first the so called big q−Hermite polynomials {h n (x|a, q)} n≥−1 whose orthogonalizing measure has density for |a| < 1. This density has a form (see [7] (3.18.2)) which can be written with the help of functions f h and ϕ h . Namely: The form of polynomials h n (x|a, q) and their relation to q−Hermite polynomials is not important for our considerations. It can be found e.g. in [7], (3.18.4) or in [13] , (2.11, 2.12). So for the sake of completeness let us remark that from (2.4) it follows immediately that for |x| ≤ 1, |a| < 1 Here and below, where we will present similar expansions convergence is almost uniform since all these expansions are in fact the Fourier series and that the Rademacher-Menshov theorem can be applied following (1.5).
Let us notice immediately that following (2.4) we have: Secondly let us mention the so called Al-Salam-Chihara polynomials {Q n (x|a, b, q)} n≥−1 that are orthogonal with respect to the measure that for |a| , |b| < 1 has the density of the form (compare [7], (3.8.2)) We have the following Lemma that illustrates our method as well as to will give a very simple proof of well known so called Poisson-Mehler formula as a corollary.
Proof. Following (2.6) and (1.14) we have : Now we use (1.6) and (1.1) and change the order of summation getting: As an immediate corollary of our result we have: Remark 5. Let a = ρe iη , b = ρe −iη and denote y = cos η. Then i) S n (a, b|q) = ρ n h n (y|q) , As a slightly more complicated corollary implied by Lemma 1 we have the following famous Poisson-Mehler (PM) expansion formula: Proof. We take a = ρe iη , b = ρe −iη and denote y = cos η. Now we use (2.6) and Remark 5, ii) to get left hand side multiplied by f h . Then we apply Lemma 1, and Remark 5, i) to get right hand side of our PM formula also multiplied by f h . Finally we cancel out f h which is positive on (−1, 1).  [1] or recently obtained very short in [3]. In fact the formula (2.8) can be dated back to Carlitz who in [17] formulated it for Rogers-Szegö polynomials. The one presented above seems to be one of the shortest, was obtained as a by-product and as already mentioned is almost the same as the one presented in [8].
Notice that considering (2.7) with a = ρe iη , b = ρe −iη and y = cos η leads in view of Remark 5, i) to a nice symmetric formula that appeared in [2] in probabilistic context. Its probabilistic interpretation was exploited further in [11].
Third in our sequence of families of polynomials that constitute AW scheme are the so called continuous dual Hahn (C2H) polynomials. Again their relationship to other sets of polynomials is not important. From [7], (3.3.2) it follows that the density of measure that makes them orthogonal is given by the following formula.
We have the following lemma.
Proof. Is shifted to Section 4.

Now it remains to change the index of summation in the second sum, use (2.2) and use the fact that
Corollary 2. For |a|, |b| , |c| < 1 : (a, b, c|q) .

Proof. Elementary.
Fourth family of polynomials that constitute AW scheme are the celebrated Askey-Wilson polynomials. Again their form and relationship to other families of polynomials of AW scheme is not important for our considerations. Recently a relatively rich study of these relationships was done in [13] hence it may serve as the reference. We need only the form of AW density. It is given e.g. in [7], (3.1.2) and after necessary adaptation to our notation is presented below: for |x| ≤ 1, |a| , |b| , |c| , |d| < 1. Our main result concerns this density and is the following: are symmetric functions of a, b, c, d.
Proof. is shifted to section 4.
As immediate corollaries we have the following fact. Proof. Follows directly from (2.10).
(2.13) is nothing else but the celebrated AW integral. Notice also that recently there appeared at least two papers [15], [14] where (2.13) was derived from much more advanced theorems.
Remark 9. Notice also that (2.12) allows calculation of all moments of AW density. This is so since one knows the form of polynomials h n . Moments of AW density were calculated by Corteel et. al. in 2010 in [5] using combinatorial means. For complex a, b, c, d but forming conjugate pairs this formula was also obtained independently about the same time. Namely it was done in [4] where also an elegant expansion of σ (4) n ρ 1 e iη , ρ 1 e −iη , ρ 2 e iθ , ρ 2 e −iθ |q in terms of h n (y|q) and h n (z|q) , where cos η = y and cos θ = z was presented.

Generalization and open questions
3.1. Generalization. The presented above results allow the following generalization. The cases |q| < 1 and q = 1 will be treated separately. First let us consider |q| < 1.
Let us denote a (k) = (a 1 , . . . , a k ) , k = 0, 1, . . . . We will assume that |x| ≤ 1 and that all parameters a i have absolute values less that 1. Let us denote where functions f h and ϕ h were defined by (1.16) and (1.14) respectively.
We have the following general result. such that for |a k | < 1, k = 1, . . . , n : Notice that this space is spanned by the polynomials {h j (x|q)} j≥0 . Visibly, under our assumptions and by (1.15), n j=1 ϕ h (x|a i , q) ∈ G. Now notice that T A n a (n) , q is the value of are symmetric follows the observations that n j=1 ϕ h (x|a i , q) is symmetric.
Using formula (1.9) we can write g n in the following way where h j are q−Hermite polynomials defined by (1.4). Functions A n a (n) , q and T (n) j a (n) , q j≥0 have the following interpretation: for n, j ≥ 0. We have the following easy Proposition giving recursions that are satisfied by functions A n and T Then i) Proof. Proof is shifted to section 4.

Remark 10. The integral
1 −1 g n x|a (n) , q dx has been calculated in [9] (see also theorem 15.3.1 in [8]) by combinatorial methods. Obtained formula is however very complicated. Besides above mentioned Theorem 15.3.1 of [8] does not provide expansion (3.1) which is automatically obtained in our approach.

a 4 |q)
For q = 0 the calculations presented in (3.3) can be carried out completely and the concise form can be obtained. This is possible due to the following simplified form of (2.11).
For q = 1 the problem of finding sequences A n a (n) |1 and T (n) j a (n) , 1 j≥0 can be solved completely and trivially. Namely we have: Proof. Using Remark 3 we get:

Questions.
• What are the compact forms of functions T (n) j a (n) , q j≥0,n≥5 and A n a (n) , q n≥5 ? • What is the compact form of these functions for q = 0 (free probability case) ? • Following formula for 1 −1 g 5 x|a (5) , 0 dx given in assertion ii) of Theorem 2 is it true that: Notice that for a 5 = 0 it would reduce to AW integral. • It would be valuable to get values A n a (n) , q for n = 8, 12 and so on for complex values of parameters a (n) but forming conjugate pairs. It would be also fascinating to find polynomials that would be orthogonalized by so obtained densities. This problem follows the probabilistic interpretation of Askey-Wilson density rescaled, with complex parameters. Such interpretation for finite Markov chains of length at least 3 was presented in [4], [13]. Let {X 1 , X 2 , X 3 } denote this finite Markov chain. Then recall that then AW density can be interpreted as the conditional density of X 2 |X 1 , X 3 .
It would be exciting to find out if for say n = 8 similar probabilistic interpretation could be established. That is if we could have defined 5−dimensional random vector (X 1 , . . . , X 5 ) with normalized function g 8 x|a (8) , q as the conditional density X 3 |X 1 , X 2 , X 4 , X 5 . Note that then the chain (X 1 , . . . , X 5 ) could not be Markov.

3.2.2.
Unsolved related problems and direction of further research. In [10] we find Theorem 10.8.2 which is due Gasper and Rahman (1990) and which can be stated in our notation. For max 1 ≤j≤5 |a j | < 1, |q| < 1 we have: This result suggests considering the following functions where a (n) and b (m) are certain vectors of dimensions respectively n and m, find its integrals over [−1, 1] and expansions similar to (3.1).

Proofs
Proof of Lemma 2. We have k,n,m≥0 m+j−k (a, b|q) .
Since obviously S (2) n (a, b|q) = a n w n (b/a|q) we get: Now we apply formula (1.7) and get: Now we use (1.12)and Proposition 2, ii) and get: Proof of Theorem 1. We have k,n,m,j≥0 and further k,n,m,j≥0 (ac) j (q) j w k+j (b/a|q) w j+n−k (d/c|q) a m 1 and A 1 (a 1 , q) = 1. Next notice that: g n+1 x|a (n+1) , q = g n x|a (n) , q ϕ h (x|a n+1 , q) , where we understand a (n+1) = (a 1 , . . . , a n , a n+1 ) . So by induction assumption the left hand side of (3.1) is equal to: while the right hand side to A n a (n) , q f h (x|q) j,k≥0 T (n) j a (n) , q a k n+1 (q) j (q) k h j (x|q) h k (x|q) .