Askey-Wilson integral and its generalizations
© Szabłowski; licensee Springer. 2014
Received: 25 August 2014
Accepted: 26 November 2014
Published: 15 December 2014
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 a system of recurrences that if solved successfully gives a sequence of generalized AW densities with more and more parameters.
MSC: 33D45, 05A30, 05E05.
1 Introduction and preliminaries
where , functions and defined by (1.16) and (1.14) denote respectively the density of measure that makes the so-called continuous q-Hermite polynomials orthogonal and the generating function of these polynomials calculated at points , . Naturally functions are symmetric with respect to vectors .
Our elementary but crucial for this paper observation is that examples of such functions are proportional to the densities of measures that make orthogonal respectively the so-called continuous q-Hermite (q-Hermite, [, Eq. (14.26.2)]), big q-Hermite (bqH, [, Eq. (14.18.2)]), Al-Salam-Chihara (ASC, [, Eq. (14.8.2)]), continuous dual Hahn (C2H, [, Eq. (14.3.2)]), Askey-Wilson (AW, [, Eq. (14.1.2)]) polynomials. This observation makes functions important and, what is more exciting, allows possible generalization of both AW integral and AW polynomials, i.e., go beyond .
Let us notice that this is a second attempt to generalize AW polynomials. The other one was made in  by generalizing certain properties of generating functions of q-Hermite, bqH, ASC, C2H and AW polynomials.
On the other hand, by the observation that these functions are symmetric in variables we enter the fascinating world of symmetric functions.
The paper is organized as follows. Next Section 1.2 presents notation that will be used and basic families of orthogonal polynomials that will appear in the sequel. We also present here important properties of these polynomials.
and symbol is explained at the beginning of the next subsection.
We do this effectively for , obtaining known results in a new way. In Section 3 we show that sequences defined above do exist, and we present the way how to obtain them recursively. We are unable, however, to present nice compact forms of these sequences resembling those obtained for , thus posing several open questions (see Section 3.2) and leaving the field to younger and more talented researchers.
The partially legible, although not very compact, form was obtained for (see (3.4)).
For , the case important for the rapidly developing so-called free probability, we give a simple, compact form for (see Theorem 2(ii)) paving the way to conjecture the compact form of (3.4).
Tedious proofs are shifted to Section 4.
Often as well as will be abbreviated to and if that will not cause misunderstanding.
The case will be considered only when it might make sense and will be understood as the limit .
Remark 1 Notice that , , , and , , for ,
We will need the following sets of polynomials.
for and . They will play an auxiliary role in the sequel.
with , .
which shows that , where denote the so-called probabilistic Hermite polynomials, i.e., polynomials orthogonal with respect to the measure with density equal to . This observation suggests that although the case lies within our interest, it requires special approach. In fact it will be solved completely in Section 3. For now we will assume that .
will enable us to show absolute and uniform convergence of practically all series considered in the sequel.
that will be our basic tool.
It is elementary to prove the following two properties of the polynomials , hence we present them without proof.
To perform our calculations, we will also need the following two functions.
following (1.16) since for .
2 Main results
where we denoted by the so-called q-multinomial coefficient defined by .
Remark 4 Notice that .
Proof Obvious since . □
- (ii)For and ,(2.3)
Secondly recall that . Now the assertion is easy. (ii) follows either from direct calculation or (i) and the properties of generating functions. (iii) We use (2.1). □
Recall (i.e.,  or ) 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 a 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.
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).
We have the following lemma that illustrates our method and we will give a very simple proof of the well-known Poisson-Mehler formula as a corollary.
Proof (i) is an immediate consequence of (1.3). (ii) We have . □
As a slightly more complicated corollary implied by Lemma 1, we have the following famous Poisson-Mehler (PM) expansion formula.
Proof We take , and denote . Now we use (2.6) and Remark 5(ii) to get the left-hand side multiplied by . Then we apply Lemma 1 and Remark 5(i) to get the right-hand side of our PM formula also multiplied by . Finally we cancel out which is positive on . □
Remark 6 The calculations we have performed while proving Lemma 1 are very much like those performed in  while proving Theorem 13.1.6 concerning the Poisson kernel (or Poisson-Mehler) formula. There exist many proofs of the PM formula (see, e.g.,  or a recently obtained very short one in ). In fact formula (2.9) can be dated back to Carlitz who in  formulated it for Rogers-Szegö polynomials. The one presented above, which seems to be one of the shortest, was obtained as a by-product and, as it has already been mentioned, is almost the same as the one presented in .
We have the following lemma.
Proof 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 . □
Proof Elementary. □
for , . Our main result concerns this density and is the following.
are symmetric functions of a, b, c, d.
Proof Proof is shifted to Section 4. □
As immediate corollaries we have the following fact.
Proof Follows directly from (2.11). □
Remark 9 Notice also that (2.13) allows calculation of all moments of AW density. This is so since one knows the form of polynomials . Moments of AW density were calculated by Corteel and Williams in 2010 in  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  where also an elegant expansion of in terms of and , where and , was presented.
3 Generalization and open questions
The results presented above allow the following generalization. The cases and will be treated separately. First let us consider .
where functions and were defined by (1.16) and (1.14) respectively.
for , and for .
We have the following general result.
is the value of . Inequality (3.3) follow properties of the Fourier expansion, more precisely Perseval’s identity. The fact that and are symmetric follows the observations that is symmetric. □
We have the following easy proposition giving recursions that are satisfied by functions and .
Proof Proof is shifted to Section 4. □
Remark 10 The integral was calculated in  (see also Theorem 15.3.1 in ) by combinatorial methods. The obtained formula is, however, very complicated. Besides, the above mentioned Theorem 15.3.1 of  does not provide expansion (3.2) which is automatically obtained in our approach.
For the calculations presented in (3.4) can be carried out completely and the concise form can be obtained. This is possible due to the following simplified form of (2.12).
where denote respectively first five elementary symmetric functions of the vector . That is, .
Proof Proof is shifted to Section 4. □
For , the problem of finding sequences and can be solved completely and trivially. Namely we have the following.
3.2 Unsolved problems and open questions
What are the compact forms of functions and ?
What are the compact forms of these functions for (free probability case)?
Following formula for given in assertion (ii) of Theorem 2 is it true that
Notice that for it would reduce to AW integral.
It would be valuable to get values for and so on for complex values of parameters but forming conjugate pairs. It would be also fascinating to find polynomials that would be orthogonalized by densities obtained in this way.
This problem follows the probabilistic interpretation of Askey-Wilson density rescaled with complex parameters. Such an interpretation of finite Markov chains of length at least 3 was presented in [9, 10]. Let denote this finite Markov chain. Recall that then AW density can be interpreted as the conditional density of , .
It would be exciting to find out if, for say , a similar probabilistic interpretation could be established. That is, if we could define five-dimensional random vector with normalized function as the conditional density , , , . Note that then the chain could not be Markov.
Similar questions apply to the case .
3.2.2 Unsolved related problems and direction of further research
- 1.In  we find Theorem 10.8.2 which is due to Gasper and Rahman (1990) and which can be stated in our notation. For , , we have
Recently a paper  on q-Laplace transform, where many analogies to ordinary case were indicated, has appeared. What would a q-Laplace transform of the distributions that were considered above be?
Now we use formula (2.3). Then we replace a by , b by and so on. Finally we use formulae (3.4) and (2.2) remembering that leads to our integral formula. □
The author is very grateful to an unknown referee for pointing out additional references and just evaluation of the paper.
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