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Askey-Wilson integral and its generalizations
Advances in Difference Equations volume 2014, Article number: 316 (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
We consider a sequence of nonnegative, integrable functions defined by the formula
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.
Section 2 is devoted to expanding functions in the series of the form
where denote q-Hermite polynomials, are the sequences of certain symmetric functions and finally are the values of the integrals
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.
q is a parameter. We will assume that unless otherwise stated. Let us define , , , with and
We will use the so-called q-Pochhammer symbol for ,
Often as well as will be abbreviated to and if that will not cause misunderstanding.
It is easy to notice that and that
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.
The Rogers-Szegö polynomials that are defined by the equality
for and . They will play an auxiliary role in the sequel.
In particular one shows (see, e.g., ) that the polynomials defined by
where , satisfy the following 3-term recurrence:
with , .
Remark 2 Notice that equals the n th Chebyshev polynomial of the second kind. More about these polynomials, one can find in, e.g., . To analyze the case , let us consider rescaled polynomials , i.e., . Then equation (1.4) takes a form
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 .
In the sequel the following identities discovered by Carlitz (see Exercise 12.3(b), (c) of ), true for ,
will enable us to show absolute and uniform convergence of practically all series considered in the sequel.
We have also the so-called linearization formula [, Eq. (13.1.25)], which can be dated back in fact to Rogers and Carlitz (see [, Eq. (10.11.10)] with or  for Rogers-Szegö polynomials), as follows:
that will be our basic tool.
We will use the following two formulae of Carlitz presented in  that concern properties of Rogers-Szegö polynomials. Let us define two sets of functions
where functions are polynomials that are defined by
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.
where . Notice that for and that from (1.5) it follows that series in (1.3) converges for . Notice also that from (1.5) it follows that
where denotes Kronecker’s delta, and
where . Notice that
following (1.16) since for .
Remark 3 We have
After proper rescaling and normalization similar to the one performed in Remark 2, the case leads to
2 Main results
Since in our approach symmetric polynomials will appear, let us introduce the following set of symmetric polynomials of k variables:
where we denoted by the so-called q-multinomial coefficient defined by .
Remark 4 Notice that .
Proof Obvious since . □
Proposition 2 Let , then:
For and ,(2.3)
If , then
Proof (i) Notice that
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.
So let us mention first the so-called big q-Hermite polynomials whose orthogonalizing measure has density for . The density of the orthogonalizing measure has a form (see [, Eq. (14.18.2)]) which can be written with the help of functions and , namely
The form of polynomials and their relation to q-Hermite polynomials is not important for our considerations. It can be found, e.g., in [, Eq. (14.26.1)] or in [, Eqs. (2.11), (2.12)]. So, for the sake of completeness, let us remark that from (2.4) it follows immediately that for , ,
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
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.
Lemma 1 For , , we have
Proof Following (2.6) and (1.14) we have
Now we use (1.6) and (1.1) and change the order of summation to get
As an immediate corollary of our result, we have
Remark 5 Let , and denote . Then
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.
Corollary 1 For , , we have
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 .
Notice that considering (2.8) with , and leads in view of Remark 5(i) to
Third in our sequence of families of polynomials that constitute AW scheme is the so-called continuous dual Hahn (C2H) polynomials. Again their relationship to other sets of polynomials is not important. From [, Eq. (14.3.2)] it follows that the density of measure that makes them orthogonal is given by the following formula:
We have the following lemma.
Proof Proof is shifted to Section 4. □
Remark 7 Notice that for ,
Proof Using (2.10) we have
Now it remains to change the index of summation in the second sum, use (2.2) and use the fact that . □
Corollary 2 For ,
Proof Elementary. □
The 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 are not important for our considerations. Recently a relatively rich study of these relationships was done in , hence it may serve as the reference. We need only the form of AW density. It is given, e.g., in [, Eq. (14.1.2)] and, after necessary adaptation to our notation, is presented below,
for , . Our main result concerns this density and is the following.
Theorem 1 For , ,
are symmetric functions of a, b, c, d.
Proof Proof is shifted to Section 4. □
As immediate corollaries we have the following fact.
Corollary 3 For ,
Proof Follows directly from (2.11). □
Remark 8 Notice that from (2.11) in fact the value of AW integral follows since we see that , which means that the integral
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 .
Let us denote , . We will assume that and that all parameters have absolute values less than 1. Let us denote
where functions and were defined by (1.16) and (1.14) respectively.
We remark following (1.15) and (1.17) that
for , and for .
We have the following general result.
Lemma 3 For every , there exist a symmetric function of and a sequence of symmetric in functions such that for , ,
Proof Let be the space of functions such that . Notice that this space is spanned by the polynomials . Visibly, under our assumptions and by (1.15), . Now notice that are coefficients of the Fourier expansion of the function in with respect to . Since
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. □
Using formula (1.9) we can write in the following way where are q-Hermite polynomials defined by (1.4). Functions and have the following interpretation:
We have the following easy proposition giving recursions that are satisfied by functions and .
Proposition 3 Let us define a new sequence of functions of n variables as follows:
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.
Remark 11 Notice also that following Proposition 3(i) we get for , ,
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).
Theorem 2 Under , , we have
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.
Proof Using Remark 3 we get
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
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
This result suggests considering the following functions:
where and are certain vectors of dimensions respectively n and m, find its integrals over and expansions similar to (3.2).
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?
Proof of Lemma 2 We have
Since obviously , 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 Applying (2.7) we get
Now we apply Carlitz formulae (1.10) and (1.11) to get
Thus it remains to show that for every ,
This fact is a result of the following calculations:
Now we introduce new indices of summation , . We have then
Proof of Proposition 3 Notice that for our formulae are true since we have . So and . Next notice that
where we understand . So by induction assumption the left-hand side of (3.2) is equal to
while the right-hand side to
We apply again (1.6) to get
Next we introduce new indices of summation and and get
Proof of Theorem 2 We use (2.12) and utilizing Remark 1 we get
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. □
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The author is very grateful to an unknown referee for pointing out additional references and just evaluation of the paper.
The author declares that he has no competing interests.