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 .
Similar observations were made in fact in  when commenting on formula (10.11.19). Hence one can say that we are developing a certain idea of .
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 , .
These polynomials are the so-called continuous q-Hermite polynomials. A lot is known about their properties. For good reference, see [1, 4] or . In particular we know that for ,
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
defined for , and n, m being nonnegative integers. Carlitz proved ([, Eq. (3.2)], after correcting an obvious misprint) that
where functions are polynomials that are defined by
and that ([, Eq. (1.4)], case also given in [, Exercise 12.3(d)])
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.
The generating function of the q-Hermite polynomials that is given by the formula below (see [, Eq. (14.26.1)]):
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
The density of the measure with respect to which polynomials are orthogonal is given in, e.g., [, Eq. (14.26.2)]. Following it we have
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
for , as respectively the density of orthogonalizing measure and the generating function. For details, see  or .