Q-Hermite Polynomials and Classical Orthogonal Polynomials ^{∗}

Christian Berg and Mourad E. H. Ismail

February 17, 1995

Abstract

We use generating functions to express orthogonality relations in the form of q-beta integrals. The integrand of such a q-beta integral is then used as a weight function for a new set of orthogonal or biorthogonal functions. This method is applied to the continuous q-Hermite polynomials, the Al-Salam-Carlitz polynomials, and the polynomials of Szegő and leads naturally to the Al-Salam-Chihara polynomials then to the Askey-Wilson polynomials, the big q-Jacobi polynomials and the biorthogonal rational functions of Al-Salam and Verma, and some recent biorthogonal functions of Al-Salam and Ismail.

Running title: Classical Orthogonal Polynomials.

1990 Mathematics Subject Classification: Primary 33D45, Secondary 33A65, 44A60.

em Key words and phrases. Askey-Wilson polynomials, q-orthogonal polynomials, orthogonality relations, q-beta integrals, q-Hermite polynomials, biorthogonal rational functions.

1. Introduction and Preliminaries. The q-Hermite polynomials seem to be at the bottom of a hierarchy of the classical q-orthogonal polynomials, [6]. They contain no parameters, other than q, and one can get them as special or limiting cases of other orthogonal polynomials.

The purpose of this work is to show how one can systematically build the classical q-orthogonal polynomials from the q-Hermite polynomials using a simple procedure of attaching generating functions to measures.

Let {p_{n}(x)} be orthogonal polynomials with respect to a positive measure μ with moments of any order and infinite support such that ∫_{−∞}^{∞} p_{n}(x)p_{m}(x) dμ(x) = ζ_{n}δ_{m,n}. (1.1)

∗Research partially supported by NSF grant DMS 9203659

Assume that we know a generating function for {p_{n}(x)}, that is we have

∑_{n=0}^{∞} p_{n}(x)t^{n}/c_{n} = G(x, t), (1.2)

for a suitable numerical sequence of nonzero elements {c_{n}}. This implies that the orthogonality relation (1.1) is equivalent to

∫_{−∞}^{∞} G(x, t_{1})G(x, t_{2})dμ(x) = ∑_{0}^{∞} ζ_{n} \frac{(t_{1}t_{2})^{n}}{c_{n}^{2}} , (1.3)

provided that we can justify the interchange of integration and sums.

Our idea is to use

G(x, t_{1})G(x, t_{2}) dμ(x)

as a new measure, the total mass of which is given by (1.3), and then look for a system of functions (preferably polynomials) orthogonal or biorthogonal with respect to it. If such a system is found one can then repeat the process. The generating function in (1.2) is assumed to be an elementary function, that is a quotient of products of powers and infinite products. It it clear that we cannot indefinitely continue this process. The form of the generating function will become too complicated at a certain level, and the process will then terminate. The referee wondered whether there is a principal reason which forbids that nice explicit (bi)orthogonal systems can be found with respect to measures which are not elementary. We do not know the answer to this question especially since in the case of associated orthogonal polynomials [11], [19], [28] the weight function involves the reciprocal of the square of the absolute value of a transcendental function. Part of the difficulty is that we do not have direct proofs of the orthogonality of the associated polynomials.

If μ has compact support it will often be the case that (1.2) converges uniformly for x in the support and |t| sufficiently small. In this case the justification is obvious.

We mention the following general result with no assumptions about the support of μ. For 0 < ρ ≤ ∞ we denote by D(0, ρ) the set of z ∈ C with |z| < ρ.

Proposition 1.1 Assume that (1.1) holds and that the power series

∑_{n=0}^{∞} \frac{\sqrt{ζ_{n}}}{c_{n}} z^{n} (1.4)

has a radius of convergence ρ with 0 < ρ ≤ ∞.

(i) Then there is a μ-null set N ⊆ R such that (1.2) converges absolutely for |t| < ρ, x ∈ R\N. Furthermore (1.2) converges in L^{2}(μ) for |t| < ρ, and (1.3) holds for |t_{1}|, |t_{2}| < ρ.

(ii) If μ is indeterminate then (1.2) converges absolutely and uniformly on compact subsets of Ω = C × D(0, ρ), and G is holomorphic in Ω.

Proof. For 0 < r_{0} < r < ρ there exists C > 0 such that (\sqrt{ζ_{n}}/|c_{n}|)r^{n} ≤ C for n ≥ 0, and we find

∥ ∑_{n=0}^{N} |p_{n}(x)|\frac{r_{0}^{n}}{|c_{n}|}∥_{L^{2}(μ)} ≤ ∑_{n=0}^{N} \frac{\sqrt{ζ_{n}}}{|c_{n}|} r^{n}(\frac{r_{0}}{r})^{n} ≤ C ∑_{n=0}^{∞}(\frac{r_{0}}{r})^{n} < ∞,

which by the monotone convergence theorem implies that

∑_{n=0}^{∞} |p_{n}(x)|\frac{r_{0}^{n}}{|c_{n}|} ∈ L^{2}(μ),

and in particular the sum is finite for μ-almost all x. This implies that there is a μ-null set N ⊆ R such that ∑ p_{n}(x)(t^{n}/c_{n}) is absolutely convergent for |t| < ρ and x ∈ R \ N.

The series (1.2) can be considered as a power series with values in L^{2}(μ), and by assumption its radius of convergence is ρ. It follows that the series in (1.2) converges in L^{2}(μ) to some G(x, t) for |t| < ρ, and (1.3) is a consequence of Parseval’s formula.

If μ is indeterminate it is well known that ∑ |p_{n}(x)|^{2}/ζ_{n} converges uniformly on compact subsets of C, cf. [1], [26], and the assertion follows. □

In order to describe details of our work we will need to introduce some notations. There are three systems of q-Hermite polynomials. Two of them are orthogonal on compact subsets of the real line and the third is orthogonal on the unit circle. The two q-Hermite polynomials on the real line are the discrete q-Hermite polynomials {H_{n}(x : q)} [13] and the continuous q-Hermite polynomials {H_{n}(x|q)} of L. J. Rogers [8]. They are generated by

2xH_{n}(x|q) = H_{n+1}(x|q) + (1 − q^{n})H_{n−1}(x|q), (1.5)

xH_{n}(x : q) = H_{n+1}(x : q) + q^{n−1}(1 − q^{n})H_{n−1}(x : q), (1.6)

and the initial conditions

H_{0}(x|q) = H_{0}(x : q) = 1, H_{1}(x|q) = 2x, H_{1}(x : q) = x. (1.7)

We will describe the q-Hermite polynomials on the unit circle later in the Introduction. The discrete and continuous q-Hermite polynomials have generating functions

∑_{0}^{∞} \frac{H_{n}(x : q)}{(q; q)_{n}}t^{n} = \frac{(t, −t; q)_{∞}}{(xt; q)_{∞}} , (1.8)

and

∑_{0}^{∞} \frac{H_{n}(x|q)}{(q; q)_{n}}t^{n} = \frac{1}{(te^{iθ}, te^{−iθ}; q)_{∞}} , x = cos θ, (1.9)

respectively, where we used the notation in [16] for the q-shifted factorials

(a; q)_{0} := 1, (a; q)_{n} := ∏_{k=1}^{n}(1 − aq^{k−1}), n = 1, 2, ⋯, or ∞, (1.10)

and the multiple q-shifted factorials

(a_{1}, a_{2}, ⋯, a_{k}; q)_{n} := ∏_{j=1}^{k}(a_{j}; q)_{n}. (1.11)

A basic hypergeometric series is

(1.12) _{r}ϕ_{s} ( \begin{matrix}a_{1},…, a_{r} \\ b_{1},…, b_{s} \end{matrix} | q, z) = _{r}ϕ_{s}(a_{1},…, a_{r}; b_{1},…, b_{s}; q, z)
= ∑_{n=0}^{∞} \frac{(a_{1},…, a_{r}; q)_{n}}{(q, b_{1},…, b_{s}; q)_{n}} z^{n}((−1)^{n}q^{n(n−1)/2})^{s+1−r}.

In (1.9) e^{±iθ} is x ± \sqrt{x^{2} − 1} and the square root is chosen so that \sqrt{x^{2} − 1} ≈ x as x → ∞. This makes |e^{−iθ}| ≤ |e^{iθ}|. It is clear that the right hand sides of (1.8) and (1.9) are analytic functions of the complex variable t for |t| < 1/|x|, |t| < |e^{−iθ}|.

In Section 2 we apply the procedure outlined at the beginning of the Introduction to the continuous q-Hermite polynomials for |q| < 1 and we reach the Al-Salam-Chihara polynomials in the first step and the second step takes us to the Askey-Wilson polynomials. It is worth mentioning that the Askey-Wilson polynomials are the general classical orthogonal polynomials, [6]. As a byproduct we get a simple evaluation of the Askey-Wilson q-beta integral, [10]. This seems to be the end of the line in this direction. The case q > 1 will be studied in Section 5, see comments below. In Section 3 we apply the same procedure to the polynomials {U_{n}^{(a)}(x; q)} and {V_{n}^{(a)}(x; q)} of Al-Salam and Carlitz [2]. They are generated by the recurrences

U_{n+1}^{(a)}(x; q) = [x − (1 + a)q^{n}]U_{n}^{(a)}(x; q) + aq^{n−1}(1 − q^{n})U_{n−1}^{(a)}(x; q), n > 0, (1.13)

V_{n+1}^{(a)}(x; q) = [x − (1 + a)q^{−n}]V_{n}^{(a)}(x; q) − aq^{1−2n}(1 − q^{n})V_{n−1}^{(a)}(x; q), n > 0, (1.14)

and the initial conditions

U_{0}^{(a)}(x; q) = V_{0}^{(a)}(x; q) = 1, U_{1}^{(a)}(x; q) = V_{1}^{(a)}(x; q) = x − 1 − a, (1.15)

[2], [14]. It is clear that U_{n}^{(a)}(x; 1/q) = V_{n}^{(a)}(x; q), so there is no loss of generality in assuming 0 < q < 1 with appropriate restrictions on a. The U_{n}’s provide a one parameter extension of the discrete q-Hermite polynomials when 0 < q < 1 corresponding to a = −1. In Section 3 we show that our attachment procedure generates the big q-Jacobi polynomials from the U_{n}’s. The big q-Jacobi polynomials were studied by Andrews and Askey in 1976. The application of our procedure to the V_{n}’s does not lead to orthogonal polynomials but to a system of biorthogonal rational functions of Al-Salam and Verma [5].

The q-analogue of Hermite polynomials on the unit circle are the polynomials

H_{n}(z; q) = ∑_{k=0}^{n} \frac{(q; q)_{n} (q^{−1/2}z)^{k}}{(q; q)_{k} (q; q)_{n−k}} . (1.16)

Szegő introduced these polynomials in [27] to illustrate his theory of polynomials orthogonal on the unit circle. Szegő used the Jacobi triple product identity to prove the orthogonality relation

\frac{1}{2π} ∫_{0}^{2π} H_{m}(e^{iθ}; q)\overline{H_{n}(e^{iθ}; q)} (q^{1/2}e^{iθ}, q^{1/2}e^{−iθ}; q)_{∞} dθ = \frac{(q; q)_{n} q^{−n}}{(q; q)_{∞}} δ_{m,n}. (1.17)

In Section 4 we show how generating functions transform (1.16) to a q-beta integral of Ramanujan. This explains the origin of the biorthogonal polynomials of Pastro [24] and the _{4}ϕ_{3} biorthogonal rational functions of Al-Salam and Ismail [4].

In section 5 we consider the q-Hermite polynomials for q > 1. They are orthogonal on the imaginary axis. For 0 < q < 1 we put h_{n}(x|q) = (−i)^{n}H_{n}(ix|1/q), and {h_{n}(x|q)} are called the q^{−1}-Hermite polynomials. They correspond to an indeterminate moment problem considered in detail in [18]. Using a q-analogue of the Mehler formula for these polynomials we derive an analogue of the Askey-Wilson integral valid for all the solutions to the indeterminate moment problem. Our derivation, which is different from the one in [18], is based on Parseval’s formula.

The attachment procedure for the q^{−1}-Hermite polynomials leads to a special case of the Al-Salam-Chihara polynomials corresponding to q > 1, more precisely to the polynomials

u_{n}(x; t_{1}, t_{2}) = v_{n}(−2x; q, −(t_{1} + t_{2})/q, t_{1}t_{2}q^{−2}, −1), (1.18) cf. [9]. We prove that for any positive orthogonality measure μ for the q^{−1}-Hermite polynomials

dν_{μ}(sinh ξ; t_{1}, t_{2}) := \frac{(−t_{1}e^{ξ}, t_{1}e^{−ξ}, −t_{2}e^{ξ}, t_{2}e^{−ξ}; q)_{∞}}{(−t_{1}t_{2}/q; q)_{∞}} dμ(sinh ξ), (1.19)

is an orthogonality measure for {u_{n}}.

The attachment procedure applied to {u_{n}} leads to the biorthogonal rational functions

φ_{n}(sinh ξ; t_{1}, t_{2}, t_{3}, t_{4}) := _{4}ϕ_{3} ( \begin{matrix}q^{−n}, −t_{1}t_{2}q^{n−2}, −t_{1}t_{3}/q, −t_{1}t_{4}/q \\ −t_{1}e^{ξ}, t_{1}e^{−ξ}, t_{1}t_{2}t_{3}t_{4}q^{−3} \end{matrix} q, q) . (1.20)

of Ismail and Masson [18] in the special case t_{3} = t_{4} = 0.

The referee raised the question of what determines the starting point in our process. In each case we used the polynomials with fewest possible parameters. In the cases of the polynomials in Sections 2, 4 and 5 we started with polynomials with no parameters, other than q. In Section 3 we used the 1-parameter family of Al-Salam-Carlitz polynomials. We cannot set a = 0 in the Al-Salam-Carlitz polynomials and maintain their orthogonality, so it seems that the full Al-Salam-Carlitz polynomials are the correct starting point. We do not have a canonical answer. The referee also remarked that the Askey scheme in [22] contains many q-polynomials at the lowest level of the classification and wondered if these q-polynomials are Hermite like and when we apply our procedure to such Hermite like polynomials we may obtain other results. We plan to investigate this point in a future work.

2. The Continuous q-Hermite Ladder. Here we assume −1 < q < 1. The orthogonality relation for the continuous q-Hermite polynomials is

∫_{0}^{π} H_{m}(cos θ|q)H_{n}(cos θ|q)(e^{2iθ}, e^{−2iθ}; q)_{∞}dθ = \frac{2π(q; q)_{n}}{(q; q)_{∞}} δ_{m,n}, (2.1)

and follows easily from the Jacobi triple product identity [16]. The series in (1.9) converges for |t| < 1 uniformly in θ ∈ [0, π]. The reason is that

H_{n}(cos θ|q) = ∑_{k=0}^{n} \frac{(q; q)_{n}}{(q; q)_{k}(q; q)_{n−k}} e^{i(n−2k)θ}

implies |H_{n}(x|q)| ≤ H_{n}(1|q) for x ∈ [−1, 1] and (1.9) converges at x = 1. Thus (1.2), (1.3) and the generating function (1.9) imply

∫_{0}^{π} \frac{(e^{2iθ}, e^{−2iθ}; q)_{∞}}{(t_{1}e^{iθ}, t_{1}e^{−iθ}, t_{2}e^{iθ}, t_{2}e^{−iθ}; q)_{∞}} dθ = \frac{2π}{(q, t_{1}t_{2}; q)_{∞}} , |t_{1}|, |t_{2}| < 1, (2.2)

where we used the q-binomial theorem [16, (II.3)]

∑_{0}^{∞} \frac{(a; q)_{n}}{(q; q)_{n}} z^{n} = \frac{(az; q)_{∞}}{(z; q)_{∞}} , (2.3)

with a = 0.

The next step is to find polynomials {p_{n}(x)} orthogonal with respect to the weight function

w_{1}(x; t_{1}, t_{2}) := \frac{(e^{2iθ}, e^{−2iθ}; q)_{∞}}{(t_{1}e^{iθ}, t_{1}e^{−iθ}, t_{2}e^{iθ}, t_{2}e^{−iθ}; q)_{∞}} \frac{1}{\sqrt{1 − x^{2}}} , x = cos θ, (2.4)

which is positive for t_{1}, t_{2} ∈ (−1, 1). Here we follow a clever technique of attachment which was used by Askey and Andrews and by Askey and Wilson in [10]. Write {p_{n}(x)} in the form

p_{n}(x) = ∑_{k=0}^{n} \frac{(q^{−n}, t_{1}e^{iθ}, t_{1}e^{−iθ}; q)_{k}}{(q; q)_{k}} a_{n,k}, (2.5)

then determine a_{n,k} such that p_{n}(x) is orthogonal to (t_{2}e^{iθ}, t_{2}e^{−iθ}; q)_{j}, j = 0, 1, ⋯, n − 1. Note that (ae^{iθ}, ae^{−iθ}; q)_{k} is a polynomial in x of degree k, since

(2.6) (ae^{iθ}, ae^{−iθ}; q)_{k} = ∏_{j=0}^{k−1}(1 − 2axq^{j} + a^{2}q^{2j})
= (−2a)^{k}q^{k(k−1)/2}x^{k} + lower order terms.

The reason for choosing the bases {(t_{1}e^{iθ}, t_{1}e^{−iθ}; q)_{k}} and {(t_{2}e^{iθ}, t_{2}e^{−iθ}; q)_{j}} is that they attach nicely to the weight function and (2.2) enables us to integrate (t_{1}e^{iθ}, t_{1}e^{−iθ}; q)_{k}(t_{2}e^{iθ}, t_{2}e^{−iθ}; q)_{j} against the weight function w_{1}(x; t_{1}, t_{2}). Indeed

(t_{1}e^{iθ}, t_{1}e^{−iθ}; q)_{k}(t_{2}e^{iθ}, t_{2}e^{−iθ}; q)_{j}w_{1}(x; t_{1}, t_{2}) = w_{1}(x; t_{1}q^{k}, t_{2}q^{j}).

Therefore

∫_{−1}^{1}(t_{2}e^{iθ}, t_{2}e^{−iθ}; q)_{j}p_{n}(x)w_{1}(x; t_{1}, t_{2})dx

= ∑_{k=0}^{n} \frac{(q^{−n}; q)_{k}}{(q; q)_{k}} a_{n,k} ∫_{0}^{π} \frac{(e^{2iθ}, e^{−2iθ}; q)_{∞} dθ}{(t_{1}q^{k}e^{iθ}, t_{1}q^{k}e^{−iθ}, t_{2}q^{j}e^{iθ}, t_{2}q^{j}e^{−iθ}; q)_{∞}}

= \frac{2π}{(q; q)_{∞}} ∑_{k=0}^{n} \frac{(q^{−n}; q)_{k}a_{n,k}}{(q; q)_{k}(t_{1}t_{2}q^{k+j}; q)_{∞}}

= \frac{2π}{(q, t_{1}t_{2}q^{j}; q)_{∞}} ∑_{k=0}^{n} \frac{(q^{−n}, t_{1}t_{2}q^{j}; q)_{k}}{(q; q)_{k}} a_{n,k}.

At this stage we look for a_{n,k} as a quotient of products of q-shifted factorials in order to make the above sum vanish for 0 ≤ j < n. The q-Chu-Vandermonde sum [16, (II.6)]

2ϕ_{1}(q^{−n}, a; c; q, q) = \frac{(c/a; q)_{n}}{(c; q)_{n}} a^{n} (2.7)

suggests

a_{n,k} = q^{k}/(t_{1}t_{2}; q)_{k}.

Therefore

∫_{−1}^{1}(t_{2}e^{iθ}, t_{2}e^{−iθ}; q)_{j}p_{n}(x)w_{1}(x; t_{1}, t_{2})dx = \frac{2π(q^{−j}; q)_{n}}{(q, t_{1}t_{2}q^{j}; q)_{∞} (t_{1}t_{2}; q)_{n}} (t_{1}t_{2}q^{j})^{n}.

It follows from (2.5) and (2.6) that the coefficient of x^{n} in p_{n}(x) is

(−2t_{1})^{n}q^{n(n+1)/2}(q^{−n}; q)_{n}/(q, t_{1}t_{2}; q)_{n} = (2t_{1})^{n}/(t_{1}t_{2}; q)_{n}. (2.8)

This leads to the orthogonality relation

∫_{−1}^{1} p_{m}(x)p_{n}(x)w_{1}(x; t_{1}, t_{2})dx = \frac{2π(q; q)_{n}t_{1}^{2n}}{(q, t_{1}t_{2}; q)_{∞}(t_{1}t_{2}; q)_{n}} δ_{m,n}. (2.9)

Furthermore the polynomials are given by

p_{n}(x) = _{3}ϕ_{2} ( \begin{matrix}q^{−n}, t_{1}e^{iθ}, t_{1}e^{−iθ} \\ t_{1}t_{2}, 0 \end{matrix} | q, q) . (2.10)

The polynomials we have just found are the Al-Salam-Chihara polynomials and were first identified by W. Al-Salam and T. Chihara [3]. Their weight function was given in [9] and [10].

One might hope that it is possible to make other choices for the coefficients a_{n,k} and possibly use summation theorems other than (2.7). We went through the summation theorems in [16] and found that (2.7) is the only summation theorem that works in the case at hand. It is also worth mentioning that the generating function (1.9) is the only elementary generating function for the q-Hermite polynomials known to us, [8].

Observe that the orthogonality relation (2.9) and the uniqueness of the polynomials orthogonal with respect to a positive measure show that t_{1}^{−n}p_{n}(x) is symmetric in t_{1} and t_{2}. This gives the known transformation

3ϕ_{2} ( \begin{matrix}q^{−n}, t_{1}e^{iθ}, t_{1}e^{−iθ} \\ t_{1}t_{2}, 0 \end{matrix} | q, q) = (t_{1}/t_{2})^{n}_{3}ϕ_{2} ( \begin{matrix}q^{−n}, t_{2}e^{iθ}, t_{2}e^{−iθ} \\ t_{1}t_{2}, 0 \end{matrix} | q, q) (2.11)

as a byproduct of our analysis.

Our next task is to repeat the process with the Al-Salam-Chihara polynomials as our starting point. The representation (2.10) needs to be transformed to a form more amenable to generating functions. This can be done using an idea of Ismail and Wilson [21]. First write the _{3}ϕ_{2} as a sum over k then replace k by n − k. Applying

(a; q)_{n−k} = \frac{(a; q)_{n}}{(q^{1−n}/a; q)_{k}} (−q/a)^{k}q^{−kn+k(k−1)/2} (2.12)

we obtain

p_{n}(x) = \frac{(t_{1}e^{iθ}, t_{1}e^{−iθ}; q)_{n}}{(t_{1}t_{2}; q)_{n}} q^{−n(n−1)/2}(−1)^{n} ∑_{k=0}^{n} \frac{(−t_{2}/t_{1})^{k} (q^{−n}, q^{1−n}/t_{1}t_{2}; q)_{k}}{(q, q^{1−n}e^{iθ}/t_{1}, q^{1−n}e^{−iθ}/t_{1}; q)_{k}} q^{k(k+1)/2}. (2.13)

Applying the q-analogue of the Pfaff-Kummer transformation [16, (III.4)]

∑_{n=0}^{∞} \frac{(A, C/B; q)_{n}}{(q, C, Az; q)_{n}} q^{n(n−1)/2}(−Bz)^{n} = \frac{(z; q)_{∞}}{(Az; q)_{∞}}_{2}ϕ_{1}(A, B; C; q, z), (2.14)

with

A = q^{−n}, B = t_{2}e^{iθ}, C = q^{1−n}e^{iθ}/t_{1}, z = qe^{−iθ}/t_{1}

to (2.13), we obtain the representation

p_{n}(x) = \frac{(t_{1}e^{−iθ}; q)_{n}t_{1}^{n}e^{inθ}}{(t_{1}t_{2}; q)_{n}} _{2}ϕ_{1} ( \begin{matrix}q^{−n}, t_{2}e^{iθ} \\ q^{1−n}e^{iθ}/t_{1} \end{matrix} | q, qe^{−iθ}/t_{1}) .

Using (2.12) we express a multiple of p_{n} as a Cauchy product of two sequences. The result is

p_{n}(x) = \frac{(q; q)_{n}t_{1}^{n}}{(t_{1}t_{2}; q)_{n}} ∑_{k=0}^{n} \frac{(t_{2}e^{iθ}; q)_{k}}{(q; q)_{k}} e^{−ikθ} \frac{(t_{1}e^{−iθ}; q)_{n−k}}{(q; q)_{n−k}} e^{i(n−k)θ}.

This and the q-binomial theorem (2.3) establish the generating function

∑_{n=0}^{∞} \frac{(t_{1}t_{2}; q)_{n}}{(q; q)_{n}} p_{n}(x) (t/t_{1})^{n} = \frac{(tt_{1}, tt_{2}; q)_{∞}}{(te^{−iθ}, te^{iθ}; q)_{∞}} . (2.15)

The orthogonality relation (2.9), the q-binomial theorem and the generating function (2.15) imply the Askey-Wilson q-beta integral, [10], [16]

∫_{0}^{π} \frac{(e^{2iθ}, e^{−2iθ}; q)_{∞}}{∏_{j=1}^{4}(t_{j}e^{iθ}, t_{j}e^{−iθ}; q)_{∞}} dθ = \frac{2π (t_{1}t_{2}t_{3}t_{4}; q)_{∞}}{(q; q)_{∞} ∏_{1≤j<k≤4}(t_{j}t_{k}; q)_{∞}} . (2.16)

The polynomials orthogonal with respect to the weight function whose total mass is given by (2.16) are the Askey-Wilson polynomials. Their explicit representation and orthogonality follow from (2.16) and the q-analogue of the Pfaff-Saalschütz theorem, [16, (II.12)]. The details of this calculation are in [10]. The polynomials are

p_{n}(x; t_{1}, t_{2}, t_{3}, t_{4}|q) = t_{1}^{−n}(t_{1}t_{2}, t_{1}t_{3}, t_{1}t_{4}; q)_{n} _{4}ϕ_{3} ( \begin{matrix}q^{−n}, t_{1}t_{2}t_{3}t_{4}q^{n−1}, t_{1}e^{iθ}, t_{1}e^{−iθ} \\ t_{1}t_{2}, t_{1}t_{3}, t_{1}t_{4} \end{matrix} | q, q) . (2.17)

The orthogonality relation of the Askey-Wilson polynomials is [10, (2.3)-(2.5)]

∫_{0}^{π} p_{m}(cos θ; t_{1}, t_{2}, t_{3}, t_{4}|q)p_{n}(cos θ; t_{1}, t_{2}, t_{3}, t_{4}|q)w(cos θ; t_{1}, t_{2}, t_{3}, t_{4})dθ (2.18)

= \frac{2π (t_{1}t_{2}t_{3}t_{4}q^{2n}; q)_{∞} (t_{1}t_{2}t_{3}t_{4}q^{n−1}; q)_{n}}{(q^{n+1}; q)_{∞} ∏_{1≤j<k≤4}(t_{j}t_{k}q^{n}; q)_{∞}} δ_{m,n},

for max{|t_{1}|, |t_{2}, |t_{3}|, |t_{4}|} < 1, and the weight function is given by

w(cos θ; t_{1}, t_{2}, t_{3}, t_{4}) = \frac{(e^{2iθ}, e^{−2iθ}; q)_{∞}}{∏_{j=1}^{4}(t_{j}e^{iθ}, t_{j}e^{−iθ}; q)_{∞}} . (2.19)

Observe that the weight function in (2.19) and the right-hand side of (2.18) are symmetric functions of t_{1}, t_{2}, t_{3}, t_{4}. The weight function in (2.18) is positive when max{|t_{1}|, |t_{2}, |t_{3}|, |t_{4}|} < 1, and the uniqueness of the polynomials orthogonal with respect to a positive measure shows that the Askey-Wilson polynomials are symmetric in the four parameters t_{1}, t_{2}, t_{3}, t_{4}. This symmetry is the Sears transformation [16, (III.15)], a fundamental transformation in the theory of basic hypergeometric functions. The Sears transformation may be stated in the form

4ϕ_{3} ( \begin{matrix}q^{−n}, a, b, c \\ d, e, f \end{matrix} | q, q) = (\frac{bc}{d})^{n} \frac{(\frac{de}{bc}, \frac{df}{bc}; q)_{n}}{(e, f; q)_{n}} _{4}ϕ_{3} ( \begin{matrix}q^{−n}, a, d/b, d/c \\ d, \frac{de}{bc}, \frac{df}{bc} \end{matrix} | q, q) , (2.20)

where abc = defq^{n−1}.

Ismail and Wilson [21] used the Sears transformation to establish the generating function

∑_{n=0}^{∞} \frac{p_{n}(cos θ; t_{1}, t_{2}, t_{3}, t_{4}|q)}{(q, t_{1}t_{2}, t_{3}t_{4}; q)_{n}} t^{n} (2.21)

= _{2}ϕ_{1} ( \begin{matrix}t_{1}e^{iθ}, t_{2}e^{iθ} \\ t_{1}t_{2} \end{matrix} | q, te^{−iθ}) _{2}ϕ_{1} ( \begin{matrix}t_{3}e^{−iθ}, t_{4}e^{−iθ} \\ t_{3}t_{4} \end{matrix} | q, te^{iθ}) .

Thus (2.18) leads to the evaluation of the following integral

∫_{0}^{π} ∏_{j=5}^{6}_{2}ϕ_{1} ( \begin{matrix}t_{1}e^{iθ}, t_{2}e^{iθ} \\ t_{1}t_{2} \end{matrix} | q, t_{j}e^{−iθ}) _{2}ϕ_{1} ( \begin{matrix}t_{3}e^{−iθ}, t_{4}e^{−iθ} \\ t_{3}t_{4} \end{matrix} | q, t_{j}e^{iθ}) (2.22)

×\frac{(e^{2iθ}, e^{−2iθ}; q)_{∞}}{∏_{j=1}^{4}(t_{j}e^{iθ}, t_{j}e^{−iθ}; q)_{∞}} dθ

= \frac{2π (t_{1}t_{2}t_{3}t_{4}; q)_{∞}}{(q; q)_{∞} ∏_{1≤j<k≤4}(t_{j}t_{k}; q)_{∞}}

×_{6}ϕ_{5} ( \begin{matrix}\sqrt{t_{1}t_{2}t_{3}t_{4}/q}, −\sqrt{t_{1}t_{2}t_{3}t_{4}/q}, t_{1}t_{3}, t_{1}t_{4}, t_{2}t_{3}, t_{2}t_{4} \\ \sqrt{t_{1}t_{2}t_{3}t_{4}q}, −\sqrt{t_{1}t_{2}t_{3}t_{4}q}, t_{1}t_{2}, t_{3}t_{4}, t_{1}t_{2}t_{3}t_{4}/q \end{matrix} | q, t_{5}t_{6}) ,

valid for max{|t_{1}|, |t_{2}|, |t_{3}|, |t_{4}|, |t_{5}|, |t_{6}|} < 1.

In [20] the Askey-Wilson integral (2.16) was evaluated using Rogers’s linearization formula of products of continuous q-Hermite polynomials, [8], without going through the Al-Salam-Chihara polynomials. The approach in this work is different and does not use Rogers’s formula. In fact Rogers’s linearization formula is the special case t_{4} = 0 of (2.16).

3. The Discrete q-Hermite Ladder. Here we assume 0 < q < 1. Instead of using the discrete q-Hermite polynomials directly we will use the Al-Salam-Carlitz q-polynomials which are a one parameter generalization of the discrete q-Hermite polynomials. The Al-Salam-Carlitz polynomials {U_{n}^{(a)}(x; q)} have the generating function [2], [14]

G(x; t) := ∑_{n=0}^{∞} U_{n}^{(a)}(x; q)\frac{t^{n}}{(q; q)_{n}} = \frac{(t, at; q)_{∞}}{(tx; q)_{∞}} , a < 0, 0 < q < 1, (3.1)

and satisfy the orthogonality relation

∫_{−∞}^{∞} U_{m}^{(a)}(x; q)U_{n}^{(a)}(x; q) dμ^{(a)}(x) = (−a)^{n}q^{n(n−1)/2}(q; q)_{n}δ_{m,n}, (3.2)

with μ^{(a)} a discrete probability measure on [a, 1] given by

μ^{(a)} = ∑_{n=0}^{∞} [\frac{q^{n}}{(q, q/a; q)_{n}(a; q)_{∞}} ε_{q^{n}} + \frac{q^{n}}{(q, aq; q)_{n}(1/a; q)_{∞}} ε_{aq^{n}}] . (3.3)

In (3.3) ε_{y} denotes a unit mass supported at y. The form of the orthogonality relation (3.2)-(3.3) given in [2] and [14] contained a complicated form of a normalization constant. The value of the constant was simplified in [17]. Since the radius of convergence of (1.4) is ρ = ∞ we can apply Proposition 1.1 and get

∫_{−∞}^{∞} G(x; t_{1}) G(x; t_{2}) dμ^{(a)}(x) = ∑_{n=0}^{∞} \frac{(−at_{1}t_{2})^{n}}{(q; q)_{n}} q^{n(n−1)/2} = (at_{1}t_{2}; q)_{∞}, t_{1}, t_{2} ∈ C, (3.4)

where the last expression follows by Euler’s theorem [16, (II.2)]

∑_{n=0}^{∞} z^{n}q^{n(n−1)/2}/(q; q)_{n} = (−z; q)_{∞}. (3.5)

This establishes the integral

∫_{−∞}^{∞} \frac{dμ^{(a)}(x)}{(xt_{1}, xt_{2}; q)_{∞}} = \frac{(at_{1}t_{2}; q)_{∞}}{(t_{1}, t_{2}, at_{1}, at_{2}; q)_{∞}} . (3.6)

When we substitute for μ^{(a)} from (3.3) in (3.4) or (3.6) we discover the nonterminating Chu-Vandermonde sum, [16, (II.23)],

\frac{(Aq/C, Bq/C; q)_{∞}}{(q/C; q)_{∞}}_{2}ϕ_{1} ( \begin{matrix}A, B \\ C \end{matrix} | q, q) + \frac{(A, B; q)_{∞}}{(C/q; q)_{∞}}_{2}ϕ_{1} ( \begin{matrix}Aq/C, Bq/C \\ q^{2}/C \end{matrix} | q, q) (3.7)

= (ABq/C; q)_{∞}.

We now restrict the attention to t_{1}, t_{2} ∈ (a^{−1}, 1) in the case of which 1/(xt_{1}, xt_{2}; q)_{∞} is a positive weight function on [a, 1]. The next step is to find polynomials orthogonal with respect to dμ^{(a)}(x)/(xt_{1}, xt_{2}; q)_{∞}. Define P_{n}(x) by

P_{n}(x) = ∑_{k=0}^{n} \frac{(q^{−n}, xt_{1}; q)_{k}}{(q; q)_{k}} q^{k}a_{n,k} (3.8)

where a_{n,k} will be chosen later. Using (3.6) it is easy to see that

∫_{−∞}^{∞} P_{n}(x)\frac{(xt_{2}; q)_{m}}{(xt_{1}, xt_{2}; q)_{∞}} dμ^{(a)}(x) = ∑_{k=0}^{n} \frac{(q^{−n}; q)_{k}}{(q; q)_{k}} q^{k}a_{n,k} \frac{(at_{1}t_{2}q^{k+m}; q)_{∞}}{(t_{1}q^{k}, at_{1}q^{k}, t_{2}q^{m}, at_{2}q^{m}; q)_{∞}}

= \frac{(at_{1}t_{2}q^{m}; q)_{∞}}{(t_{1}, at_{1}, t_{2}q^{m}, at_{2}q^{m}; q)_{∞}} ∑_{k=0}^{n} \frac{(q^{−n}, t_{1}, at_{1}; q)_{k}}{(q, at_{1}t_{2}q^{m}; q)_{k}} a_{n,k}q^{k}.

The choice a_{n,k} = (λ; q)_{k}/(t_{1}, at_{1}; q)_{k} allows us to apply the q-Chu-Vandermonde sum (2.7). The choice λ = at_{1}t_{2}q^{n−1} leads to

∫_{−∞}^{∞} P_{n}(x)\frac{(xt_{2}; q)_{m}}{(xt_{1}, xt_{2}; q)_{∞}} dμ^{(a)}(x) = \frac{(at_{1}t_{2}q^{m}; q)_{∞} (q^{m+1−n}; q)_{n} (at_{1}t_{2}q^{n−1})^{n}}{(t_{1}, at_{1}, t_{2}q^{m}, at_{2}q^{m}; q)_{∞} (at_{1}t_{2}q^{m}; q)_{n}} . (3.9)

The right-hand side of (3.9) vanishes for 0 ≤ m < n. The coefficient of x^{n} in P_{n}(x) is

\frac{(q^{−n}, at_{1}t_{2}q^{n−1}; q)_{n}}{(q, t_{1}, at_{1}; q)_{n}} (−t_{1})^{n}q^{n(n+1)/2} = \frac{(at_{1}t_{2}q^{n−1}; q)_{n}}{(t_{1}, at_{1}; q)_{n}} t_{1}^{n}.

Therefore

P_{n}(x) = φ_{n}(x; a, t_{1}, t_{2}) = _{3}ϕ_{2} ( \begin{matrix}q^{−n}, at_{1}t_{2}q^{n−1}, xt_{1} \\ t_{1}, at_{1} \end{matrix} | q, q) , (3.10)

satisfies the orthogonality relation

∫_{−∞}^{∞} φ_{m}(x; a, t_{1}, t_{2})φ_{n}(x; a, t_{1}, t_{2})\frac{dμ^{(a)}(x)}{(xt_{1}, xt_{2}; q)_{∞}} (3.11)

= \frac{(q, t_{2}, at_{2}, at_{1}t_{2}q^{n−1}; q)_{n} (at_{1}t_{2}q^{2n}; q)_{∞}}{(t_{1}, at_{1}, t_{2}, at_{2}; q)_{∞} (t_{1}, at_{1}; q)_{n}} (−at_{1}^{2})^{n} q^{n(n−1)/2} δ_{m,n}.

The polynomials {φ_{n}(x; a, t_{1}, t_{2})} are the big q-Jacobi polynomials of Andrews and Askey [6] in a different normalization. The Andrews-Askey normalization is

P_{n}(x; α, β, γ : q) = _{3}ϕ_{2} ( \begin{matrix}q^{−n}, αβq^{n+1}, x \\ αq, γq \end{matrix} | q, q) . (3.12)

Note that we may rewrite the orthogonality relation (3.11) in the form

∫_{−∞}^{∞} t_{1}^{−m}(t_{1}, at_{1}; q)_{m}φ_{m}(x; a, t_{1}, t_{2}) t_{1}^{−n}(t_{1}, at_{1}; q)_{n}φ_{n}(x; a, t_{1}, t_{2})\frac{dμ^{(a)}(x)}{(xt_{1}, xt_{2}; q)_{∞}} (3.13)

= \frac{(q, t_{1}, at_{1}, t_{2}, at_{2}, at_{1}t_{2}q^{n−1}; q)_{n} (at_{1}t_{2}q^{2n}; q)_{∞}}{(t_{1}, at_{1}, t_{2}, at_{2}; q)_{∞}} (−a)^{n} q^{n(n−1)/2} δ_{m,n}.

Since dμ^{(a)}(x)/(xt_{1}, xt_{2}; q)_{∞} and the right-hand side of (3.13) are symmetric in t_{1} and t_{2}, then

t_{1}^{−n}(t_{1}, at_{1}; q)_{n} φ_{n}(x; a, t_{1}, t_{2})

must be symmetric in t_{1} and t_{2}. This gives the known _{3}ϕ_{2} transformation

3ϕ_{2} ( \begin{matrix}q^{−n}, at_{1}t_{2}q^{n−1}, xt_{1} \\ t_{1}, at_{1} \end{matrix} | q, q) = \frac{t_{1}^{n}(t_{2}, at_{2}; q)_{n}}{t_{2}^{n}(t_{1}, at_{1}; q)_{n}}_{3}ϕ_{2} ( \begin{matrix}q^{−n}, at_{1}t_{2}q^{n−1}, xt_{2} \\ t_{2}, at_{2} \end{matrix} | q, q) . (3.14)

We now consider the polynomials {V_{n}^{(a)}(x; q)} and restrict the parameters to 0 < a, 0 < q < 1, in which case they are orthogonal with respect to a positive measure, cf. [14, VI.10]. The corresponding moment problem is determinate if and only if 0 < a ≤ q or 1/q ≤ a. In the first case the unique solution is

m^{(a)} = (aq; q)_{∞} ∑_{n=0}^{∞} \frac{a^{n}q^{n^{2}}}{(q, aq; q)_{n}} ε_{q^{−n}}, (3.15)

and in the second case it is

σ^{(a)} = (q/a; q)_{∞} ∑_{n=0}^{∞} \frac{a^{−n}q^{n^{2}}}{(q, q/a; q)_{n}} ε_{aq^{−n}}, (3.16)

cf. [12]. The total mass of these measures was evaluated to 1 in [17].

If q < a < 1/q the problem is indeterminate and both measures are solutions. In [12] the following one-parameter family of solutions with an analytic density was found

ν(x; a, q, γ) = \frac{γ|a − 1|(q, aq, q/a; q)_{∞}}{πa[(x/a; q)_{∞}^{2} + γ^{2}(x; q)_{∞}^{2}]}, γ > 0. (3.17)

In the above a = 1 has to be excluded. For a similar formula when a = 1 see [12]. If μ is one of the solutions of the moment problem we have the orthogonality relation

∫_{−∞}^{∞} V_{m}^{(a)}(x; q)V_{n}^{(a)}(x; q)dμ(x) = a^{n}q^{−n^{2}}(q; q)_{n}δ_{m,n}. (3.18)

The polynomials have the generating function [2], [14]

V (x; t) := ∑_{n=0}^{∞} V_{n}^{(a)}(x; q)\frac{q^{n(n−1)/2}}{(q; q)_{n}} (−t)^{n} = \frac{(xt; q)_{∞}}{(t, at; q)_{∞}} , |t| < min (1, 1/a), (3.19)

and ζ_{n} = q^{−n^{2}}a^{n}(q; q)_{n}.

The power series (1.4) has the radius of convergence \sqrt{q/a}, and therefore (1.3) becomes

∫_{−∞}^{∞} \frac{(xt_{1}, xt_{2}; q)_{∞} dμ(x)}{(t_{1}, at_{1}, t_{2}, at_{2}; q)_{∞}} = ∫_{−∞}^{∞} V (x, t_{1}) V (x, t_{2})dμ(x)
= ∑_{n=0}^{∞} \frac{(at_{1}t_{2}/q)^{n}}{(q; q)_{n}}
= \frac{1}{(at_{1}t_{2}/q; q)_{∞}} , |t_{1}|, |t_{2}| < \sqrt{q/a}.

This identity with μ = m^{(a)} or μ = σ^{(a)} is nothing but the q-analogue of the Gauss theorem,

2ϕ_{1}(a, b; c; q, c/ab) = \frac{(c/a, c/b; q)_{∞}}{(c, c/ab; q)_{∞}} , |c| < |ab|, (3.20)

[16, (II.8)].

Specializing to the density (3.17) we get

∫_{−∞}^{∞} \frac{(xt_{1}, xt_{2}; q)_{∞} dx}{(x/a; q)_{∞}^{2} + γ^{2}(x; q)_{∞}^{2}} = \frac{πa(t_{1}, at_{1}, t_{2}, at_{2}; q)_{∞}}{|a − 1|γ(q, aq, q/a, at_{1}t_{2}/q; q)_{∞}} , (3.21)

valid for q < a < 1/q, a ≠ 1, γ > 0. Formula (3.21) seems to be new.

We now seek polynomials or rational functions that are orthogonal with respect to the measure

dν(x) = (xt_{1}, xt_{2}; q)_{∞} dμ(x), (3.22)

where μ satisfies (3.18). It is clear that we can integrate 1/[(xt_{1}; q)_{k}(xt_{2}; q)_{j}] with respect to the measure ν. Set

ψ_{n}(x; a, t_{1}, t_{2}) := ∑_{k=0}^{n} \frac{(q^{−n}; q)_{k}}{(q; q)_{k}} \frac{q^{k}a_{n,k}}{(xt_{1}; q)_{k}} . (3.23)

The rest of the analysis is similar to our treatment of the U_{n}’s. We get

∫_{−∞}^{∞} \frac{ψ_{n}(x; a, t_{1}, t_{2})}{(xt_{2}; q)_{m}} dν(x) = ∑_{k=0}^{n} \frac{(q^{−n}; q)_{k}}{(q; q)_{k}} q^{k}a_{n,k} ∫_{−∞}^{∞}(xt_{1}q^{k}, xt_{2}q^{m}; q)_{∞} dμ(x),

and if we choose a_{n,k} = (t_{1}, at_{1}; q)_{k}/(at_{1}t_{2}/q; q)_{k} the above expression is equal to

\frac{(t_{1}, at_{1}, t_{2}q^{m}, at_{2}q^{m}; q)_{∞}}{(at_{1}t_{2}q^{m−1}; q)_{∞}} _{2}ϕ_{1} ( \begin{matrix}q^{−n}, at_{1}t_{2}q^{m−1} \\ at_{1}t_{2}/q \end{matrix} | q, q) ,

= \frac{(t_{1}, at_{1}, t_{2}q^{m}, at_{2}q^{m}; q)_{∞}(q^{−m}; q)_{n}}{(at_{1}t_{2}q^{m−1}; q)_{∞}(at_{1}t_{2}/q; q)_{n}} (at_{1}t_{2}q^{m−1})^{n},

which is 0 for m < n. We have used the Chu-Vandermonde sum (2.7). Since ν is symmetric in t_{1}, t_{2}, this leads to the biorthogonality relation

∫_{−∞}^{∞} ψ_{m}(x; a, t_{2}, t_{1})ψ_{n}(x; a, t_{1}, t_{2}) dν(x) = \frac{(t_{1}, at_{1}, t_{2}, at_{2}; q)_{∞}(q; q)_{n}}{(at_{1}t_{2}/q; q)_{∞}(at_{1}t_{2}/q; q)_{n}} (at_{1}t_{2}/q)^{n}δ_{m,n}. (3.24)

The ψ_{n}’s are given by

ψ_{n}(x; a, t_{1}, t_{2}) = _{3}ϕ_{2} ( \begin{matrix}q^{−n}, t_{1}, at_{1} \\ xt_{1}, at_{1}t_{2}/q \end{matrix} q, q) . (3.25)

They are essentially the rational functions studied by Al-Salam and Verma in [5]. Al-Salam and Verma used the notation

R_{n}(x; α, β, γ, δ; q) = _{3}ϕ_{2} ( \begin{matrix}β, αγ/δ, q^{−n} \\ βγ/q, αqx \end{matrix} q, q) . (3.26)

The translation between the two notations is ψ_{n}(x; a, t_{1}, t_{2}) = R_{n}(βxq^{−1}/α; α, β, γ, δ; q), (3.27)

with

t_{1} = β, t_{2} = βδ/qα, a = αγ/βδ. (3.28)

Note that R_{n} has only three free variables since one of the parameters α, β, γ, δ can be absorbed by scaling the independent variable.

4. The Szegő Ladder. Here we assume 0 < q < 1.

As already mentioned in the introduction Szegő [27] used the Jacobi triple product identity to prove (1.17). The explicit form (1.16) and the q-binomial theorem (2.3) give

H(z, t) := ∑_{n=0}^{∞} H_{n}(z; q)\frac{t^{n}}{(q; q)_{n}} = 1/(t, tzq^{−1/2}; q)_{∞}, (4.1)

for |t| < 1, |tz| < q^{1/2}. ¿From (1.16) it follows that |H_{n}(z; q)| ≤ H_{n}(|z|; q). Furthermore Darboux’s method [23] and (4.1) give

H_{n}(1; q) ≈ q^{−n/2}/(q^{−1/2}; q)_{∞}.

Thus (1.17) and (4.1) imply the Ramanujan q-beta integral [16]

\frac{1}{2πi} ∫_{|z|=1} \frac{(q^{1/2}z, q^{1/2}/z; q)_{∞}}{(t_{1}q^{−1/2}z, t_{2}q^{−1/2}/z; q)_{∞}} \frac{dz}{z} = \frac{(t_{1}, t_{2}; q)_{∞}}{(q, t_{1}t_{2}/q; q)_{∞}} , (4.2)

for |t_{1}| < q^{1/2}, |t_{2}| < q^{1/2}, since (1.4) has radius of convergence q^{1/2} and the series in (4.1) converges uniformly in z for z on the unit circle. This can be proved by estimating H_{n}(z; q) directly from (1.16).

Putting

Ω(z) = \frac{(q, t_{1}t_{2}q, q^{1/2}z, q^{1/2}/z; q)_{∞}}{(t_{1}q, t_{2}q, t_{1}q^{1/2}z, t_{2}q^{1/2}/z; q)_{∞}} (4.3)

and applying the attachment technique to (4.2) we find that the polynomials

p̃_{n}(z, t_{1}, t_{2}) := _{3}ϕ_{2} ( \begin{matrix}q^{−n}, t_{1}q^{1/2}z, t_{1}q \\ 0, t_{1}t_{2}q \end{matrix} | q, q) . (4.4)

satisfy the biorthogonality relation

\frac{1}{2πi} ∫_{|z|=1} p̃_{m}(z, t_{1}, t_{2})\overline{p̃_{n}(z, \overline{t_{2}}, \overline{t_{1}})} Ω(z) \frac{dz}{z} = \frac{(q; q)_{n}}{(t_{1}t_{2}q; q)_{n}} (t_{1}t_{2}q)^{n}δ_{m,n}. (4.5)

Using the transformation [16, (III.7)] we see that

p̃_{n}(z, t_{1}, t_{2}) = \frac{(q; q)_{n}}{(t_{1}t_{2}q; q)_{n}} (t_{1}q)^{n}p_{n}(z, t_{1}, t_{2}), (4.6)

where

p_{n}(z, a, b) = \frac{(b; q)_{n}}{(q; q)_{n}}_{2}ϕ_{1}(q^{−n}, aq; q^{1−n}/b; q, q^{1/2}z/b) = ∑_{k=0}^{n} \frac{(aq; q)_{k} (b; q)_{n−k}}{(q; q)_{k} (q; q)_{n−k}} (q^{−1/2}z)^{k}, (4.7)

are the polynomials considered by Pastro [24] and for which the biorthogonality relation reads

\frac{1}{2πi} ∫_{|z|=1} p_{m}(z, t_{1}, t_{2})\overline{p_{n}(z, \overline{t_{2}}, \overline{t_{1}})} Ω(z) \frac{dz}{z} = \frac{(t_{1}t_{2}q; q)_{n}}{(q; q)_{n}} q^{−n}δ_{m,n}. (4.8)

The special case when a and b are real was considered by Askey in his comments on [27] in Szegő’s Collected Papers. Al-Salam and Ismail [4] used (4.8) and the generating function

∑_{n=0}^{∞} p_{n}(z; a, b)t^{n} = \frac{(atzq^{1/2}, bt; q)_{∞}}{(tzq^{−1/2}, t; q)_{∞}} (4.9)

to establish a q-beta integral and found the rational functions biorthogonal to its integrand. The interested reader is referred to [4] for details.

5. The q^{−1}-Hermite Ladder. When q > 1 in (1.5) the polynomials {H_{n}(x|q)} become orthogonal on the imaginary axis. The result of replacing x by ix and q by 1/q put the orthogonality on the real line and the new q is now in (0, 1), [7]. Denote (−i)^{n}H_{n}(ix|1/q) by h_{n}(x|q). In this new notation the recurrence relation (1.5) and the initial conditions (1.7) become

h_{n+1}(x|q) = 2xh_{n}(x|q) − q^{−n}(1 − q^{n})h_{n−1}(x|q), n > 0, (5.1)

h_{0}(x|q) = 1, h_{1}(x|q) = 2x. (5.2)

The polynomials {h_{n}(x|q)} are called the q^{−1}-Hermite polynomials, [18]. The corresponding moment problem is indeterminate. Let V_{q} be the set of probability measures which solve the problem. For any μ ∈ V_{q} we have

∫_{−∞}^{∞} h_{m}(x|q)h_{n}(x|q) dμ(x) = q^{−n(n+1)/2}(q; q)_{n}δ_{m,n}. (5.3)

The h′_{n}s have the generating function, [18],

∑_{n=0}^{∞} h_{n}(x|q)\frac{t^{n}}{(q; q)_{n}} q^{n(n−1)/2} = (−te^{ξ}, te^{−ξ}; q)_{∞}, x = sinh ξ, t, ξ ∈ C. (5.4)

By Proposition 1.1 it is clear that (5.3) and (5.4) imply ∫_{−∞}^{∞}(−t_{1}e^{ξ}, t_{1}e^{−ξ}, −t_{2}e^{ξ}, t_{2}e^{−ξ}; q)_{∞}dμ(x) = (−t_{1}t_{2}/q; q)_{∞}, t_{1}, t_{2} ∈ C, μ ∈ V_{q} (5.5) since the power series (1.4) has radius of convergence ρ = ∞. Incidentally the function χ_{t}(x) = (−te^{ξ}, te^{−ξ}; q)_{∞} = (−t(\sqrt{x^{2} + 1} + x), t(\sqrt{x^{2} + 1} − x); q)_{∞} (5.6) belongs to L^{2}(μ) for any μ ∈ V_{q} and any t ∈ C. Therefore the complex measure ν_{μ}(t_{1}, t_{2}) defined by

dν_{μ}(x; t_{1}, t_{2}) := \frac{χ_{t_{1}}(x)χ_{t_{2}}(x)}{(−t_{1}t_{2}/q; q)_{∞}} dμ(x), μ ∈ V_{q}, t_{1}, t_{2} ∈ C, t_{1}t_{2} ≠ −q^{1−k}, k ≥ 0 (5.7)

has total mass 1, and it is non-negative if t_{1} = \overline{t_{2}}.

Note that (−te^{ξ}, te^{−ξ}; q)_{k} is a polynomial of degree k in x = sinh ξ for each fixed t ≠ 0. Since

(−te^{ξ}/q^{k}, te^{−ξ}/q^{k}; q)_{k}χ_{t}(x) = χ_{t/q^{k}}(x), we see that the non-negative polynomial |(−te^{ξ}/q^{k}, te^{−ξ}/q^{k}; q)_{k}|^{2} of degree 2k is ν_{μ}(t, \overline{t})-integrable. This implies that ν_{μ}(t, \overline{t}) has moments of any order, and by the Cauchy-Schwarz inequality every polynomial is ν_{μ}(t_{1}, t_{2})-integrable for all t_{1}, t_{2} ∈ C, μ ∈ V_{q}.

Introducing the orthonormal polynomials

h̃_{n}(x|q) = \frac{h_{n}(x|q)}{\sqrt{(q; q)_{n}}} q^{n(n+1)/4} (5.8)

the q-Mehler formula, cf. [18], reads

∑_{n=0}^{∞} h̃_{n}(sinh ξ|q)h̃_{n}(sinh η|q)z^{n} = \frac{(−zqe^{ξ+η}, −zqe^{−ξ−η}, zqe^{ξ−η}, zqe^{−ξ+η}; q)_{∞}}{(z^{2}q; q)_{∞}} (5.9)

valid for ξ, η ∈ C, |z| < 1/\sqrt{q}.

Applying the Darboux method [23] to (5.9) Ismail and Masson [18] found the asymptotic behavior of h̃_{n}(sinh η|q), and from their result it follows that (h̃_{n}(sinh η|q)z^{n}) ∈ l^{2} for |z| < q^{−1/4}, η ∈ C. In this case the right-hand side of (5.9) belongs to L^{2}(μ) as a function of x = sinh ξ and the formula is its orthogonal expansion. Putting t = qze^{η}, s = −qze^{−η}, we have z^{2} = −stq^{−2}, so if |st| < q^{3/2} we have |z| < q^{−1/4} and the right-hand side of (5.9) becomes χ_{t}(sinh ξ)χ_{s}(sinh ξ)/(−st/q; q)_{∞}, which belongs to L^{2}(μ). Using this observation we can give a simple proof of the following formula from [18].

Proposition 5.1 Let μ ∈ V_{q} and let t_{i} ∈ C, i = 1, ⋯, 4 satisfy |t_{1}t_{3}|, |t_{2}t_{4}| < q^{3/2}. (This holds in particular if |t_{i}| < q^{3/4}, i = 1, ⋯, 4). Then ∏_{i=1}^{4} χ_{t_{i}} ∈ L^{1}(μ) and

∫ ∏_{i=1}^{4} χ_{t_{i}} dμ = \frac{∏_{1≤j<k≤4}(−t_{j}t_{k}/q; q)_{∞}}{(t_{1}t_{2}t_{3}t_{4}q^{−3}; q)_{∞}} . (5.10)

Proof. We write

qz_{1}e^{η_{1}} = t_{1}, qz_{2}e^{η_{2}} = t_{2}, −qz_{1}e^{−η_{1}} = t_{3}, −qz_{2}e^{−η_{2}} = t_{4},

noting that z_{1}^{2} = −t_{1}t_{3}q^{−2}, z_{2}^{2} = −t_{2}t_{4}q^{−2}, so the equations have solutions z_{i}, η_{i}, i = 1, 2 if t_{i} ≠ 0 for i = 1, ⋯, 4. We next apply Parseval’s formula to the two L^{2}(μ)-functions χ_{t_{1}}χ_{t_{3}}, χ_{t_{2}}χ_{t_{4}} and get

∫ ∏_{i=1}^{4} χ_{t_{i}} dμ = (−t_{1}t_{3}/q, −t_{2}t_{4}/q; q)_{∞} ∑_{n=0}^{∞} h̃_{n}(sinh η_{1}|q)h̃_{n}(sinh η_{2}|q)(z_{1}z_{2})^{n},

which by the q-Mehler formula gives the right-hand side of (5.10).

If t_{1} = 0 and t_{2}t_{3}t_{4} ≠ 0 we apply Parseval’s formula to χ_{t_{3}} and χ_{t_{2}}χ_{t_{4}}, and if two of the parameters are zero the formula reduces to (5.5). □

We shall now look at orthogonal polynomials for the measures ν_{μ}(t_{1}, t_{2}). When q > 1 the Al-Salam-Chihara polynomials are orthogonal on (−∞, ∞) and their moment problem may be indeterminate [9], [3], [15]. If one replaces q by 1/q in the Al-Salam-Chihara polynomials, they can be renormalized to polynomials {v_{n}(x; q, a, b, c)} satisfying

(1 − q^{n+1})v_{n+1}(x; q, a, b, c) = (a − xq^{n})v_{n}(x; q, a, b, c) − (b − cq^{n−1})v_{n−1}(x; q, a, b, c), (5.11)

where 0 < q < 1, and a, b, c are complex parameters. We now consider the special case u_{n}(x; t_{1}, t_{2}) = v_{n}(−2x; q, −(t_{1} + t_{2})/q, t_{1}t_{2}q^{−2}, −1), (5.12)

where t_{1}, t_{2} are complex parameters. The corresponding monic polynomials û_{n}(x) satisfy the recurrence relation determined from (5.11)

xû_{n}(x) = û_{n+1}(x) + \frac{1}{2}(t_{1} + t_{2})q^{−n−1}û_{n}(x) + \frac{1}{4}(t_{1}t_{2}q^{−2n−1} + q^{−n})(1 − q^{n})û_{n−1}(x), (5.13)

so by Favard’s theorem, cf. [14], {u_{n}(x; t_{1}, t_{2})} are orthogonal with respect to a complex measure α(t_{1}, t_{2}) if and only if t_{1}t_{2} ≠ −q^{n+1}, n ≥ 1, and with respect to a probability measure α(t_{1}, t_{2}) if and only if t_{2} = \overline{t_{1}} ∈ C \ R or t_{1}, t_{2} ∈ R and t_{1}t_{2} ≥ 0. In the affirmative case

∫_{−∞}^{∞} u_{m}(x; t_{1}, t_{2})u_{n}(x; t_{1}, t_{2}) dα(x; t_{1}, t_{2}) = \frac{q^{n(n−3)/2}}{(q; q)_{n}} (−t_{1}t_{2}q^{−n−1}; q)_{n}δ_{m,n}. (5.14)

It follows from (3.77) in [9] that the moment problem is indeterminate if t_{1} = \overline{t_{2}}. If t_{1}, t_{2} are real, different and t_{1}t_{2} ≥ 0, we can assume |t_{1}| < |t_{2}| without loss of generality, and in this case the moment problem is indeterminate if and only if |t_{1}/t_{2}| > q.

The generating function (3.70) in [9] takes the form (x = sinh ξ)

∑_{n=0}^{∞} u_{n}(x; t_{1}, t_{2})t^{n} = \frac{(−te^{ξ}, te^{−ξ}; q)_{∞}}{(−t_{1}t/q, −t_{2}t/q; q)_{∞}} , for |t| < min\{q/|t_{1}|, q/|t_{2}|\}. (5.15)

By the method used in [9] we can derive formulas for u_{n} in the following way: Use the qbinomial theorem to write the right-hand side of (5.15) as a product of two power series in t and equate coefficients of t^{n} to get

u_{n}(x; t_{1}, t_{2}) = ∑_{k=0}^{n} \frac{(qe^{ξ}/t_{1}; q)_{k}}{(q; q)_{k}} (−t_{1}/q)^{k} \frac{(−qe^{−ξ}/t_{2}; q)_{n−k}}{(q; q)_{n−k}} (−t_{2}/q)^{n−k}. (5.16)

Application of the identity (I.11) in [16] gives the explicit representation

u_{n}(x; t_{1}, t_{2}) = (−t_{2}/q)^{n} \frac{(−qe^{−ξ}/t_{2}; q)_{n}}{(q; q)_{n}}_{2}ϕ_{1} ( \begin{matrix}q^{−n}, qe^{ξ}/t_{1} \\ −t_{2}e^{ξ}/q^{n} \end{matrix} | q, −t_{1}e^{ξ}) , (5.17)

which by (III.8) in [16] can be transformed to

u_{n}(x; t_{1}, t_{2}) = \frac{(−q^{2}/(t_{1}t_{2}); q)_{n}}{(q; q)_{n}} (−t_{2}/q)^{n}_{3}ϕ_{1} ( \begin{matrix}q^{−n}, qe^{ξ}/t_{1}, −qe^{−ξ}/t_{1} \\ −q^{2}/(t_{1}t_{2}) \end{matrix} | q, (t_{1}/t_{2})q^{n}) . (5.18)

Writing the _{3}ϕ_{1} as a finite sum and applying the formula

(a; q)_{k} = (q^{1−k}/a; q)_{k}(−a)^{k}q^{k(k−1)/2},

we see that (5.18) can be transformed to

u_{n}(x; t_{1}, t_{2}) = (−1/t_{1})^{n} \frac{(−t_{1}t_{2}/q^{n+1}; q)_{n}}{(q; q)_{n}} q^{n(n+1)/2} ∑_{k=0}^{n} \frac{(q^{−n}, −t_{1}e^{ξ}/q^{k}, t_{1}e^{−ξ}/q^{k}; q)_{k}}{(q, −t_{1}t_{2}/q^{k+1}; q)_{k}} q^{nk}. (5.19)

By symmetry of t_{1}, t_{2} a similar formula holds for t_{1} and t_{2} interchanged.

Theorem 5.2 For μ ∈ V_{q} and t_{1}, t_{2} ∈ C such that t_{1}t_{2} ≠ −q^{n+1}, n ≥ 1 the Al-Salam-Chihara polynomials {u_{n}(x; t_{1}, t_{2})} are orthogonal with respect to the complex measure ν_{μ}(t_{1}, t_{2}) given by (5.7).

Proof. It follows by (5.13) that û_{n}(x; 0, 0) = 2^{−n}h_{n}(x|q) so the assertion is clear for t_{1} = t_{2} = 0.

Assume now that t_{1} ≠ 0. By the three term recurrence relation it suffices to prove that

∫ u_{n}(x; t_{1}, t_{2}) dν_{μ}(x; t_{1}, t_{2}) = 0 for n ≥ 1. (5.20)

By (5.19) and (5.7) we get

∫ u_{n}(x; t_{1}, t_{2}) dν_{μ}(x; t_{1}, t_{2})

= (−1/t_{1})^{n} \frac{(−t_{1}t_{2}/q^{n+1}; q)_{n}}{(q; q)_{n}} q^{n(n+1)/2} ∑_{k=0}^{n} q^{nk} \frac{(q^{−n}; q)_{k}}{(q; q)_{k}} ∫ \frac{χ_{t_{1}/q^{k}}(x)χ_{t_{2}}(x)}{(−t_{1}t_{2}/q^{k+1}; q)_{∞}} dμ(x).

By (5.5) the integral is 1, and the sum is equal to _{1}ϕ_{0}(q^{−n}; −; q, q^{n}), which is equal to 0 for n ≥ 1 by (II.4) in [16]. □

In particular, if t_{2} = \overline{t_{1}} then ν_{μ}(t_{1}, \overline{t_{1}}) is a positive measure and the Al-Salam-Chihara moment problem corresponding to \{u_{n}(x; t_{1}, \overline{t_{1}})\} is indeterminate. The set

\{ν_{μ}(t_{1}, \overline{t_{1}}) | μ ∈ V_{q}\}

is a compact convex subset of the full set C(t_{1}) of solutions to the \{u_{n}(x; t_{1}, \overline{t_{1}})\}-moment problem.

If t_{1} = t_{2} ∈ R then

\{ν_{μ}(t_{1}, \overline{t_{1}}) | μ ∈ V_{q}\} ≠ C(t_{1})

since the measures on the left can have no mass at the zeros of χ_{t_{1}}(x).

If t_{1} = t ∈ (q, 1) and t_{2} = 0 then the Al-Salam-Chihara moment problem is determinate and the set {ν_{μ}(t, 0) | μ ∈ V_{q}} contains exactly one positive measure namely the one coming from μ ∈ V_{q} being the N-extremal solution corresponding to the choice a = q/t in (6.27) and (6.30) of [18], i.e. μ is the discrete measure with mass m_{n} at x_{n} for n ∈ Z, where

x_{n} = \frac{1}{2} (\frac{t}{q^{n+1}} − \frac{q^{n+1}}{t} )

and

m_{n} = \frac{(q/t)^{4n}q^{n(2n−1)}(1 + q^{2n+2}/t^{2})}{(−q^{2}/t^{2}, −t^{2}/q, q; q)_{∞}}.

The function χ_{t}(x) vanishes for x = x_{n} when n < 0 and we get

ν_{μ}(t, 0) = ∑_{n=0}^{∞} c_{n}ε_{x_{n}},

where

c_{n} = \frac{q^{3n(n+1)/2}(1 + q^{2n+2}/t^{2})(−q^{2}/t^{2}; q)_{n}}{t^{2n}(q; q)_{n}(−q^{2}/t^{2}; q)_{∞}} .

We now go back to (5.5) and integrate 1/(−t_{1}e^{ξ}, t_{1}e^{−ξ}; q)_{k} against the integrand in (5.5). Here again the attachment method works and we see that

φ_{n}(sinh ξ; t_{1}, t_{2}) := _{3}ϕ_{2} ( \begin{matrix}q^{−n}, −t_{1}t_{2}q^{n−2}, 0 \\ −t_{1}e^{ξ}, t_{1}e^{−ξ} \end{matrix} | q, q) . (5.21)

satisfies the biorthogonality relation

∫_{−∞}^{∞} φ_{m}(x; t_{1}, t_{2})φ_{n}(x; t_{2}, t_{1})χ_{t_{1}}(x)χ_{t_{2}}(x) dμ(x) (5.22)

= \frac{1 + t_{1}t_{2}q^{n−2}}{1 + t_{1}t_{2}q^{2n−2}} (−t_{1}t_{2}q^{n−1}; q)_{∞}(q; q)_{n}q^{n(n−3)/2} (t_{1}t_{2})^{n} δ_{m,n}.

The biorthogonal rational functions (5.21) are the special case t_{3} = t_{4} = 0 of the biorthogonal rational functions

φ_{n}(sinh ξ; t_{1}, t_{2}, t_{3}, t_{4}) := _{4}ϕ_{3} ( \begin{matrix}q^{−n}, −t_{1}t_{2}q^{n−2}, −t_{1}t_{3}/q, −t_{1}t_{4}/q \\ −t_{1}e^{ξ}, t_{1}e^{−ξ}, t_{1}t_{2}t_{3}t_{4}q^{−3} \end{matrix} q, q) . (5.23)

of Ismail and Masson [18]. We have not been able to apply a generating function technique to (5.21) because we have not been able to find a suitable generating function for the rational functions (5.21).

We now return to the Al-Salam-Chihara polynomials {u_{n}(x; t_{1}, t_{2})} in the positive definite case and reconsider the generating function (5.15). The radius of convergence of (1.4) is ρ = q^{3/2}/\sqrt{t_{1}t_{2}}, and we get by Proposition 1.1, (5.14) and the q-binomial theorem

∫_{−∞}^{∞} χ_{t_{3}}(x)χ_{t_{4}}(x)dα(x; t_{1}, t_{2}) = \frac{(−t_{1}t_{3}/q, −t_{1}t_{4}/q, −t_{2}t_{3}/q, −t_{2}t_{4}/q, −t_{3}t_{4}/q; q)_{∞}}{(t_{1}t_{2}t_{3}t_{4}q^{−3}; q)_{∞}} , (5.24)

valid for |t_{3}|, |t_{4}| < ρ.

Applying this to the measures α(t_{1}, t_{2}) = ν_{μ}(t_{1}, t_{2}), we get a new proof of (5.10), now under slightly different assumptions on t_{1}, ⋯, t_{4}.

The attachment procedure works in this case, and we prove the biorthogonality relation of [18] under the same assumptions as in Proposition 5.1:

∫_{−∞}^{∞} φ_{m}(x; t_{1}, t_{2}, t_{3}, t_{4}) φ_{n}(x; t_{2}, t_{1}, t_{3}, t_{4}) ∏_{i=1}^{4} χ_{t_{i}}(x) dμ(x) (5.25)

= \frac{1 + t_{1}t_{2}q^{n−2}}{1 + t_{1}t_{2}q^{2n−2}} \frac{(t_{1}t_{2}t_{3}t_{4}q^{−3})^{n}(q, −q^{2}/t_{3}t_{4}; q)_{n}(−t_{1}t_{2}q^{n−1}; q)_{∞}}{(t_{1}t_{2}t_{3}t_{4}q^{−3}; q)_{n}} \frac{∏_{1≤j<k≤4}(−t_{j}t_{k}/q; q)_{∞}}{(t_{1}t_{2}t_{3}t_{4}q^{−3}; q)_{∞}} δ_{m,n}.

Acknowledgments We greatly appreciate the referee queries, suggestions and the many interesting points he/she raised.

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Mathematics Institute, Copenhagen University, Universitetsparken 5, DK-2100 Copenhagen Ø, Denmark.

Department of Mathematics, University of South Florida, Tampa, Florida 33620-5700.

e-mail  berg@math.ku.dk  and  ismail@math.usf.edu