AN EIGENVALUE PROBLEM FOR THE ASSOCIATED ASKEY–WILSON POLYNOMIALS

ANDREA BRUDER, CHRISTIAN KRATTENTHALER^{†}, AND SERGEI K. SUSLOV

Abstract. To derive an eigenvalue problem for the associated Askey–Wilson polynomials, we consider an auxiliary function in two variables which is related to the associated Askey–Wilson polynomials introduced by Ismail and Rahman. The Askey–Wilson operator, applied in each variable separately, maps this function to the ordinary Askey–Wilson polynomials with different sets of parameters. A third Askey–Wilson operator is found with the help of a computer algebra program which links the two, and an eigenvalue problem is stated.

1. Introduction

Throughout this paper, we use the standard notation for the q-shifted factorials:

(a; q)_{n} := ∏_{j=0}^{n−1} (1 − aq^{j}) ,  (a_{1}, a_{2}, …, a_{r}; q)_{n} := ∏_{k=1}^{r} (a_{k}; q)_{n} ,

(a; q)_{∞} := lim_{n→∞} (a; q)_{n} ,  (a_{1}, a_{2}, …, a_{r}; q)_{∞} := ∏_{k=1}^{r} (a_{k}; q)_{∞} ,

provided |q| < 1. The basic hypergeometric series is defined by (cf. [9])

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

If 0 < |q| < 1, the series converges absolutely for all z if r ≤ s, and for |z| < 1 if r = s + 1.

The Askey–Wilson polynomials are the most general extension of the classical orthogonal polynomials [1], [2], [6], [11], [12], [18]. They are most conveniently given in terms of a _{4}φ_{3}-series,

p_{n}(x) = p_{n}(x; a, b, c, d) = p_{n}(x; a, b, c, d|q)

= a^{−n} (ab, ac, ad; q)_{n} _{4}φ_{3}(\begin{matrix}q^{−n}, abcdq^{n−1}, az, a/z \\ ab, ac, ad \end{matrix}; q, q) ,

where x = (z + z^{−1}) /2, and |z| < 1. In this normalization, the Askey–Wilson polynomials are symmetric in all four parameters due to Sears’ transformation [6].

Date: March 16, 2012. 1991 Mathematics Subject Classification. 33D45, 42C10; 33D15. Key words and phrases. Basic hypergeometric functions, q-orthogonal polynomials, Askey–Wilson polynomials,

associated Askey–Wilson polynomials, eigenvalue problem.

†Research partially supported by the Austrian Science Foundation FWF, grants Z130-N13 and S9607-N13, the latter in the framework of the National Research Network “Analytic Combinatorics and Probabilistic Number Theory”.

2 ANDREA BRUDER, CHRISTIAN KRATTENTHALER^{†}, AND SERGEI K. SUSLOV

The Askey–Wilson polynomials satisfy the 3-term recurrence relation

2x p_{n}(x; a, b, c, d) = A_{n} p_{n+1}(x; a, b, c, d) + B_{n} p_{n}(x; a, b, c, d) + C_{n} p_{n−1}(x; a, b, c, d),  (1.1)

where

A_{n} = \frac{a^{−1}(1 − abq^{n})(1 − acq^{n})(1 − adq^{n})(1 − abcdq^{n−1})}{(1 − abcdq^{2n−1})(1 − abcdq^{2n} − q^{2n})},  (1.2)
C_{n} = \frac{a(1 − bcq^{n−1})(1 − bdq^{n−1})(1 − cdq^{n−1})(1 − q^{n})}{(1 − abcdq^{2n−1})(1 − abcdq^{2n})},  (1.3)
B_{n} = a + a^{−1} − A_{n} − C_{n}.  (1.4)

The weight function with respect to which the polynomials p_{n}(x) are orthogonal was found by Askey and Wilson in [6]. The Askey–Wilson divided difference operator is defined by

L(x)u := L (s; a, b, c, d) u (s)

= \frac{σ (−s) ∇x (s) u (s + 1) + σ (s) Δx (s) u (s − 1) − [σ (s) Δx (s) + σ (−s) ∇x (s)] u (s)}{Δx (s) ∇x (s) ∇x_{1} (s)}, (1.5)

where σ (s) = q^{−2s} (q^{s} − a) (q^{s} − b) (q^{s} − c) (q^{s} − d) and, by definition,

x(s) = \frac{1}{2} (q^{s} + q^{−s}) ,  x_{1}(s) = x (s + \frac{1}{2}) ,
Δf(s) = f(s + 1) − f(s),  ∇f(s) = f(s) − f(s − 1).

(We follow the notation in [7] and [8].) We will make use of an analogue of the power series expansion method, where a function is expanded in terms of generalized powers. For a positive integer m, the generalized powers are defined by

[x(s) − x(z)]^{(m)} = ∏_{n=0}^{m−1}[x_{n}(s) − x_{n}(z − k)],  x_{n}(z) = x (z + \frac{n}{2})  (1.6)

(see [16, Exercises 2.9–2.11, 2.25] and [17] for more details).

2. The Associated Askey–Wilson Polynomials

The associated Askey–Wilson polynomials, p_{n}^{α}(x) = p_{n}^{α}(x; a, b, c, d) = p_{n}^{α}(x; a, b, c, d|q), were introduced by Ismail and Rahman in [10]. They are solutions of the 3-term recurrence relation

2x p_{n}^{α}(x; a, b, c, d) = A_{n+α} p_{n+1}^{α}(x; a, b, c, d) + B_{n+α} p_{n}^{α}(x; a, b, c, d) + C_{n+α} p_{n−1}^{α}(x; a, b, c, d), (2.1)

where 0 < α < 1, with initial values p_{−1}^{α}(x) = 0, p_{0}^{α}(x) = 1, and A_{n+α}, B_{n+α}, C_{n+α} are given as in (1.2)–(1.4) with n replaced by n + α. The two linearly independent solutions to (1.1) found in [10] are

R_{n+α} = \frac{(abq^{n+α}, acq^{n+α}, adq^{n+α}, bcdq^{n+α}/z; q)_{∞}}{(bcq^{n+α}, bdq^{n+α}, cdq^{n+α}, azdq^{n+α}; q)_{∞}} (\frac{a}{z})^{n+α}

× _{8}W_{7}(bcd/qz; b/z, c/z, d/z, abcdq^{n+α−1}, q^{−α−n}; q, qz/a) (2.2)

and

S_{n+α} = \frac{(abcdq^{2n+2α}, bzq^{n+α+1}, czq^{n+α+1}, dzq^{n+α+1}, bcdzq^{n+α+1}; q)_{∞}}{(bcq^{n+α}, bdq^{n+α}, cdq^{n+α}, q^{n+α+1}, bcdzq^{2n+2α+1}; q)_{∞}} (az)^{n+α}

× _{8}W_{7}(bcdzq^{2n+2α}; bcq^{n+α}, bdq^{n+α}, cdq^{n+α}, q^{n+α+1}, zq/a; q, az). (2.3)

The weight function for the associated Askey–Wilson polynomials and an explicit polynomial representation were found by Ismail and Rahman in [10]. The latter is given by

p_{n}^{α}(x) = p_{n}^{α}(x; a, b, c, d|q)

= ∑_{k=0}^{n} \frac{(q^{−n}, abcdq^{2α+n−1}, abcdq^{2α−1}, ae^{iθ}, ae^{−iθ}; q)_{k}}{(q, abq^{α}, acq^{α}, adq^{α}, abcdq^{α−1}; q)_{k}} q^{k}

× _{10}W_{9}(abcdq^{2α+k−1}; q^{α}, bcq^{α−1}, bdq^{α−1}, cdq^{α−1}, q^{k+1}, abcdq^{2α+n+k−1}, q^{k−n}; q, a^{2}). (2.4)

There is another useful representation of the associated Askey–Wilson polynomials in terms of a double series due to Rahman,

p_{n}^{α}(x) = p_{n}^{α}(x; a, b, c, d|q)

= \frac{(abcdq^{2α−1}, q^{α+1}; q)_{n}}{(q, abcdq^{α−1}; q)_{n}} q^{−αn} ∑_{k=0}^{n} \frac{(q^{−n}, abcdq^{2α+n−1}; q)_{k}}{(q^{α+1}, abq^{α}; q)_{k}}

× \frac{(aq^{α}e^{iθ}, aq^{α}e^{−iθ}; q)_{k}}{(acq^{α}, acq^{α}; q)_{k}} ∑_{j=0}^{k} \frac{(q^{α}, abq^{α−1}, acq^{α−1}, adq^{α−1}; q)_{j}}{(q, abcdq^{2α−2}, aq^{α}e^{iθ}, aq^{α}e^{−iθ}; q)_{j}} q^{j},  (2.5)

where x = cos θ (see [9, Exercises 8.26–8.27] and [14], [15]). This formula will be the starting point for our investigation.

3. An Overview of the Main Result

To construct an eigenvalue problem for the associated Askey–Wilson polynomials, let us consider an auxiliary function u_{n}^{α}(x, y) in two variables, which for x = y coincides with the associated Askey–Wilson polynomials (up to a factor). We observe that the Askey–Wilson operator L_{0}(x) (in one variable x) maps u_{n}^{α}(x, y) to the n-th degree ordinary Askey–Wilson polynomial (up to some factors). A similar result is obtained for the operator L_{1}(y) applied to u_{n}^{α}(x, y) with respect to the second independent variable y. We will find an operator L_{2}(x), which maps certain multiples of (L_{1}(y) + λ) u_{n}^{α}(x, y) to (L_{0}(x) + λ)u_{n}^{α}(x, y). As a result, we obtain an eigenvalue problem of the form

\frac{(aq^{s}, aq^{−s}; q)_{∞}}{(aq^{α+s−1}, aq^{α−s−1}; q)_{∞}} (L_{2}(x) + λ)\frac{(aq^{α+s}, aq^{α−s}; q)_{∞}}{(aq^{s}, aq^{−s}; q)_{∞}} (L_{1}(y) + μ_{α}) u_{n}^{α}(x, y)

= \frac{4q^{9/2}}{(1 − q)^{2}γ} (L_{0}(x) + λ_{α+n})u_{n}^{α}(x, y) (3.1)

related to the associated Askey–Wilson polynomials of Ismail and Rahman (see Theorem 1 below for an exact statement). We shall use the normalization

p_{n}(x; a, b, c, d) = _{4}φ_{3}(\begin{matrix}q^{−n}, abcdq^{n−1}, aq^{s}, aq^{−s} \\ ab, ac, ad \end{matrix}; q, q)  (3.2)

for the ordinary Askey–Wilson polynomials throughout this paper.

Lemma 1. Let u_{n}^{α}(x, y) be the function in the two variables x and y defined by

u_{n}^{α}(x, y) := \frac{(aq^{s}, aq^{−s}, aq^{α+z}, aq^{α−z}; q)_{∞}}{(aq^{α+s}, aq^{α−s}, aq^{z}, aq^{−z}; q)_{∞}}

× ∑_{m=0}^{n} \frac{(q^{−n}, γq^{2α+n−1}, aq^{α+s}, aq^{α−s}; q)_{m}}{(q^{α+1}, abq^{α}, acq^{α}, adq^{α}; q)_{m}} q^{m}

× ∑_{k=0}^{m} \frac{(q^{α}, abq^{α−1}, acq^{α−1}, adq^{α−1}; q)_{k}}{(q, γq^{2α−2}, aq^{α+z}, aq^{α−z}; q)_{k}} q^{k},  (3.3)

with x(s) = (q^{s} + q^{−s})/2 and y(z) = (q^{z} + q^{−z})/2. Then u_{n}^{α}(x, y) satisfies an equation of the form

(L_{0}(x) + λ_{α+n})u_{n}^{α}(x, y) = f_{n}^{α}(x, y),  (3.4)

where L_{0}(x) = L (s; a, b, c, d) is the Askey–Wilson divided difference operator in the variable x given by (1.5). Here,

f_{n}^{α}(x, y) = −\frac{4q^{3/2−α}}{(1 − q)^{2}} \frac{(aq^{s}, aq^{−s}, aq^{α+z}, aq^{α−z}; q)_{∞}}{(aq^{α+s−1}, aq^{α−s−1}, aq^{z}, aq^{−z}; q)_{∞}}
× (q^{α}, abq^{α−1}, acq^{α−1}, adq^{α−1}; q)_{1}
× p_{n}(x; aq^{α−1}, bcdq^{α−1}, q^{1+z}, q^{1−z}),

and

λ_{α+n} = \frac{4q^{3/2}}{(1 − q)^{2}} (1 − q^{−α−n}) (1 − γq^{α+n−1}) , γ = abcd.

Note that f_{n}^{α}(x, y) contains the n-th degree ordinary Askey–Wilson polynomial of the form (3.2) in the variable x. Our function u_{n}^{α}(x, y) is the Askey–Wilson polynomial when α = 0 and a constant multiple of the associated Askey–Wilson polynomial if x = y.

Lemma 2. The function u_{n}^{α}(x, y) satisfies another equation, namely

(L_{1}(y) + μ_{α})u_{n}^{α}(x, y) = g_{n}^{α}(x, y),

where L_{1}(y) := L (y; q/a, q/b, q/c, q/d) is the Askey–Wilson divided difference operator in y. Here,

g_{n}^{α}(x, y) = −\frac{4q^{9/2−α}}{(1 − q)^{2}γ} \frac{(aq^{s}, aq^{−s}, aq^{α+z+1}, aq^{α−z+1}; q)_{∞}}{(aq^{α+s}, aq^{α−s}, aq^{z}, aq^{−z}; q)_{∞}}
× (q^{α}, abq^{α−1}, acq^{α−1}, adq^{α−1}; q)_{1}
× p_{n}(x; aq^{α}, bcdq^{α−2}, q^{1+z}, q^{1−z})

and

μ_{α} = \frac{4q^{3/2}}{(1 − q)^{2}} (1 − q^{α}) (1 − q^{3−α}/γ) .

Note that g_{n}^{α}(x, y) contains another n-th degree Askey–Wilson polynomial (3.2) in the same variable x.

Lemma 3. The difference differentiation formula

(L (x) + λ)p_{n}(x; a, b, c, d) = λp_{n}(x; a/q, bq, c, d)  (3.5)

holds for the Askey–Wilson polynomials given by (3.2). Here, L (x) = L (s; a, a/q, c, d) is the Askey– Wilson divided difference operator (1.5) and

λ = \frac{4q^{3/2}}{(1 − q)^{2}} (1 − ac/q) (1 − ad/q) .

Lemmas 1–3 allow us to establish the eigenvalue problem (3.1) for the associated Askey–Wilson functions (3.3), see the next section.

4. Main Result

With the help of Lemmas 1–3, we now identify an operator L_{2}(x) linking (L_{0}(x) + λ_{α+n})u_{n}^{α}(x, y) and (L_{1}(y) + λ_{−α})u_{n}^{α}(x, y) in such a way that an eigenvalue problem is formulated.

Theorem 1. Let L_{2}(x) = L(s; aq^{α}, aq^{α−1}, q^{1+z}, q^{1−z}) be the Askey–Wilson divided difference operator defined by (1.5) with

σ(s) = q^{−2s} (q^{s} − aq^{α}) (q^{s} − aq^{α−1}) (q^{s} − q^{1+z}) (q^{s} − q^{1−z})

and

λ = \frac{4q^{3/2}}{(1 − q)^{2}} (1 − aq^{α−z}) (1 − aq^{α+z}) .

Then an eigenvalue problem for the associated Askey–Wilson functions u_{n}^{α}(x, y) can be stated as

\frac{γ}{q^{3}} \begin{matrix}(aq^{s}, aq^{−s}; q) & ∞ \\ (aq^{α+s−1}, aq^{α−s−1}; q) & ∞ \end{matrix} (L_{2}(x) + λ) \begin{matrix}(aq^{α+s}, aq^{α−s}; q) & ∞ \\ (aq^{s}, aq^{−s}; q) & ∞ \end{matrix} (L_{1}(y) + μ_{α}) u_{n}^{α}(x, y)

= \frac{4q^{3/2}}{(1 − q)^{2}} (L_{0}(x) + λ_{α+n}) u_{n}^{α}(x, y), (4.1)

where L_{0}, L_{1}, λ_{α+n}, μ_{α} and u_{n}^{α}(x, y) are defined as in Lemmas 1–3.

Computational details are left to the reader. The explicit form of the difference operator in two variables on the left-hand side of the last equation has also been calculated, but it is too long to be displayed here.

5. Proofs

5.1. Proof of Lemma 1. Let λ_{ν} be an arbitrary number. We are looking for solutions of a generalization of the equation (3.4), namely,

(L_{0}(x) + λ_{ν})u_{n}^{α}(x, y) = f_{n}^{α}(x, y),

in terms of generalized powers (see (1.6) for the definition)

u_{n}^{α}(x, y) = ∑_{m=0}^{n} c_{m}v_{m}[x(s) − x(ξ)]^{(α+m)},

where

v_{m} = v_{m}(y) = \frac{(aq^{α+z}, aq^{α−z}; q)_{∞}}{(aq^{z}, aq^{−z}; q)_{∞}} ∑_{k=0}^{n} \frac{(q^{α}, abq^{α−1}, acq^{α−1}, adq^{α−1}; q)_{k}}{(q, γq^{2α−2}, aq^{α+z}, aq^{α−z}; q)_{k}} q^{k},

and γ = abcd. (This is an analogue of the power series expansion; see [7], [16, Exercises 2.9–2.11], and [17] for properties of the generalized powers.)

Apply the Askey–Wilson operator to u_{n}^{α}(x, y) to obtain

(L_{0}(x) + λ_{ν})u_{n}^{α}(x, y) = λ_{ν} ∑_{m=0}^{n} c_{m}v_{m}[x(s) − x(ξ)]^{(α+m)} + ∑_{m=0}^{n} c_{m}v_{m} L_{0}(x)[x(s) − x(ξ)]^{(α+m)},

since v_{m} is independent of x. By [7], we have

L_{0}(x)[x(s) − x(ξ)]^{(α+m)} = γ(α + m)γ(α + m − 1)σ(ξ − α − m + 1)[x(s) − x(ξ − 1)]^{(α+m−2)}
+ γ(α + m)τ_{α+m−1}(ξ − α − m + 1)[x(s) − x(ξ − 1)]^{(α+m−1)}
− λ_{α+m}[x(s) − x(ξ)]^{(α+m)}.

We use the same notations as in [7], [16, Exercise 2.25], or [17]. Choose a_{0} := ξ − α − m + 1 to be a root of the equation σ(a_{0}) = 0. Then ξ = a_{0} + α + m − 1, and one obtains

(L_{0}(x) + λ_{ν})u_{n}^{α}(x, y) = ∑_{m=0}^{n} c_{m}v_{m}γ(α + m)τ_{α+m−1}(a_{0})[x(s) − x(a_{0} + α + m − 2)]^{(α+m−1)}

+ ∑_{m=0}^{n} c_{m}v_{m}(λ_{ν} − λ_{α+m})[x(s) − x(a_{0} + α + m − 1)]^{(α+m)}

= c_{0}v_{0}γ(α)τ_{α−1}(a_{0})[x(s) − x(a_{0} + α − 2)]^{(α−1)}

+ ∑_{m=1}^{n} c_{m}v_{m}γ(α + m)τ_{α+m−1}(a_{0})[x(s) − x(a_{0} + α + m − 2)]^{(α+m−1)}

+ ∑_{m=0}^{n} c_{m}v_{m}(λ_{ν} − λ_{α+m})[x(s) − x(a_{0} + α + m − 1)]^{(α+m)}.

Letting m = k + 1, we get

(L_{0}(x) + λ_{ν})u_{n}^{α}(x, y) = c_{0}v_{0}γ(α)τ_{α−1}(a_{0})[x(s) − x(a_{0} + α − 2)]^{(α−1)}

+ ∑_{k=0}^{n−1} c_{k+1}v_{k+1}γ(α + k + 1)τ_{α+k}(a_{0})[x(s) − x(a_{0} + α + k − 1)]^{(α+k)}

+ ∑_{k=0}^{n} c_{k}v_{k}(λ_{ν} − λ_{α+k})[x(s) − x(a_{0} + α + k − 1)]^{(α+k)}. (5.1)

Note that for

v_{k} = ∑_{l=0}^{k} e_{l}, e_{l} := \frac{(aq^{α+z}, aq^{α−z}; q)_{∞}}{(aq^{z}, aq^{−z}; q)_{∞}} \frac{(q^{α}, abq^{α−1}, acq^{α−1}, adq^{α−1}; q)_{l}}{(q, γq^{2α−2}, aq^{α+z}, aq^{α−z}; q)_{l}} q^{l}

one has

v_{k+1} = v_{k} + e_{k+1} and v_{0} = e_{0.}

AN EIGENVALUE PROBLEM 7

After choosing λ_{ν} = λ_{α+n}, equation (5.1) becomes

(L_{0}(x) + λ_{α+n})u_{n}^{α}(x, y) = ∑_{k=−1}^{n−1} c_{k+1}e_{k+1}γ(α + k + 1)τ_{α+k}(a_{0})[x(s) − x(a_{0} + α + k − 1)]^{(α+k)}

+ ∑_{k=0}^{n−1} c_{k+1}v_{k}γ(α + k + 1)τ_{α+k}(a_{0})[x(s) − x(a_{0} + α + k − 1)]^{(α+k)}

+ ∑_{k=0}^{n−1} c_{k}v_{k}(λ_{α+n} − λ_{α+k})[x(s) − x(a_{0} + α + k − 1)]^{(α+k)}.

The latter two sums vanish if

c_{k+1}γ(α + k + 1)τ_{α+k}(a_{0}) = c_{k}(λ_{α+n} − λ_{α+k}).

Therefore,

(L_{0}(x) + λ_{α+n})u_{n}^{α}(x, y) = ∑_{k=−1}^{n−1} c_{k+1}e_{k+1}γ(α + k + 1)τ_{α+k}(a_{0})[x(s) − x(a_{0} + α + k − 1)]^{(α+k)}

= ∑_{m=0}^{n} c_{m}e_{m}γ(α + m)τ_{α+m−1}(a_{0})[x(s) − x(a_{0} + α + m − 2)]^{(α+m−1)} =: f_{n}^{α}(x, y).

Finally, we show that the function f_{n}^{α}(x, y) is, up to a factor, the n-th ordinary Askey–Wilson polynomial. The generalized powers have the property (see [16])

[x(s) − x(z)]^{(n+1)} = [x(s) − x(z)][x(s) − x(z − 1)]^{(n)},

which leads to

f_{n}^{α}(x, y) = ∑_{m=0}^{n} c_{m}e_{m}γ(α + m)τ_{α+m−1}(a_{0})\frac{[x(s) − x(a_{0} + α + m − 1)]^{(α+m)}}{[x(s) − x(a_{0} + α + m − 1)]} .

Moreover,

c_{m}[x(s) − x(a_{0} + α + m − 1)]^{(α+m)}

= c_{0} \frac{(q^{−n}, γq^{2α+n−1}; q)_{m}}{(q^{α+1}, abq^{α}, acq^{α}, adq^{α}; q)_{m}} q^{m} [x(s) − x(a_{0} + α + m − 1)]^{(α+m)}

= c_{0} φ_{m}(x) [x(s) − x(a_{0} + α − 1)]^{(α)},

where, by definition,

φ_{m}(x) := \frac{(aq^{s}, aq^{−s}; q)_{∞}}{(aq^{α+s}, aq^{α−s}; q)_{∞}} \frac{(q^{−n}, γq^{2α+n−1}, aq^{α+s}, aq^{α−s}; q)_{m}}{(q^{α+1}, abq^{α}, acq^{α}, adq^{α}; q)_{m}} q^{m}.

Therefore,

f_{n}^{α}(x, y) = \frac{(aq^{s}, aq^{−s}; q)_{∞}}{(aq^{α+s}, aq^{α−s}; q)_{∞}} \frac{(q^{−n}, γq^{2α+n−1}, aq^{α+s}, aq^{α−s}; q)_{m}}{(q^{α+1}, abq^{α}, acq^{α}, adq^{α}; q)_{m}} q^{m}

× \frac{(aq^{α+z}, aq^{α−z}; q)_{∞}}{(aq^{z}, aq^{−z}; q)_{∞}} \frac{(q^{α}, abq^{α−1}, acq^{α−1}, adq^{α−1}; q)_{m}}{(q, γq^{2α−2}, aq^{α+z}, aq^{α−z}; q)_{m}} q^{m}

8 ANDREA BRUDER, CHRISTIAN KRATTENTHALER^{†}, AND SERGEI K. SUSLOV

× \frac{γ(α + m)τ_{α+m−1}(a_{0})}{[x(s) − x(a_{0} + α + m − 1)]}.  (5.2)

Recall that a = q^{a_{0}} and

γ(α + m) = q^{− \frac{α+m−1}{2}} \frac{1 − q^{α+m}}{1 − q} ,
x(s) − x(a_{0} + α + m − 1) = −\frac{1}{2a}q^{−α−m+1}(1 − aq^{α−s+m−1})(1 − aq^{α+s+m−1}),
τ_{α+m−1}(a_{0}) = \frac{2}{a(1 − q)}q^{−2(α+m−1)+ \frac{α+m}{2}} (1 − abq^{α+m−1})(1 − acq^{α+m−1})(1 − adq^{α+m−1}),

which allows us to simplify the last term of (5.2) to

q^{m} \frac{γ(α + m)τ_{α+m−1}(a_{0})}{[x(s) − x(a_{0} + α + m − 1)]} = −4q^{\frac{3}{2} −α} \frac{1 − q^{α+m}}{1 − q} \frac{(1 − abq^{α+m−1})(1 − acq^{α+m−1})(1 − adq^{α+m−1})}{(1 − aq^{α−s+m−1})(1 − aq^{α+s+m−1})}.

Thus f_{n}^{α}(x, y) becomes

f_{n}^{α}(x, y) = \frac{−4q^{\frac{3}{2} −α}}{(1 − q)^{2}} \frac{(aq^{s}, aq^{−s}, aq^{α+z}, aq^{α−z}; q)_{∞}}{(aq^{α+s−1}, aq^{α−s−1}, aq^{z}, aq^{−z}; q)_{∞}} (q^{α}, abq^{α−1}, acq^{α−1}, adq^{α−1}; q)_{1}  (5.3)

× ∑_{m=0}^{n} \frac{(q^{−n}, γq^{2α+n−1}, aq^{α+s−1}, aq^{α−s−1}; q)_{m}}{(q, γq^{2α−2}, aq^{α+z}, aq^{α−z}; q)_{m}} q^{m}

= \frac{−4q^{\frac{3}{2} −α}}{(1 − q)^{2}} \frac{(aq^{s}, aq^{−s}, aq^{α+z}, aq^{α−z}; q)_{∞}}{(aq^{α+s−1}, aq^{α−s−1}, aq^{z}, aq^{−z}; q)_{∞}} (q^{α}, abq^{α−1}, acq^{α−1}, adq^{α−1}; q)_{1}

× p_{n}(x; aq^{α−1}, bcdq^{α−1}, q^{1+z}, q^{1−z}),

which completes the proof of the lemma. 5.2. Proof of Lemma 2. Consider the equation

(L_{1}(y) + λ_{ν})u_{n}^{α}(x, y) = g_{n}^{α}(x, y),

and rewrite u_{n}^{α}(x, y) in the form

u_{n}^{α}(x, y) = ∑_{m=0}^{n} c_{m}^{α}(aq^{α+s}, aq^{α−s}; q)_{m} \frac{(aq^{s}, aq^{−s}; q)_{∞}}{(aq^{α+s}, aq^{α−s}; q)_{∞}} v_{m}^{α}(y),

where

c_{m}^{α} = \frac{(q^{−n}, γq^{2α+n−1}; q)_{m}}{(q^{α+1}, abq^{α}, acq^{α}, adq^{α}; q)_{m}} q^{m}, γ = abcd,

and

v_{m}^{α}(y) = \frac{(aq^{α+z}, aq^{α−z}; q)_{∞}}{(aq^{z}, aq^{−z}; q)_{∞}} ∑_{k=0}^{m} \frac{(q^{α}, abq^{α−1}, acq^{α−1}, adq^{α−1}; q)_{k}}{(q, γq^{2α−2}, aq^{α+z}, aq^{α−z}; q)_{k}} q^{k}.

Apply the Askey–Wilson operator L_{1}(y) := L (y; q/a, q/b, q/c, q/d) to u_{n}^{α}(x, y) to obtain

(L_{1}(y) + λ_{ν})u_{n}^{α}(x, y) = ∑_{m=0}^{n} c_{m}^{α}(aq^{α+s}, aq^{α−s}; q)_{m} \frac{(aq^{s}, aq^{−s}; q)_{∞}}{(aq^{α+s}, aq^{α−s}; q)_{∞}} (L_{1}(y) + λ_{ν}) v_{m}^{α}(y).

AN EIGENVALUE PROBLEM 9

Let

v_{m}^{α}(y) := ∑_{k=0}^{m} \frac{c_{k}}{[x(s) − x(ξ)]^{(α+k)}}

in analogy with [7]. Then

(L_{1}(y) + λ_{ν})v_{m}^{α}(y) = λ_{ν} ∑_{k=0}^{m} \frac{c_{k}}{[x(s) − x(ξ)]^{(α+k)}} + ∑_{k=0}^{m} c_{k} L_{1}(y) (\frac{1}{[x(s) − x(ξ)]^{(α+k)}} ) .

By [7], we have

L_{1}(y) (\frac{1}{[x(s) − x(ξ)]^{(α+k)}} ) = \frac{γ(α + k)γ(α + k + 1)σ(ξ + 1)}{[x(z) − x(ξ + 1)]^{(α+k+2)}}

− \frac{γ(α + k)τ_{−α−k−1}(ξ + 1)}{[x(z) − x(ξ)]^{(α+k+1)}} − \frac{λ_{−α−k}}{[x(z) − x(ξ)]^{(α+k)}}

(see also [16, Exercise 2.25]). Upon choosing a_{0} := ξ + 1 to be a root of the equation σ(a_{0}) = 0, we obtain

(L_{1}(y) + λ_{ν})v_{m}^{α}(y) = λ_{ν} ∑_{k=0}^{m} \frac{c_{k}}{[x(s) − x(a_{0})]^{(α+k)}}

− ∑_{k=0}^{m} c_{k} (\frac{γ(α + k)τ_{−α−k−1}(a_{0})}{[x(z) − x(a_{0} − 1)]^{(α+k+1)}} + \frac{λ_{−α−k}}{[x(z) − x(a_{0} − 1)]^{(α+k)}} )

= ∑_{k=0}^{m} \frac{c_{k} (λ_{ν} − λ_{−α−k})}{[x(z) − x(a_{0} − 1)]^{(α+k)}} − ∑_{k=0}^{m} \frac{c_{k} γ(α + k)τ_{−α−k−1}(a_{0})}{[x(z) − x(a_{0} − 1)]^{(α+k+1)}}

= \frac{c_{0} (λ_{ν} − λ_{−α})}{[x(z) − x(a_{0} − 1)]^{(α)}} + ∑_{k=1}^{m} \frac{c_{k} (λ_{ν} − λ_{−α−k})}{[x(z) − x(a_{0} − 1)]^{(α+k)}}

− \frac{c_{m} γ(α + m)τ_{−α−m−1}(a_{0})}{[x(z) − x(a_{0} − 1)]^{(α+m+1)}} − ∑_{k=0}^{m−1} \frac{c_{k} γ(α + k)τ_{−α−k−1}(a_{0})}{[x(z) − x(a_{0} − 1)]^{(α+k+1)}} .

Now choose λ_{ν} = λ_{−α} and let k = l + 1. Then we obtain

(L_{1}(y) + λ_{ν})v_{m}^{α}(y) = −\frac{c_{m} γ(α + m)τ_{−α−m−1}(a_{0})}{[x(z) − x(a_{0} − 1)]^{(α+m+1)}}

+ ∑_{l=0}^{m−1} \frac{c_{l+1} (λ_{−α} − λ_{−α−l−1})}{[x(z) − x(a_{0} − 1)]^{(α+l+1)}} − ∑_{l=0}^{m−1} \frac{c_{l} γ(α + l)τ_{−α−l−1}(a_{0})}{[x(z) − x(a_{0} − 1)]^{(α+l+1)}} .

The latter two sums vanish if

c_{l+1} (λ_{−α} − λ_{−α−l−1}) = c_{l} γ(α + l)τ_{−α−l−1}(a_{0}).

In that case, we have

(L_{1}(y) + λ_{ν})v_{m}^{α}(y) = −\frac{c_{m} γ(α + m)τ_{−α−m−1}(a_{0})}{[x(z) − x(a_{0} − 1)]^{(α+m+1)}} = −\frac{c_{m+1} (λ_{−α} − λ_{−α−m−1})}{[x(z) − x(a_{0} − 1)]^{(α+m+1)}} =: h_{m}^{α}(y).

10 ANDREA BRUDER, CHRISTIAN KRATTENTHALER^{†}, AND SERGEI K. SUSLOV

Here,

\frac{c_{m+1}}{[x(z) − x(a_{0} − 1)]^{(α+m+1)}} = \frac{c_{0}}{[x(z) − x(a_{0} − 1)]^{(α)}} φ_{m+1}(z),
φ_{m+1}(z) = \frac{(q^{α}, abq^{α−1}, acq^{α−1}, adq^{α−1}; q)_{m+1}}{(q, γq^{2α−2}, aq^{α+z}, aq^{α−z}; q)_{m+1}} q^{m+1},
\frac{c_{0}}{[x(z) − x(a_{0} − 1)]^{(α)}} = \frac{(aq^{α+z}, aq^{α−z}; q)_{∞}}{(aq^{z}, aq^{−z}; q)_{∞}} ,
λ_{−α} − λ_{−α−m−1} = \frac{4}{(1 − q)^{2}γ} q^{\frac{7}{2} −α−m}(1 − q^{m+1})(1 − γq^{2α+m−2})

and

h_{m}^{α}(y) = −\frac{4q^{\frac{9}{2} −α}}{(1 − q)^{2}γ} \frac{(aq^{α+z}, aq^{α−z}; q)_{∞}}{(aq^{z}, aq^{−z}; q)_{∞}} \frac{(q^{α}, abq^{α−1}, acq^{α−1}, adq^{α−1}; q)_{m+1}}{(q, γq^{2α−2}; q)_{m}(aq^{α+z}, aq^{α−z}; q)_{m+1}} .

Therefore,

(L_{1}(y) + λ_{ν})u_{n}^{α}(x, y)

= ∑_{m=0}^{n} c_{m}^{α}(aq^{α+s}, aq^{α−s}; q)_{m} \frac{(aq^{s}, aq^{−s}; q)_{∞}}{(aq^{α+s}, aq^{α−s}; q)_{∞}} L_{1}(y)v_{m}^{α}(y)

= − ∑_{m=0}^{n} c_{m}^{α}(aq^{α+s}, aq^{α−s}; q)_{m} \frac{(aq^{s}, aq^{−s}; q)_{∞}}{(aq^{α+s}, aq^{α−s}; q)_{∞}}

× \frac{4q^{\frac{9}{2} −α}}{(1 − q)^{2}γ} \frac{(aq^{α+z}, aq^{α−z}; q)_{∞}}{(aq^{z}, aq^{−z}; q)_{∞}} \frac{(q^{α}, abq^{α−1}, acq^{α−1}, adq^{α−1}; q)_{m+1}}{(q, γq^{2α−2}; q)_{m}(aq^{α+z}, aq^{α−z}; q)_{m+1}}

= −\frac{4q^{\frac{9}{2} −α}}{(1 − q)^{2}γ} \frac{(aq^{s}, aq^{−s}, aq^{α+z+1}, aq^{α−z+1}; q)_{∞}}{(aq^{α+s}, aq^{α−s}, aq^{z}, aq^{−z}; q)_{∞}}

× ∑_{m=0}^{n} \frac{(q^{−n}, γq^{2α+n−1}, aq^{s}, aq^{−s}; q)_{m}}{(q^{α+1}, abq^{α}, acq^{α}, adq^{α}; q)_{m}} q^{m}

× \frac{(1 − q^{α})(1 − abq^{α−1})(1 − acq^{α−1})(1 − adq^{α−1})}{(q, γq^{2α−2}; q)_{m}(aq^{α+z+1}, aq^{α−z+1}; q)_{m}}

= −\frac{4q^{\frac{9}{2} −α}}{(1 − q)^{2}γ} \frac{(aq^{s}, aq^{−s}, aq^{α+z+1}, aq^{α−z+1}; q)_{∞}}{(aq^{α+s}, aq^{α−s}, aq^{z}, aq^{−z}; q)_{∞}}

× (q^{α}, abq^{α−1}, acq^{α−1}, adq^{α−1}; q)_{1}

× _{4}φ_{3}(\begin{matrix}q^{−n}, γq^{2α+n−1}, aq^{α+s}, aq^{α−s} \\ γq^{2α−2}, aq^{α+z+1}, aq^{α−z+1} \end{matrix}; q, q)

= −\frac{4q^{\frac{9}{2} −α}}{(1 − q)^{2}γ} \frac{(aq^{s}, aq^{−s}, aq^{α+z+1}, aq^{α−z+1}; q)_{∞}}{(aq^{α+s}, aq^{α−s}, aq^{z}, aq^{−z}; q)_{∞}}

× (q^{α}, abq^{α−1}, acq^{α−1}, adq^{α−1}; q)_{1} × p_{n}(x; aq^{α}, bcdq^{α−2}, q^{1+z}, q^{1−z}).

This completes the proof of the lemma.

5.3. Proof of Lemma 3. The structure of the Askey–Wilson operator in (1.5) and the basic hypergeometric series representation (3.2) suggest to look for a 4-term relation of the form

K_{1} _{4}φ_{3}(\begin{matrix}A, B, C, D \\ F, G, H \end{matrix}; q, q) + K_{2} _{4}φ_{3}(\begin{matrix}A, B, Cq, D/q \\ F, G, H \end{matrix}; q, q)

+ K_{3} _{4}φ_{3}(\begin{matrix}A, B, C/q, Dq \\ F, G, H \end{matrix}; q, q) + K_{4} _{4}φ_{3}(\begin{matrix}A, B, C/q, D/q \\ F, G/q, H/q \end{matrix}; q, q) = 0, (5.4)

for some undetermined coefficients K_{1}, K_{2}, K_{3} and K_{4} (up to a common factor). Doing a term-wise comparison, we may hope to find K_{1}, K_{2}, K_{3}, K_{4} which satisfy

K_{1}(1 − C)(1 − D)(1 − Cq^{k−1})(1 − Dq^{k−1})(1 − G/q)(1 − H/q)
+ K_{2}(1 − Cq^{k})(1 − Cq^{k−1})(1 − D/q)(1 − D)(1 − G/q)(1 − H/q)
+ K_{3}(1 − Dq^{k})(1 − Dq^{k−1})(1 − C/q)(1 − C)(1 − G/q)(1 − H/q)
+ K_{4}(1 − C/q)(1 − C)(1 − D/q)(1 − D)(1 − Gq^{k−1})(1 − Hq^{k−1}) = 0.

If we are successful, then the above equation does indeed imply the contiguous relation (5.4). In the equation, we compare coefficients of powers of q^{k}. This yields a system of 3 linear equations in the 4 unknowns K_{1}, K_{2}, K_{3}, K_{4}. With the help of Mathematica, we obtain the solution

K_{1} = \frac{(C − q)(D − q)(−GH − CDq + CGq + DGq + CHq + DHq − GHq − CDq^{2})}{(G − q)(H − q)(Cq − D)(Dq − C)},
K_{2} = \frac{(C − 1)(D − G)(D − H)(C − q)q}{(D − C)(G − q)(H − q)(Cq − D)},
K_{3} = \frac{(D − 1)(C − G)(C − H)(D − q)q}{(C − D)(G − q)(H − q)(Dq − C)},

where the free parameter K_{4} was chosen to be 1 (see Appendix A for the Mathematica code). The required 4-term contiguous relation is then given by

\frac{(C − q)(D − q)(−GH − CDq + CGq + DGq + CHq + DHq − GHq − CDq^{2})}{(G − q)(H − q)(Cq − D)(Dq − C)}

× _{4}φ_{3}(\begin{matrix}A, B, C, D \\ F, G, H \end{matrix}; q, q) +_{4}φ_{3}(\begin{matrix}A, B, C/q, D/q \\ F, G/q, H/q \end{matrix}; q, q)

+ \frac{(C − 1)(D − G)(D − H)(C − q)q}{(D − C)(G − q)(H − q)(Cq − D)} _{4}φ_{3}(\begin{matrix}A, B, Cq, D/q \\ F, G, H \end{matrix}; q, q)

+ \frac{(D − 1)(C − G)(C − H)(D − q)q}{(C − D)(G − q)(H − q)(Dq − C)} _{4}φ_{3}(\begin{matrix}A, B, C/q, Dq \\ F, G, H \end{matrix}; q, q) = 0.  (5.5)

(This 4-term contiguous relation for the _{4}φ_{3}-functions can be extended to an arbitrary _{r}ψ_{s}-function, see Appendix A for more details.)

When qABCD = F GH, in view of the structure of the Askey–Wilson operator in (1.5), equation (5.5) should become

(L(x) + λ) _{4}φ_{3}(\begin{matrix}A, B, Cq, D/q \\ F, G, H \end{matrix}; q, q)

12 ANDREA BRUDER, CHRISTIAN KRATTENTHALER^{†}, AND SERGEI K. SUSLOV

= \frac{σ(−s)}{Δx(s)∇x_{1}(s)} _{4}φ_{3}(\begin{matrix}A, B, Cq, D/q \\ F, G, H \end{matrix}; q, q) + \frac{σ(s)}{∇x(s)∇x_{1}(s)} _{4}φ_{3}(\begin{matrix}A, B, C/q, Dq \\ F, G, H \end{matrix}; q, q)

+ \frac{λΔx(s)∇x(s)∇x_{1}(s) − σ(s)Δx(s) − σ(−s)∇x(s)}{Δx(s)∇x(s)∇x_{1}(s)} _{4}φ_{3}(\begin{matrix}A, B, C, D \\ F, G, H \end{matrix}; q, q) .

Equating coefficients, one obtains

(1 − C/q)(1 − D/q)(D − C)(GH + CDq − CGq − DGq − CHq − DHq + GHq + CDq^{2})

= \frac{2qa^{3}}{1 − q} (σ(−s)∇x(s) + σ(s)Δx(s) − λΔx(s)∇x(s)∇x_{1}(s))

and

(D − C)(D − C/q)(−C + D/q) = \frac{2aq^{1/2}}{1 − q} ∇x_{1}(s)\frac{2a}{1 − q} Δx(s)\frac{2a}{1 − q} ∇x(s),
(C − 1)(G − D)(H − D)(q − C) = −qa^{2}σ(−s),
(D − C)(−C + D/q) = \frac{2aq^{1/2}}{1 − q} ∇x_{1}(s)\frac{2a}{1 − q} Δx(s),
(D − 1)(G − C)(H − C)(q − D) = −qa^{2}σ(s),
(D − C)(D − C/q) = \frac{2aq^{1/2}}{1 − q} ∇x_{1}(s)\frac{2a}{1 − q} ∇x(s),
(G − q)(H − q) = q^{2} \frac{(1 − q)^{2}}{4q^{3/2}} λ.

This gives the required formula (3.5) for the Askey–Wilson operator with

σ(s) = q^{−2s} (q^{s} − a) (q^{s} − a/q) (q^{s} − c) (q^{s} − d) , λ = \frac{4q^{3/2}}{(1 − q)^{2}} (1 − ac/q) (1 − ad/q) .

The proof of the lemma is complete.

Appendix A. 4-Term Contiguous Relations

In order to derive the contiguous relation (5.5), one can use the following Mathematica program:^{1}

1A corresponding Mathematica notebook is available on the article’s website http://www.mat.univie.ac.at/~kratt/artikel/AssAWPols.html.

AN EIGENVALUE PROBLEM 13

In[1]:= X1 = K1*(1 - C) (1 - D) (1 - C*K/q) (1 - D*K/q) (1 - G/q) (1 - H/q) +
             K2*(1 - C*K) (1 - C*K/q) (1 - D/q) (1 - D) (1 - G/q) (1 - H/q) +
             K3*(1 - D*K) (1 - D*K/q) (1 - C/q) (1 - C) (1 - G/q) (1 - H/q) +
             K4*(1 - C/q) (1 - C) (1 - D/q) (1 - D) (1 - G*K/q) (1 - H*K/q) ;
        X1 = Table[Coefficient[X1, K, i] == 0, i, 0, 2];
        X1 = Solve[X1, K1, K2, K3, K4];
        X1 = {K1 -> Factor[K1/.X1[[1]]], K2 -> Factor[K2/.X1[[1]]],
              K3 -> Factor[K3/.X1[[1]]], K4 -> Factor[K4/.X1[[1]]]}
Out[1]= {K1  > (K4 (C - q) (D - q)
>           (G H + C D q - C G q - D G q - C H q - D H q + G H q + C D q^{∧}2)) /
>           ((G - q) (H - q) (-D + C q) (C - D q)),
>        K2  > - ((-1 + C) (D - G) (D - H) K4 (C - q) q) /
>           ((C - D) (G - q) (H - q) (-D + C q)),
>        K3  > - ((-1 + D) (C - G) (C - H) K4 (D - q) q) /
>           ((C - D) (G - q) (H - q) (C - D q)),
>        K4  > K4}

It is evident from the proof of (5.5) that, actually, an extension for bilateral series (see [9, equation (5.1.1)] for the definition) with an arbitrary number of parameters holds, namely:

\begin{matrix}(c − q) & (d − q) & (−gh − cdq + cgq + dgq + chq + dhq − ghq − cdq^{2}) \\ (g − q) & (h − q) & (cq − d) & (dq − c) \end{matrix}

× _{r}ψ_{s}(\begin{matrix}a_{1}, …, a_{i}, c, d \\ b_{0}, …, b_{k}, g, h \end{matrix}; q, t)

+ \frac{(c − 1) (d − g) (d − h) (c − q) q}{(d − c) (g − q) (h − q) (cq − d)} _{r}ψ_{s}(\begin{matrix}a_{1}, …, a_{i}, cq, d/q \\ b_{0}, …, b_{k}, g, h \end{matrix}; q, t)

+ \frac{(d − 1) (c − g) (c − h) (d − q) q}{(c − d) (g − q) (h − q) (dq − c)} _{r}ψ_{s}(\begin{matrix}a_{1}, …, a_{i}, c/q, dq \\ b_{0}, …, b_{k}, g, h \end{matrix}; q, t)

+ _{r}ψ_{s}(\begin{matrix}a_{1}, …, a_{i}, c/q, d/q \\ b_{0}, …, b_{k}, g/q, h/q \end{matrix}; q, t) = 0.  (A.1)

Furthermore, in the same way, the following variation can be obtained:^{2}

\begin{matrix}(g − 1) & (h − 1) & (−gh − cdq + cgq + dgq + chq + dhq − ghq − cdq^{2}) \\ (c − 1) & (d − 1) & (gq − h) & (hq − g) \end{matrix}

× _{r}ψ_{s}(\begin{matrix}a_{1}, …, a_{i}, c, d \\ b_{0}, …, b_{k}, g, h \end{matrix}; q, t)

2Again, a corresponding Mathematica notebook is available on the article’s website http://www.mat.univie.ac.at/~kratt/artikel/AssAWPols.html.

+ \begin{matrix}(c − g) & (d − g) & (h − 1) & (h − q) \\ (c − 1) & (d − 1) & (h − g) & (gq − h) \end{matrix} _{r}ψ_{s}(\begin{matrix}a_{1}, …, a_{i}, c, d \\ b_{0}, …, b_{k}, gq, h/q \end{matrix}; q, t)

+ \begin{matrix}(c − h) & (d − h) & (g − 1) & (g − q) \\ (c − 1) & (d − 1) & (g − h) & (hq − g) \end{matrix} _{r}ψ_{s}(\begin{matrix}a_{1}, …, a_{i}, c, d \\ b_{0}, …, b_{k}, g/q, hq \end{matrix}; q, t)

+ _{r}ψ_{s}(\begin{matrix}a_{1}, …, a_{i}, cq, dq \\ b_{0}, …, b_{k}, gq, hq \end{matrix}; q, t) = 0.  (A.2)

Appendix B. An Inverse of the Askey–Wilson Operator

The Askey–Wilson divided difference operator on the left-hand side of equation (3.5) can be inverted by the method of Ref. [5]. The end result is

\frac{(q, q^{2}; q)_{∞}}{2π} ∫_{−1}^{1} L (x, y) p_{n} (x; a, b, c, d) ρ (x; a, b, c, d) dx = p_{n} (x; aq, b/q, c, d) ,  (B.1)

where ρ (x; a, b, c, d) is the weight function of the Askey–Wilson polynomials (3.2) and the kernel is given by

L (x, y) = (ac, ad, qce^{iφ}, qde^{−iφ}; q)_{1} \begin{matrix}(be^{iθ}, be^{−iθ}, qde^{iθ}, qde^{−iθ}, qae^{iφ}, qae^{−iφ}, qce^{iφ}, qce^{−iφ}; q) & ∞ \\ (qe^{iθ+iφ}, qe^{iθ−iφ}, qe^{iφ−iθ}, qe^{−iθ−iφ}; q) & ∞ \end{matrix}

× _{8}φ_{7}(\begin{matrix}qde^{−iφ}, q\sqrt{qde^{−iφ}}, −q\sqrt{qde^{−iφ}}, qe^{iθ−iφ}, qe^{−iθ−iφ}, qd/c, q \\ \sqrt{qde^{−iφ}}, \sqrt{qde^{−iφ}}, qde^{−iθ}, qde^{iθ}, q^{2}, qce^{−iφ}, qde^{−iφ} \end{matrix}; q, ce^{iφ}) .

Here, x = cos θ and y = cos φ. Computational details are left to the reader.

Acknowledgment. We thank Mizan Rahman for valuable discussions and encouragement.

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[15] M. Rahman: The Associated Classical Orthogonal Polynomials; in “Special Functions 2000”, J. Bustoz, M. E. H. Ismail and S. K. Suslov, eds., Kluwer Academic Publishers, Dordrecht, Boston, London, 2001.

[16] S. K. Suslov: The theory of difference analogues of special functions of hypergeometric type; Russian Math. Surveys 44 (1989), 227–278.

[17] S. K. Suslov: An introduction to basic Fourier series; Kluwer Series “Developments in Mathematics”, Vol. 9, Kluwer Academic Publishers, Dordrecht, Boston, London, 2003.

[18] G. Szegő: Orthogonal polynomials; Fourth Edition, Amer. Math. Soc. Colloq. Publ., Vol. 23, Providence, R. I., 1975.

Mathematics and Computer Science, Colorado College, Tutt Science Center, 14 E. Cache la Poudre St., Colorado Springs, CO 80903, U.S.A.

  E-mail address: Andrea.Bruder@coloradocollege.edu

Fakultät für Mathematik, Universität Wien, Nordbergstraße 15, A-1090 Vienna, Austria

  URL: http://www.mat.univie.ac.at/~kratt

School of Mathematical and Statistics Sciences & Mathematical. Computational and Modeling Sciences Center, Arizona State University, Tempe, AZ 85287-1804, U.S.A.

E-mail address: sks@asu.edu

  URL: http://hahn.la.asu.edu/~suslov/index.html