Applications of q-Taylor theorems ^{∗}

Mourad E.H. Ismail Department of Mathematics University of South Florida Tampa, FL 33620-5700

Dennis Stanton School of Mathematics University of Minnesota Minneapolis, MN 55455

Abstract We establish two new q-analogues of a Taylor series expansion for polynomials using special Askey-Wilson polynomial bases. Combining these expansions with an earlier expansion theorem we derive inverse relations and evaluate certain linearization coefficients. Byproducts include new summation theorems, new results on a q-exponential function, and quadratic transformations for q-series.

Running Title: q-Taylor Theorems

Mathematics Subject Classification. Primary 05A30, 41A58. Secondary 33C45.

Key words and phrases. Polynomial bases, Askey-Wilson operators, qexponential function, summation theorems, connection relations.

1 Introduction

The Taylor theorem for polynomials f(x) evaluates the coefficients f_{k} in the expansion

f(x) = ∑_{k=0}^{∞} f_{k}(x − c)^{k}, f_{k} = \frac{f^{(k)}(c)}{k!}. (1.1)

It is possible to generalize (1.1) by considering other polynomial bases and suitable operators. One such example, which has been previously considered [5], replaces (x − c)^{k} by

∗Research partially supported by NSF grants DMS 99-70865, DMS 99-70627, and Liu Bie Ju Center for Mathematical Sciences.

ϕ_{k}(x; a) = (ae^{iθ}, ae^{−iθ}; q)_{k} = ∏_{i=0}^{k−1}(1 − 2axq^{i} + a^{2}q^{2i}).

Since

lim_{q→1} ϕ_{k}(x; a) = (1 − 2ax + a^{2})^{k}

we can consider ϕ_{k}(x; a) as a q-analogue of (x − c)^{k} for c = a + 1/a. The Taylor theorem for ϕ_{k}(x; a) is stated as Theorem 1.1 below. In this paper we consider two other q-analogues of (1.1): Theorem 2.1 which has a qanalogue of (x − 1)^{k} and Theorem 2.2 for a q-analogue of x^{k}. We shall follow the notations and terminology in [1] and [4].

All three theorems use polynomial bases and the Askey-Wilson operator. We first define the Askey-Wilson operator D_{q}. Given a function f we set f̆(e^{iθ}) := f(x), x = cos θ, that is

f̆(z) = f((z + 1/z)/2), z = e^{iθ}.

In other words we think of f(cos θ) as a function of e^{iθ}. In this notation the Askey-Wilson divided difference operator D_{q} is defined by

(D_{q}f)(x) = \frac{f̆(q^{1/2}e^{iθ}) − f̆(q^{−1/2}e^{iθ})}{(q^{1/2} − q^{−1/2}) i sin θ}, x = cos θ. (1.2)

It easy to see that the action of D_{q} on Chebyshev polynomials is given by

D_{q}T_{n}(x) = \frac{q^{n/2} − q^{−n/2}}{q^{1/2} − q^{−1/2}} U_{n−1}(x),

hence D_{q} reduces the degree of a polynomial by one and

lim_{q→1} D_{q} = \frac{d}{dx}.

In the calculus of the Askey-Wilson operator [3, p. 32] the basis {ϕ_{n}(x; a) : n ≥ 0} plays the role played by the monomials {(1 − 2ax + a^{2})^{n} : n ≥ 0} in the differential and integral calculus. In fact

D_{q}(ae^{iθ}, ae^{−iθ}; q)_{n} = −\frac{2a(1 − q^{n})}{1 − q}(aq^{1/2}e^{iθ}, aq^{1/2}e^{−iθ}; q)_{n−1}. (1.3)

Ismail [5] proved the following Taylor theorem for polynomials f(x).

Theorem 1.1. If f(x) is a polynomial in x of degree n, then

f(x) = ∑_{k=0}^{n} f_{k}ϕ_{k}(x; a),

where

f_{k} = \frac{(q − 1)^{k}}{(2a)^{k}(q; q)_{k}} q^{−k(k−1)/4}(D_{q}^{k}f)(x_{k})

and x_{k} is given by

x_{k} := \frac{1}{2} (aq^{k/2} + q^{−k/2}/a) .

In our recent work [7] it was realized that the basis {ϕ_{n}(x) : n ≥ 0},

ϕ_{n}(cos θ) = (q^{1/4}e^{iθ}, q^{1/4}e^{−iθ}; q^{1/2})_{n}, (1.4)

plays an important role in the calculus of the Askey-Wilson operators and basic hypergeometric functions. Rahman [11], [3, p. 23] had previously used this basis for expressing continuous q-Jacobi polynomials [3] in the basis {ϕ_{n}(x)}. Since

lim_{q→1} ϕ_{n}(x) = 2^{n}(1 − x)^{n},

we consider ϕ_{n}(x) as a q-analogue of (x − 1)^{n}. We shall also see that the basis {ψ_{n}(x) : n ≥ 0},

ψ_{n}(cos θ) = (1 + e^{2iθ})(−q^{2−n}e^{2iθ}; q^{2})_{n−1}e^{−inθ}, (1.5)

has a nice relationship to the Askey-Wilson operator. Since

lim_{q→1} ψ_{n}(x) = 2^{n}x^{n}

we consider ψ_{n}(x) as a q-analogue of x^{n}. Note that as functions of θ the polynomials {ψ_{n}(cos θ)} are essentially partial theta functions [14] since

ψ_{2n}(cos θ) = q^{n(1−n)}(−e^{2iθ}, −e^{−2iθ}; q^{2})_{n},
ψ_{2n+1}(cos θ) = 2q^{−n^{2}} cos θ (−qe^{2iθ}, −qe^{−2iθ}; q^{2})_{n}.

In this paper we give q-Taylor theorems for polynomials using {ϕ_{n}(x)} and {ψ_{n}(x)} in Theorems 2.1 and 2.2. We then explore consequences of these results to connection coefficient problems. They are applied to obtain connection coefficient results in Theorems 2.3 and 2.4. These two theorems are then used to give a simple proof of our new [7] representation (Corollary 2.5) for E_{q}, the addition theorem for the q-exponential function E_{q}, (Corollary 2.6) [6], [13], and another new representation for E_{q} (Corollary 2.7). In §3 we explicitly evaluate the coefficients in the expansion of a product ϕ_{m}(x; a)ϕ_{n}(x; b) in terms of {ϕ_{k}(x; c)}. In §4 all three theorems are used to derive expansions for the continuous q-ultraspherical polynomials (Propositions 4.1, 4.2, and 4.3) which include known quadratic transformations. Section 5 contains remarks on representations of continuous q-ultraspherical polynomials and implications of Theorems 1.1 and 2.1.

We note that the q-Taylor expansions derived and applied here are different from the recent results in [10], [12].

2 More q-Taylor theorems

In this section we give the version of Theorem 1.1 for {ϕ_{n}(x)} and {ψ_{n}(x)}, which are Theorem 2.1 and Theorem 2.2. We apply the resulting facts to the q-exponential function E_{q}, in Corollaries 2.5, 2.6 and 2.7.

It is straightforward to see that

D_{q}ϕ_{n}(x) = −2q^{1/4} \frac{1 − q^{n}}{1 − q} ϕ_{n−1}(x). (2.1)

Theorem 2.1. If f(x) is a polynomial in x of degree n, then

f(x) = ∑_{k=0}^{n} f_{k}ϕ_{k}(x),

where

f_{k} = \frac{(q − 1)^{k}}{2^{k}q^{k/4}(q; q)_{k}} (D_{q}^{k} f)(ζ_{0})

and ζ_{0} = (q^{1/4} + q^{−1/4})/2.

Proof. The proof is an immediate consequence of (2.1), since ϕ_{n}(ζ_{0}) = 0 for n ≥ 1.

It is important to contrast the series expansion of Theorem 1.1 with evaluations at variable points x_{k} and that of Theorem 2.1, where the coefficients depend on evaluations at a fixed point ζ_{0}.

For {ψ_{n}(x)} we have the following result, which uses

D_{q}ψ_{n}(x) = 2q^{(1−n)/2} \frac{1 − q^{n}}{1 − q} ψ_{n−1}(x). (2.2)

Theorem 2.2. If f(x) is a polynomial in x of degree n, then

f(x) = ∑_{k=0}^{n} f_{k}ψ_{k}(x),

where

f_{k} = \frac{q^{(k^{2}−k)/4}(1 − q)^{k}}{2^{k}(q; q)_{k}} (D_{q}^{k} f)(0) .

Proof. The proof is an immediate consequence of (2.2), since if e^{2iθ} = −1, namely θ = π/2, cos θ = 0, ψ_{n}(0) = 0 for n ≥ 1.

The first application of Theorem 2.1 is to expand ϕ_{n}(x; a) in terms of {ϕ_{k}(x) : 0 ≤ k ≤ n}. Theorem 2.3. The following summation theorem holds

\frac{(ae^{iθ}, ae^{−iθ}; q)_{n}}{(aq^{−1/4}; q^{1/2})_{2n}}

= _{4}ϕ_{3} ( \begin{matrix}q^{−n/2}, −q^{−n/2}, q^{1/4}e^{iθ}, q^{1/4}e^{−iθ} \\ −q^{1/2}, aq^{−1/4}, q^{−n+3/4}/a \end{matrix} | q^{1/2}, q^{1/2}) .

Proof. Use Theorem 2.1 and (1.3).

We now give a direct proof of Theorem 2.3. Proof. Use the Sears transformation [4, (III.15)] with

A = −q^{−n/2}, B = q^{1/4}e^{iθ}, C = q^{1/4}e^{−iθ}

D = aq^{−1/4}, E = −q^{1/2}, F = q^{−n+3/4}/a.

Then use the quadratic transformation [4, (III.21)] with C = q^{−n/2}, D = −q^{−n/2}, A^{2} = aq^{−1/2}e^{−iθ}, B^{2} = aq^{−1/2}e^{iθ}.

The result is that the right-hand side of the equation in Theorem 2.3 is

\frac{(q^{(n+1)/2}, −q^{(−n+3/2)/2}/a; q^{1/2})_{n}}{(−q^{1/2}, q^{−n+3/4}/a; q^{1/2})_{n}} (−1)^{n}q^{−n^{2}/2}

×_{3}ϕ_{2} ( \begin{matrix}aq^{−1/2}e^{iθ}, aq^{−1/2}e^{−iθ}, q^{−n} \\ a^{2}q^{−1/2}, q^{−n+1/2} \end{matrix} | q, q) .

The _{3}ϕ_{2} is balanced and its sum is (ae^{iθ}, ae^{−iθ}; q)_{n}/(a^{2}q^{−1/2}, q^{1/2}; q)_{n}, see [4, (II.12)].

Although Theorem 2.3 can be proved from the existing literature we nevertheless believe it is interesting and is worth recording as a sum of a special balanced _{4}ϕ_{3}.

Since Theorem 2.2 is just an expansion of ϕ_{n}(x; a) in terms of {ϕ_{k}(x)}, it is natural to record the inverse relation expanding ϕ_{n}(x) in {ϕ_{k}(x; a)}. The result is

\frac{(q^{1/4}e^{iθ}, q^{1/4}e^{−iθ}; q^{1/2})_{n}}{(q^{1/4}a, q^{1/4}/a; q^{1/2})_{n}} (2.3)

= _{3}ϕ_{2} ( \begin{matrix}q^{−n}, ae^{iθ}, ae^{−iθ} \\ aq^{(1−2n)/4}, aq^{(3−2n)/4} \end{matrix} | q, q) .

The _{3}ϕ_{2} in (2.3) is balanced, hence it can be summed by [4, (II.12)], proving the result.

We next expand ϕ_{n}(x; a) in terms of {ψ_{k}(x) : 0 ≤ k ≤ n}.

Theorem 2.4. The following summation theorem holds

(ae^{iθ}, ae^{−iθ}; q)_{n} = = ∑_{k=0}^{n} \left[{n \atop k}\right]_{q}q^{\binom{k}{2}}(−a)^{k}(−a^{2}q^{k}; q^{2})_{n−k}ψ_{k}(cos θ).

Proof. Use Theorem 2.2 and (1.3).

We next give several corollaries to Theorem 2.3 and Theorem 2.4, which concern the q-exponential functions of [9]

(2.4)  E_{q}(cos θ; t) = \frac{(t^{2}; q^{2})_{∞}}{(qt^{2}; q^{2})_{∞}} ∑_{n=0}^{∞} \frac{(−it)^{n}}{(q; q)_{n}} q^{n^{2}/4}
×(−iq^{(1−n)/2}e^{iθ}, −iq^{(1−n)/2}e^{−iθ}; q)_{n},

E_{q}(cos θ, cos ϕ; t) (2.5)

= \frac{(t^{2}; q^{2})_{∞}}{(qt^{2}; q^{2})_{∞}} ∑_{n=0}^{∞}(−e^{i(ϕ+θ)}q^{(1−n)/2}, −e^{i(ϕ−θ)}q^{(1−n)/2}; q)_{n}

× \frac{(te^{−iϕ})^{n}}{(q; q)_{n}} q^{n^{2}/4}.

Note that (2.4) is an expansion for E_{q}(x; t) in terms of {ϕ_{n}(x; −iq^{(1−n)/2})}, so it reasonable to find the expansion in the bases {ϕ_{n}(x)} and {ψ_{n}(x)}. Corollary 2.5 gives the {ϕ_{n}(x)} expansion, and Corollary 2.7 gives the {ψ_{n}(x)} expansion.

Corollary 2.5. The function E_{q}(x; t) has the representations

E_{q}(cos θ; t) = \frac{(−t; q^{1/2})_{∞}}{(qt^{2}; q^{2})_{∞}} _{2}ϕ_{1} ( \begin{matrix}q^{1/4}e^{iθ}, q^{1/4}e^{−iθ} \\ −q^{1/2} \end{matrix} | q^{1/2}, −t)
= \frac{(t; q^{1/2})_{∞}}{(qt^{2}; q^{2})_{∞}} _{2}ϕ_{1} ( \begin{matrix}−q^{1/4}e^{iθ}, −q^{1/4}e^{−iθ} \\ −q^{1/2} \end{matrix} | q^{1/2}, t) .

Corollary 2.6. We have

E_{q}(cos θ, cos ϕ; t) = E_{q}(cos θ; t)E_{q}(cos ϕ; t).

Corollary 2.7. The function E_{q}(x; t) has the expansion formula

E_{q}(cos θ; t) = ∑_{k=0}^{∞} \frac{(1 + e^{2iθ})(−e^{2iθ}q^{−k}; q^{2})_{k}}{(q; q)_{k}(1 + e^{2iθ}q^{−k})} q^{k^{2}/4}t^{k}e^{−ikθ}.

It must emphasized that the first equation in Corollary 2.5 says that the q-Taylor expansion of Theorem 2.1 holds for the function f(x) = E_{q}(x; t), because

D_{q}E_{q}(cos θ; t) = \frac{2tq^{1/4}}{1 − q} E_{q}(cos θ; t),
E_{q}(ζ_{0}; t) = \frac{(−t; q^{1/2})_{∞}}{(qt^{2}; q^{2})_{∞}} .

Similarly Theorem 2.2 holds for f(x) = E_{q}(x; t) because

E_{q}(0; t) = 1.

Corollary 2.5 is Corollary 4.3 in [7], where two other proofs are given. Corollary 2.6 is the addition theorem for E_{q}; [13], [6]; and Corollary 2.7 is new.

Proof. We prove Corollary 2.5 and 2.6 simultaneously from Theorem 2.3. We rewrite Theorem 2.3 as

ϕ_{n}(x; a) = ∑_{k=0}^{n} \frac{a^{k}q^{k(k−2)/4} (q; q)_{n}}{(q; q)_{k}(q; q)_{n−k}} (aq^{(2k−1)/4}; q^{1/2})_{2n−2k}ϕ_{k}(x).

Thus

∑_{n=0}^{∞} \frac{q^{n^{2}/4}t^{n}a^{n}}{(q; q)_{n}} ϕ_{n}(x; aq^{(1−n)/2})

= ∑_{n=0}^{∞} ∑_{k=0}^{n} \frac{a^{n+k}t^{n}q^{(n−k)^{2}/4}}{(q; q)_{k}(q; q)_{n−k}} (aq^{(k−n+1/2)/2}; q^{1/2})_{2n−2k} ϕ_{k}(x).

After replacing n by n + k the n-sum is

∑_{n=0}^{∞} \frac{a^{2n}t^{n}(−1)^{n}}{(q; q)_{n}} (aq^{1/4}, q^{1/4}/a; q^{1/2})_{n}.

Therefore

\frac{(t^{2}; q^{2})_{∞}}{(qt^{2}; q^{2})_{∞}} ∑_{n=0}^{∞} \frac{q^{n^{2}/4}t^{n}a^{n}}{(q; q)_{n}} ϕ_{n}(x; aq^{(1−n)/2}) (2.6)

= \frac{(t^{2}; q^{2})_{∞}}{(qt^{2}; q^{2})_{∞}} [∑_{n=0}^{∞} \frac{(−a^{2}t)^{n}}{(q; q)_{n}} (aq^{1/4}, q^{1/4}/a; q^{1/2})_{n}]

× [∑_{k=0}^{∞} \frac{a^{2k}t^{k}}{(q; q)_{k}} ϕ_{k}(x)]

Equation (2.6) proves both Corollary 2.5 and 2.6. If a = ±i, the left side of (2.6) is the definition (2.4) of E_{q}(cos θ; t), while the n-sum on the right side is evaluable to infinite products by the q-binomial theorem. The k-sum is the _{2}ϕ_{1} for both equations in Corollary 2.5. For Corollary 2.6, replace a by −e^{iϕ} and t by −e^{−2iϕ}t in (2.6) and use Corollary 2.5.

The identical steps may be performed using Theorem 2.4 to find the {ψ_{n}} analogue of (2.6)

\frac{(t^{2}; q^{2})_{∞}}{(qt^{2}; q^{2})_{∞}} ∑_{n=0}^{∞} \frac{q^{n^{2}/4}t^{n}a^{n}}{(q; q)_{n}} ϕ_{n}(x; aq^{(1−n)/2}) (2.7)

= \frac{(t^{2}; q^{2})_{∞}}{(qt^{2}; q^{2})_{∞}} [∑_{k=0}^{∞} \frac{(−a^{2}t)^{k}q^{k^{2}/4}}{(q; q)_{k}} ψ_{k}(x)]

× [∑_{n=0}^{∞} \frac{(−a^{2}q^{1−n}; q^{2})_{n}}{(q; q)_{n}} a^{n}t^{n}q^{n^{2}/4}]

This time the choice a = ±i in (2.7) allows the n-sum to be evaluated, and the result is Corollary 2.7. As before putting a = −e^{iϕ} and t = −e^{−2iϕ}t gives Corollary 2.6 and as bonus E_{q}(0, cos θ; t) = E_{q}(cos θ; t).

Corollary 2.7 may be also proven by splitting the definition (2.4) into the even and odd terms, applying a_{2}ϕ_{1} transformation to each, and recombining the terms.

Because the polynomials {ϕ_{n}(x; aq^{(1−n)/2})} are fundamental to the study of E_{q}(cos θ; t), it is worthwhile to record another connection coefficient result which is equivalent to the addition theorem in Corollary 2.6.

Corollary 2.8. The polynomials {ϕ(x; aq^{(1−n)/2})} satisfy the connection relation

ϕ_{n}(x; aq^{(1−n)/2}) = ∑_{k=0}^{n} \left[{n \atop k}\right]_{q} \frac{1 + a^{2}}{1 + a^{2}q^{−k}} (−a^{2}q^{−k}; q^{2})_{k}q^{−k(n−k)/2}(ia)^{n−k}
×ϕ_{n−k}(x; −iq^{(1−n+k)/2}).

Proof. If a = e^{iϕ}, ϕ_{n}(x; aq^{(1−n)/2})/a^{n} is a polynomial in y = cos ϕ of degree n due to

ϕ_{n}(cos θ; e^{iϕ}q^{(1−n)/2})e^{−inϕ} = (−1)^{n}ϕ_{n}(cos ϕ; e^{iθ}q^{(1−n)/2})e^{−inθ}. (2.8)

The result then follows from Theorem 2.2 applied to a function of y, y = cos ϕ.

3 Linearization of Products

In this section we use Theorem 1.1 to evaluate the linearization coefficients c_{k,m,n}(a, b, c) in

ϕ_{m}(x; b)ϕ_{n}(x; c) = ∑_{k=0}^{m+n} c_{k,m,n}(a, b, c)ϕ_{k}(x; a), (3.1)

our main result is Theorem 3.1.

We shall use the q-Leibniz role [5]

D_{q}^{n}(fg) = ∑_{k=0}^{n} \left[{n \atop k}\right]_{q}q^{k(k−n)/2} (η_{q}^{k}D_{q}^{n−k}f) (η_{q}^{k−n}D_{q}^{k}g) , (3.2)

where (η_{q}^{a})f(x) = f̆(q^{a/2}e^{iθ}), x = cos θ. (3.3) We see that to apply Theorem 1.1 we need to evaluate η_{q}^{s}D_{q}^{k−s}ϕ_{m}(x; b) and η_{q}^{s−k}D_{q}^{s}ϕ_{n}(x; c) at x = x_{k}. It is easy to see from (1.3) that if x = cos θ

η_{q}^{s}D_{q}^{k−s}ϕ_{m}(x; b) = \frac{(2b)^{k−s}(q; q)_{m}q^{( \begin{matrix}k−s \\ 2 \end{matrix} )/2}}{(q − 1)^{k−s}(q; q)_{m+s−k}}
×(bq^{k/2}e^{iθ}, bq^{−s+k/2}e^{−iθ}; q)_{m+s−k},
η_{q}^{s−k}D_{q}^{s}ϕ_{n}(x; c) = \frac{(2c)^{s}(q; q)_{n}q^{( \begin{matrix}s \\ 2 \end{matrix} )/2}}{(q − 1)^{s}(q; q)_{n−s}} (cq^{s−k/2}e^{iθ}, cq^{k/2}e^{−iθ}; q)_{n−s}.

This leads to

(3.4) c_{k,m,n}(a, b, c) = \frac{b^{k}(q, ab; q)_{m}(q, ac; q)_{n}}{a^{k}(ab; q)_{k}} ∑_{s=0}^{k} \frac{c^{s}(abq^{m}; q)_{s}}{b^{s}(q, ac; q)_{s}}

×\frac{(c/a; q)_{n−s}(bq^{−s}/a; q)_{s+m−k}}{(q; q)_{k−s}(q; q)_{n−s}(q; q)_{m+s−k}} q^{s(s−k)}.

Thus we have proved the following theorem. Theorem 3.1. We have the summation identity

\frac{(be^{iθ}, be^{−iθ}; q)_{m}(ce^{iθ}, ce^{−iθ}; q)_{n}}{(q, ab; q)_{m}(q, ac; q)_{n}}

= ∑_{k,s≥0} \frac{b^{k−s}c^{s}(abq^{m}; q)_{s}(c/a; q)_{n−s}(bq^{−s}/a; q)_{s+m−k}}{a^{k}(ab; q)_{k}(q, ac; q)_{s}(q; q)_{k−s}(q; q)_{n−s}(q; q)_{m+s−k}}

×q^{s(s−k)}(ae^{iθ}, ae^{−iθ}; q)_{k}.

The sum is so that 0 ≤ s ≤ min{n, k}, 0 ≤ k ≤ s + m.

Mizan Rahman kindly pointed out that the spcial case a = b = c of Theorem 3.1 is (8.1.2) of [4].

We now give another proof of Theorem 3.1. First compute the k sum by replacing k by k + s then observe that the right-hand side becomes

∑_{s≥0} \frac{c^{s}(abq^{m}; q)_{s}(c/a; q)_{n−s}}{a^{s}(ab, q, ac; q)_{s}(q; q)_{n−s}} ϕ_{s}(cos θ; a)

× ∑_{k≥0} \frac{b^{k}a^{−k}(bq^{−s}/a; q)_{m−k}}{a^{k}(abq^{s}; q)_{k}(q; q)_{k}(q; q)_{m−k}} q^{−sk}(aq^{s}e^{iθ}, aq^{s}e^{−iθ}; q)_{k}.

= ∑_{s≥0} \frac{c^{s}(abq^{m}, ae^{iθ}, ae^{−iθ}; q)_{s}(c/a; q)_{n−s}(bq^{−s}/a; q)_{m}}{a^{s}(ab, q, ac; q)_{s}(q; q)_{n−s}(q; q)_{m}}

×_{3}ϕ_{2} ( \begin{matrix}q^{−m}, aq^{s}e^{iθ}, aq^{s}e^{−iθ} \\ abq^{s}, aq^{s+1−m}/b, \end{matrix} | q, q) .

The _{3}ϕ_{2} is now summable by the q-Pfaff-Saalschütz theorem [4, (II.12)] and the result is that each side in the above expression equals

\frac{ϕ_{m}(x; a)}{(ab; q)_{m}} ∑_{s=0}^{n} \frac{c^{s}(ae^{iθ}, ae^{−iθ}; q)_{s}(c/a; q)_{n−s}}{a^{s}(q, ac; q)_{s}(q; q)_{n−s}(q; q)_{m}} ,

which is again summable by the q-Pfaff-Saalschütz theorem and Theorem 3.1 follows.

In terms of basic hypergeometric functions, Theorem 3.1 can be restated as

(3.5) c_{k,m,n}(a, b, c) = \frac{b^{k}(q, ab; q)_{m}(ac, c/a; q)_{n}(b/a; q)_{m−k}}{a^{k}(q, ab; q)_{k}(q; q)_{m−k}}

×_{4}ϕ_{3} ( \begin{matrix}q^{−n}, q^{−k}, abq^{m}, qa/b \\ ac, q^{1−n}a/c, q^{m−k+1} \end{matrix} | q, q) , m ≥ k.

For k > m the s-sum is now over s, min{k, n} ≥ s ≥ k − m, so we replace s by s + k − m. The result is

c_{k,m,n}(a, b, c) (3.6)

= \frac{b^{m}c^{k−m}(q, ac; q)_{n}(c/a; q)_{m+n−k}}{a^{k}(q, ac; q)_{k−m}(q; q)_{n+m−k}}

×_{4}ϕ_{3} ( \begin{matrix}q^{−m}, q^{k−m−n}, abq^{k}, q^{1+k−m}a/b \\ acq^{k−m}, q^{1−n−m+k}a/c, q^{k−m+1} \end{matrix} | q, q) , m ≤ k.

These _{4}ϕ_{3}’s are terminating and balanced and are in general form because they depend on six parameters. This is an interesting observation because the coefficients must possess the symmetry relation c_{k,m,n}(a, b, c) = c_{k,n,m}(a, c, b). This symmetry leads to the Sears transformation as follows. Fix k as a positive integer and a terminating parameter, then observe that c_{k,m,n}(a, b, c) = c_{k,n,m}(a, c, b) is a rational function identity valid for infinitely many integer values of m and n. This allows us to replace q^{m} and q^{n} by two general parameters M and N respectively and we see that the right-hand side of (3.5) is

\frac{b^{k}(q, ab, ac, q^{1−k}M, b/a, c/a; q)_{∞}}{a^{k}(q, ab; q)_{k}(abM, acN, cN/a, bq^{−k}M/a; q)_{∞}}

×_{4}ϕ_{3} ( \begin{matrix}q^{−k}, 1/N, abM, qa/b \\ ac, qa/(Nc), Mq^{1−k} \end{matrix} | q, q) ,

which must be symmetric under the exchange (a, M) and (c, N), hence

\frac{b^{k}(q^{1−k}M; q)_{k}}{(ab; q)_{k}(bq^{−k}M/a; q)_{k}}_{4}ϕ_{3} ( \begin{matrix}q^{−k}, 1/N, abM, qa/b \\ ac, qa/(Nc), Mq^{1−k} \end{matrix} | q, q) (3.7)

= \frac{c^{k}(q^{1−k}N; q)_{k}}{(ac; q)_{k}(cq^{−k}N/a; q)_{k}}_{4}ϕ_{3} ( \begin{matrix}q^{−k}, 1/M, acN, qa/c \\ ab, qa/(Mb), Nq^{1−k} \end{matrix} | q, q) .

Equation (3.7) is the version (III.16) in [4] of the Sears transformation and iterating it leads to the standard form of the Sears transformation [4, (III.15)].

When c = a (3.4) implies

ϕ_{m}(x; b)ϕ_{n}(x; a) (3.8)

= \frac{(q; q)_{m}(ab; q)_{m+n}}{(ab; q)_{n}} ∑_{k=0}^{m} \frac{b^{k}q^{−nk}(bq^{−n}/a; q)_{m−k}}{a^{k}(q, abq^{n}; q)_{k}(q; q)_{m−k}} ϕ_{k+n}(x; a).

This result is not new since [5, (2.2)] is

ϕ_{m}(x; b) = (q, ab; q)_{m} ∑_{k=0}^{m} \frac{b^{k}(b/a; q)_{m−k}}{a^{k}(q, ab; q)_{k}(q; q)_{m−k}} ϕ_{k}(x; a), (3.9)

so we can replace a by aq^{n}, multiply by ϕ_{n}(x; a) and apply the identity ϕ_{n+k}(x; a) = ϕ_{k}(x; a)ϕ_{n}(x; aq^{k}). This yields (3.8). This indicates that (3.8) is equivalent to the connection coefficients between two ϕ_{n}’s with different a’s, which was shown to be equivalent to the q-Pfaff Saalschütz theorem [5]. Having said that (3.8) is not new, nevertheless, the special case m = n and b = −a gives the useful connection coefficient formula

(a^{2}e^{2iθ}, a^{2}e^{−2iθ}; q^{2})_{n} (3.10)

= (a; q)_{n}(−a^{2}; q)_{2n} ∑_{k=0}^{n} \frac{(−1)^{n+k}q^{n(k−n)}(−q^{−n}; q)_{k}}{(q; q)_{k}(q; q)_{n−k}(−a^{2}; q)_{2n−k}}

×(ae^{iθ}, ae^{−iθ}; q)_{2n−k}.

4 q-Ultraspherical polynomials

In this section we apply Theorems 1.1, 2.1 and 2.2 to the q-ultraspherical polynomials which have the closed form [2], [4]

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

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

and satisfy

D_{q}C_{n}(x; β|q) = \frac{2(1 − β)}{1 − q}q^{(1−n)/2}C_{n−1}(x; qβ|q). (4.2)

Furthermore the C_{n}’s have the generating function [2]

∑_{n=0}^{∞} C_{n}(cos θ; β)t^{n} = \frac{(βte^{iθ}, βte^{−iθ}; q)_{∞}}{(te^{iθ}, te^{−iθ}; q)_{∞}} . (4.3)

We first expand C_{n}(x; β|q) in terms of {ϕ_{k}(x)}. It is clear from (4.3) that

C_{n}(ζ_{0}; β|q) = \frac{(β; q^{1/2})_{n}}{(q^{1/2}; q^{1/2})_{n}} q^{−n/4}. (4.4)

Thus Theorem 2.1, (4.2) and (4.4) imply the following proposition, which is (7.5.34) in [4].

Proposition 4.1. The continuous q-ultraspherical polynomials have the basic hypergeometric representation

C_{n}(cos θ; β|q) = q^{−n/4} \frac{(β; q^{1/2})_{n}}{(q^{1/2}; q^{1/2})_{n}}

×_{4}ϕ_{3} ( \begin{matrix}q^{−n/2}, βq^{n/2}, q^{1/4}e^{iθ}, q^{1/4}e^{−iθ} \\ −q^{1/2}, β^{1/2}q^{1/4}, −β^{1/2}q^{1/4} \end{matrix} | q^{1/2}, q^{1/2}) .

We next apply Theorem 2.2 using

C_{n}(0; β|q) = \begin{cases}0 & if n is odd, \\ \frac{(β^{2};q^{2})_{n}}{(q^{2};q^{2})_{n/2}} (−1)^{n/2} & if n is even. \end{cases}

The result is the following proposition.

Proposition 4.2. We have

C_{n}(cos θ; β|q) = \frac{(β; q)_{n}}{(q; q)_{n}} (−e^{2iθ}q^{2−n}; q)_{n−1}(1 + e^{2iθ})e^{−inθ}

×_{4}ϕ_{3} ( \begin{matrix}q^{−n}, q^{1−n}, −q^{1−n}/β, −q^{2−n}/β \\ q^{2−2n}/β^{2}, −e^{2iθ}q^{2−n}, −e^{−2iθ}q^{2−n} \end{matrix} | q^{2}, q^{2}) .

Finally we apply Theorem 1.1 to C_{n}(x; β|q). Let

C_{n}(x; β|q) = ∑_{k=0}^{n} b_{n,k}(a, β)ϕ_{k}(x; a). (4.5)

This time (4.1) and (4.2) give

b_{n,k}(a, β) = \frac{(β; q)_{k}}{(q; q)_{k}} \frac{q^{(1−n)k/2}}{a^{k}} (−1)^{k}C_{n−k}(x_{k}; βq^{k}|q)
(4.6)  = \frac{(−1)^{k}(β; q)_{n} a^{n−2k}}{(q; q)_{k}(q; q)_{n−k}} q^{k(1−k)/2}

×_{2}ϕ_{1} ( \begin{matrix}q^{k−n}, q^{k}β \\ q^{1−n}/β \end{matrix} | q, \frac{q^{1−2k}}{a^{2}β} ) .

The _{2}ϕ_{1} in (4.6) may be summed by the q-Gauss theorem [4] if β = a^{2} to obtain a quadratic transformation (see [2] or (7.5.33) in [4]). Proposition 4.3. We have

C_{n}(x; β|q)

= \frac{(β^{2}; q)_{n}}{β^{n/2}(q; q)_{n}}_{4}ϕ_{3} ( \begin{matrix}q^{−n}, β^{2}q^{n}, \sqrt{β}e^{iθ}, \sqrt{β}e^{−iθ} \\ β\sqrt{q}, −β\sqrt{q}, −β \end{matrix} | q, q) .

For our last expansion we invert (4.5), let

ϕ_{n}(x; a) = ∑_{j=0}^{n} a_{n,j}C_{j}(x; β|q). (4.7)

We shall find a_{n,j} using the orthogonality relation of the continuous qultraspherical polynomials

∫_{0}^{π} C_{m}(cos θ; β|q)C_{n}(cos θ; β|q)\frac{(e^{2iθ}, e^{−2iθ}; q)_{∞}}{(βe^{2iθ}, βe^{−2iθ}; q)_{∞}} dθ (4.8)

= \frac{2π(β, qβ; q)_{∞}}{(q, β^{2}; q)_{∞}} \frac{(β^{2}; q)_{n}(1 − β)}{(q; q)_{n}(1 − βq^{n})}δ_{m,n}

From (4.8) we get

\frac{2π(β, qβ; q)_{∞}}{(q, β^{2}; q)_{∞}} \frac{(β^{2}; q)_{j}(1 − β)}{(q; q)_{j}(1 − βq^{j})}a_{n,j}

= ∫_{0}^{π}(ae^{iθ}, ae^{−iθ}; q)_{n}C_{j}(cos θ; β|q)\frac{(e^{2iθ}, e^{−2iθ}; q)_{∞}}{(βe^{2iθ}, βe^{−2iθ}; q)_{∞}} dθ.

By writing (ae^{iθ}, ae^{−iθ}; q)_{n} as quotients of infinite products then applying (4.3) to expand it in powers of a, we see that the above expression is

= ∑_{m=0}^{∞} a^{m}q^{mn} ∫_{0}^{π} C_{m}(cos θ; q^{−n}|q)C_{j}(cos θ; β|q)\frac{(e^{2iθ}, e^{−2iθ}; q)_{∞}}{(βe^{2iθ}, βe^{−2iθ}; q)_{∞}} dθ

= ∑_{m=0}^{∞} a^{m}q^{mn} ∑_{k=0}^{[m/2]} \frac{β^{k}(q^{−n}/β; q)_{k}(q^{−n}; q)_{m−k}}{(q; q)_{k}(qβ; q)_{m−k}} \frac{1 − βq^{m−2k}}{1 − β}

× ∫_{0}^{π} C_{m−2k}(cos θ; β|q)C_{j}(cos θ; β|q)\frac{(e^{2iθ}, e^{−2iθ}; q)_{∞}}{(βe^{2iθ}, βe^{−2iθ}; q)_{∞}} dθ

Therefore

\frac{(1 − β)}{(1 − βq^{j})}a_{n,j} = a^{j}q^{nj} ∑_{k=0}^{∞} \frac{β^{k}(q^{−n}/β; q)_{k}(q^{−n}; q)_{k+j}}{(q; q)_{k}(qβ; q)_{k+j}} a^{2k}q^{2kn},

that is

a_{n,j} = a^{j}q^{nj} \frac{(q^{−n}; q)_{j}}{(β; q)_{j}} _{2}ϕ_{1} ( \begin{matrix}q^{j−n}, q^{−n}/β \\ βq^{j+1} \end{matrix} | q, a^{2}βq^{2n}) (4.9)

The _{2}ϕ_{1} can be summed if a^{2} = qβ in which case we get

ϕ_{n}(x; \sqrt{qβ}) (4.10)

= \frac{(β^{2}q^{n+1}; q)_{n}}{(qβ; q)_{n}} ∑_{j=0}^{n} \frac{(q^{−n}, qβ; q)_{j}}{(β^{2}q^{n+1}, β; q)_{j}} β^{j/2}q^{j(n+1/2)}C_{j}(x; β|q).

As another example, if a^{2} = β the _{2}ϕ_{1} is a sum of two terms and the result becomes

(4.11) ϕ_{n}(x; \sqrt{β}) = \frac{(1 − βq^{n})(β^{2}q^{n}; q)_{n}}{(1 − β^{2}q^{n})(qβ; q)_{n}}

× ∑_{j=0}^{n} \frac{(1 − β^{2}q^{2j})(q^{−n}; q)_{j}}{(1 − β)(β^{2}q^{n+1}; q)_{j}} β^{j/2}q^{nj}C_{j}(x; β|q).

5 Remarks

It is natural to ask if Theorems 1.1, 2.1, and 2.2 hold for a class of nonpolynomial functions. In [8] such theorems are given for entire functions which satisfy growth conditions.

Theorems 1.1 and 2.1 give two alternative forms for the coefficients f_{k} of ϕ_{k}(x), what results is

(−q^{1/2}; q^{1/2})_{k}q^{−k(k−1)/8}(D_{q^{1/2}}^{k}f)(ζ_{k}) = (q^{1/2} + 1)^{k}(D_{q}^{k}f)(ζ_{0}), (5.1)

where

ζ_{k} = (q^{(k+1)/4} + q^{−(k+1)/4})/2.

We do not have basic hypergeometric proofs of Theorem 2.4 and Corollary 2.8. It is likely that Theorem 2.4 could be proven by splitting the sum into even and odd terms as a sum of two balanced _{4}ϕ_{3}’s. Applying the Sears transformation to each sum could then lead to a recombined single sum that is evaluable- this type of proof establishes Corollary 2.7. Nonetheless these proposed details contrast with the ease of use of Theorems 1.1, 2.1, and 2.2.

The coefficient a_{n,j} in (4.9), the inverse to (4.6), can also be written as a multiple of the q-ultraspherical polynomial

C_{n−j}(cos ϕ; q^{−n}/b|q), a = q^{(1−n)/2}e^{−iϕ}.

Even polynomials in x may be expanded as functions of cos 2θ = 2x^{2} −1, for example [4, (7.5.40)] is

(5.2) C_{2n}(cos θ; β|q) = q^{−n/2} \frac{(β; q^{1/2})_{2n}}{(q^{1/2}; q^{1/2})_{2n}}

×_{4}ϕ_{3} ( \begin{matrix}q^{−n}, βq^{n}, q^{1/2}e^{2iθ}, q^{1/2}e^{−2iθ} \\ βq^{1/2}, −q^{1/2}, −q \end{matrix} | q, q) .

However it can be shown that (5.2) is equivalent to the reversal of Proposition 4.2, which is

(5.3) C_{2n}(cos θ; β|q) = (−1)^{n} \frac{(β^{2}; q^{2})_{n}}{(q^{2}; q^{2})_{n}}

×_{4}ϕ_{3} ( \begin{matrix}q^{−2n}, β^{2}q^{2n}, −e^{2iθ}, −e^{−2iθ} \\ −β, −βq, q \end{matrix} | q^{2}, q^{2}) .

Apply Singh’s quadratic _{4}ϕ_{3} transformation [4, (III.21)] followed by the 1balanced _{4}ϕ_{3} transformation [4, (III.15)] to show that (5.3) and (5.2) are equivalent.

Acknowledgments. Part of this work was done while the authors were visiting the Liu Bie Ju Center of Mathematical Sciences of City University of Hong Kong and they gratefully acknowledge the financial support and the hospitality of the City University of Hong Kong.

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