Functions that Generalize Askey–Wilson Polynomials - CiteSeerX

THE RAMANUJAN JOURNAL, 5, 183–218, 2001 c 2001 Kluwer Academic Publishers. Manufactured in The Netherlands. 

Some Orthogonal Very-Well-Poised 8 ϕ7 -Functions that Generalize Askey–Wilson Polynomials SERGEI K. SUSLOV∗,† Department of Mathematics, Arizona State University, Tempe, Arizona 85287-1804

[email protected]

Received May 9, 2000; Accepted April 5, 2001

Abstract. In a recent paper Ismail et al. (Algebraic Methods and q-Special Functions (J.F. van Diejen and L. Vinet, eds.) CRM Proceding and Lecture Notes, Vol. 22, American Mathematical Society, 1999, pp. 183–200) have established a continuous orthogonality relation and some other properties of a 2 ϕ1 -Bessel function on a q-quadratic grid. Dick Askey (private communication) suggested that the “Bessel-type orthogonality” found in Ismail et al. (1999) at the 2 ϕ1 -level has really a general character and can be extended up to the 8 ϕ 7 -level. Verywell-poised 8 ϕ 7 -functions are known as a nonterminating version of the classical Askey–Wilson polynomials (SIAM J. Math. Anal. 10 (1979), 1008–1016; Memoirs Amer. Math. Soc. Number 319 (1985)). Askey’s conjecture has been proved by the author in J. Phys. A: Math. Gen. 30 (1997), 5877–5885. In the present paper which is an extended version of Suslov (1997) we discuss in detail properties of the orthogonal 8 ϕ 7 -functions. Another type of the orthogonality relation for a very-well-poised 8 ϕ 7 -function was recently found by Askey et al. J. Comp. Appl. Math. 68 (1996), 25–55. Key words: q-Bessel functions, basic hypergeometric series, Askey–Wilson polynomials, orthogonal functions 2000 Mathematics Subject Classification:

1.

Primary—33D15, 33D45; Secondary—42C05, 34B24

Introduction

The Askey–Wilson polynomials [6] are pn (x) = pn (x; a, b, c, d) = a −n (ab, ac, ad; q)n 4 ϕ3

 q −n , abcdq n−1 , aeiθ , ae−iθ ab, ac, ad

 ; q, q ,

(1.1)

where x = cos θ. These polynomials are the most general known classical orthogonal polynomials (see [1, 6–8, 12, 26, 29]). ∗ The

author was supported in part by NSF grant # DMS 9803443.

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

184

SUSLOV

The symbol 4 ϕ3 in (1.1) is a special case of basic hypergeometric series [12],   a1 , a2 , . . . , ar ; q, t r ϕs (t) := r ϕs b1 , b 2 , . . . , b s =

∞  1+s−r n (a1 , a2 , . . . , ar ; q)n  (−1)n q n(n−1)/2 t . (q, b1 , b2 , . . . , bs ; q)n n=0

(1.2)

The standard notations for the q-shifted factorials are (a; q)n := (a1 , a2 , . . . , ar ; q)n :=

n  k=1 r 

(1 − aq k−1 ),

(1.3)

(ak ; q)n ,

(1.4)

k=1

and (a; q)∞ := lim (a; q)n , (a1 , a2 , . . . , ar ; q)∞ :=

(1.5)

n→∞ r 

(ak ; q)∞

(1.6)

k=1

provided |q| < 1. For an excellent account on the theory of basic hypergeometric series see [12]. We shall just mention here that the s+1 ϕs -series is called balanced if qa1 a2 · · · as+1 = b1 b2 · · · bs and t = q; it is a very-well-poised if a2 b1 = a3 b2 = · · · = as+1 bs = qa1 , and √ √ a2 = q a1 , a3 = −q a1 . Askey and Wilson found the orthogonality relation  π pn (cos θ ; a, b, c, d) pm (cos θ ; a, b, c, d) (e2iθ , e−2iθ ; q)∞ dθ (a eiθ , a e−iθ , b eiθ , b e−iθ , c eiθ , c e−iθ , d eiθ , d e−iθ ; q)∞ 0 (q, ab, ac, ad, bc, bd, cd; q)∞ = δnm 2π (abcd; q)∞ (1 − abcdq −1 ) (q, ab, ac, ad, bc, bd, cd; q)n × . (1.7) (1 − abcdq 2n−1 ) (abcdq −1 ; q)n In the fundamental memoir [6], they studied in detail many other properties of these polynomials. As is well known, the Askey–Wilson polynomials and their special and limiting cases are the simplest and the most important orthogonal solutions of the difference equation of hypergeometric type on nonuniform lattices (see, for example, [6–9, 12, 29, 35]). Recently Ismail et al. have found another type of orthogonal solutions of this difference equation [17, 18]. They considered the 2 ϕ1 -function, Jν (z, r ) = J˜ν (x(z), r | q) ν ν+1 r (q , −r 2 /4; q)∞ := 2 (q; q)∞

2 ϕ1

 q (ν+1)/2 eiθ , q (ν+1)/2 e−iθ q ν+1

r2 ; q, − 4

 , (1.8)

SOME ORTHOGONAL 8 ϕ7 -FUNCTIONS

as a q-analog on a q-quadratic grid of the Bessel function [39], ν  ∞ x (−x 2 /4)n Jν (x) = . 2 n=0 n! (ν + n + 1)

185

(1.9)

Ismail et al. established the following orthogonality property for the q-Bessel function,  π (e2iθ , e−2iθ ; q)∞ J˜ν (cos θ, r ) J˜ν (cos θ, r  ) α iθ α −iθ 1−α iθ 1−α −iθ (q e , q e , q e , q e ; q)∞ 0  (ν+1)/2 iθ (ν+1)/2 −iθ (ν+1)/2 iθ (ν+1)/2 −iθ −1 × q e ,q e ,q e ,q e ; q ∞ dθ = 0 (1.10) if r = r  ,



π

(e2iθ , e−2iθ ; q)∞ (q α eiθ , q α e−iθ , q 1−α eiθ , q 1−α e−iθ ; q)∞ 0  (ν+1)/2 iθ (ν+1)/2 −iθ (ν+1)/2 iθ (ν+1)/2 −iθ −1 × q e ,q e ,q e ,q e ; q ∞ dθ ( J˜ν (cos θ, r ))2

−4π(1 − q)q −(ν+1)/2 ∂ J˜ν (x(α), r ) ∇ J˜ν (x(α), r ) =  2 ∂r 2 ∇x(α) q, q (ν+1)/2+α , q (ν+1)/2−α+1 ; q ∞

(1.11)

if r = r  . Here r and r  are two roots of the equation Jν (α, r ) = Jν (α, r  ) = 0,

(1.12)

and Jν (z, r ) = J˜ν (x(z), r ), x(z) = 12 (q z + q −z ), (x = cos θ, if q z = eiθ ), Re ν > −1, and 0 < Re α < 1. (See [17, 18] for more details.) This is a q-version of the orthogonality relation for the classical Bessel function   1 if r = r  , 0 xJ ν (r x) Jν (r  x) d x = 1 (1.13)  (Jν+1 (r ))2 if r = r  , 0 2  under the conditions Jν (r ) = Jν (r ) = 0 [39]. Another example of the orthogonality relation of this type has been discovered recently. The following functions C(x) := Cq (x; ω)

  −q e2iθ , −q e−2iθ 2 (−ω2 ; q 2 )∞ 2 = ; q , −ω 2 ϕ1 (−qω2 ; q 2 )∞ q

(1.14)

and S(x) := Sq (x; ω) =

(−ω2 ; q 2 )∞ 2q 1/4 ω (−qω2 ; q 2 )∞ 1 − q   −q 2 e2iθ , −q 2 e−2iθ 2 2 × cos θ 2 ϕ1 ; q , −ω q3

(1.15)

186

SUSLOV

were discussed in [8] and [25] as analogs of cos ωx and sin ωx on a q-quadratic lattice, respectively. Ismail and Zhang [25] expanded the corresponding basic exponential function in terms of “q-spherical harmonics” and later, together with Rahman [22], they extended this q-analog of the expansion formula of the plane wave from q-ultraspherical polynomials to continuous q-Jacobi polynomials. Ismail et al. [20] have found more expansion formulas. “Addition” theorems for the basic trigonometric functions (1.14)–(1.15) and for the basic exponential functions were studied in [36] and [23]. Bustoz and Suslov [11] have established the orthogonality property,  π (e2iθ , e−2iθ ; q)∞ C(cos θ ; ω) C(cos θ ; ω ) 1/2 2iθ 1/2 −2iθ dθ = 0, (1.16) (q e , q e ; q)∞ 0  π (e2iθ , e−2iθ ; q)∞ S(cos θ; ω) S(cos θ ; ω ) 1/2 2iθ 1/2 −2iθ dθ = 0, (1.17) (q e , q e ; q)∞ 0  π (e2iθ , e−2iθ ; q)∞ C(cos θ ; ω) S(cos θ; ω ) 1/2 2iθ 1/2 −2iθ dθ = 0, (1.18) (q e , q e ; q)∞ 0 and



π

C 2 (cos θ; ω)

0



π

= 0

= π

(e2iθ , e−2iθ ; q)∞ 1/2 (q e2iθ , q 1/2 e−2iθ ; q)

S 2 (cos θ ; ω)

∞

dθ

(e2iθ , e−2iθ ; q)∞ 1/2 (q e2iθ , q 1/2 e−2iθ ; q)

∞

∞ (q 1/2 ; q)2∞ (−ω2 ; q 2 )∞  q k/2 . 2 2 2 (q; q)∞ (−qω ; q )∞ k=0 1 + ω2 q k

Here ω and ω are different solutions of the equation

 1  1/4 −1/4 S ; ω = 0. q +q 2

dθ (1.19)

(1.20)

Bustoz and Suslov introduced the corresponding q-Fourier series and established several important facts about the basic trigonometric system and the q-Fourier series. Numerical investigation of the basic Fourier series was recently started by Gosper and Suslov [13] with the help of the MACSYMA computer algebra system. Explicit examples of these series for some elementary and q-functions are found in [38]. Dick Askey [3] has suggested that the orthogonality relation (1.10)–(1.11) can be extended to the level of very-well-poised 8 ϕ7 -functions. The author was able to prove his conjecture in [37]. Our main objective in this paper is to study the new orthogonality property for 8 ϕ7 -function in detail. This article is an extended version of the short paper [37], originally submitted as a Letter; we include detailed proofs of the main results established in [37] to make this work as self-contained as possible. The paper is organized as follows. In Section 2 we consider difference equation of hypergeometric type on a q-quadratic grid and discuss solution of this equation in terms of a very-well-poised 8 ϕ7 -function. In the next two sections we derive a continuous orthogonality

SOME ORTHOGONAL 8 ϕ7 -FUNCTIONS

187

property for this 8 ϕ7 -function. Section 5 is devoted to investigation of zeros and asymptotics of this function and in Section 6 we evaluate the normalization constants in the orthogonality relation for the 8 ϕ7 . In Section 7 we find an analog of the Wronskian determinant for solutions of difference equation of hypergeometric type. Some special and limiting cases of our orthogonality relation are discussed in Section 8. We close this paper by estimating the number of zeros on the basis of Jensen’s theorem in Section 9. C. Krattenthaler’s MATHEMATICA package HYPQ [27] has helped us to check many formulas in this paper.

2.

Difference equation and its 8 ϕ7 -solutions

Let us consider the difference equation of hypergeometric type σ (z)



 ∇u(z) u(z) + τ (z) + λ u(z) = 0, ∇x1 (z) ∇x(z) x(z)

(2.1)

on a q-quadratic lattice x(z) = 12 (q z + q −z ) with x1 (z) = x(z + 12 ) and  f (z) = ∇ f (z + 1) = f (z + 1) − f (z). Here, in the most general case, σ (z) = q −2z (q z − a)(q z − b)(q z − c)(q z − d), (2.2) σ (−z) − σ (z) τ (z) = ∇x1 (z) 2q 1/2 = (abc + abd + acd + bcd − a − b − c − d + 2(1 − abcd) x), (2.3) 1−q 4q 3/2 λ = λν = (1 − q −ν )(1 − abcd q ν−1 ). (2.4) (1 − q)2 Equation (2.1) can be rewritten in self-adjoint form,

 ∇u(z) σ (z) ρ(z) + λ ρ(z) u(z) = 0, ∇x1 (z) ∇x(z)

(2.5)

where ρ(z) is a solution of the Pearson equation, (σ (z) ρ(z)) = τ (z) ρ(z) ∇x1 (z).

(2.6)

See [29, 35] for details. Methods of solving of the difference equation of hypergeometric type (2.1) were discussed in [8, 14, 31, 35]. As is well known, there are different kinds of solutions of this equation. For integer values of the parameter ν = n = 0, 1, 2, . . . , the famous solutions of (2.1) are the Askey–Wilson 4 ϕ3 -polynomials (see, for example, [5, 6, 9, 12]). For arbitrary values of this parameter solutions of (2.1) can be written in terms of 8 ϕ7 -functions [8, 19, 31, 35].

188

SUSLOV

Let us choose the following solution, u ν (z) = u ν (x(z); a, b, c; d), such that u ν (x; a, b, c; d) (qa/d, bcq ν , q 1−ν+z /d, q 1−ν−z /d; q)∞ = (q 1−ν a/d, bc, q 1+z /d, q 1−z /d; q)∞     −ν −ν aq aq aq −ν −ν q 1−ν q 1−ν z −z ,q , −q ,q , , , aq , aq    d d bd cd ν × 8 ϕ7  d  ; q, bcq  (2.7)    aq −ν aq −ν aq q 1−ν+z q 1−ν−z ,− , , ab, ac, , d d d d d   −ν ν−1 z −z ν 1−ν q , abcdq , aq , aq (bcq , q /ad; q)∞ = ; q, q 4 ϕ3 (bc, q/ad; q)∞ ab, ac, ad (q −ν , abcdq ν−1 , qb/d, qc/d; q)∞ (ab, ac, bc, ad/q; q)∞   q 1−ν /ad, bcq ν , q 1+z /d, q 1−z /d (aq z , aq −z ; q)∞ × 1+z ; q, q . 4 ϕ3 (q /d, q 1−z /d; q)∞ qb/d, qc/d, q 2 /ad +

(2.8)

We have used Bailey’s formula (III.36) of [12] to transform (2.7) to (2.8). Both 4 ϕ3 -functions in (2.8) are balanced and converge when |q| < 1. The 8 ϕ7 -function in (2.7) has a very-wellpoised structure, it converges when |bcq ν | < 1; (2.8) provides an analytic continuation of (2.7). This 8 ϕ7 can be transformed to another equivalent form by (III.23)–(III.24) of [12]. Solution (2.7)–(2.8) was originally found in [35] as an extension of the Askey–Wilson polynomials (see also [8]). Ismail and Rahman [19] used 8 ϕ7 -functions to introduce their associated Askey–Wilson polynomials. A similar function was discussed by Rahman [31], and in a recent paper [32] he has found a q-extension of a product formula of Watson involving this function. Rahman has also shown that   lim− u ν x; q α+1/2 , q α+3/4 , −q β+1/2 ; −q β+3/4 q→1

1−x = 2 F1 −ν, α + β + ν + 1; α + 1; . 2

(2.9)

Thus, the 8 ϕ7 -function in (2.7) can be thought as a q-extension of the hypergeometric function of Euler and Gauss. Let us discuss some properties of the 8 ϕ7 -function under consideration. It is easy to see that u ν (x; a, b, c; d) defined by (2.7)–(2.8) is a function in x = (q z + q −z )/2. Indeed, by (1.3) and (1.4), (ξ q z , ξ q −z )n =

n−1  k=0

(1 − 2ξ xq k + ξ 2 q 2k ),

SOME ORTHOGONAL 8 ϕ7 -FUNCTIONS

189

where ξ = a, q/d, q 1−ν /d and n = 1, 2, 3, . . . or ∞. It is also important to mention that (q −ν , abcdq ν−1 ; q)n n−1  = (1 − (q −ν + abcdq ν−1 )q k + abcdq 2k−1 ) k=0

=

n−1  k=0

1 − 1 + abcdq

−1



(1 − q)2 k 2k−1 − λν q + abcdq 4q 3/2

and (bcqν , q 1−ν /ad; q)n

n−1  1 bc 2k+1 1− = (q −ν + abcdq ν−1 )q k+1 + q ad ad k=0



n−1  1 (1 − q)2 bc 2k+1 −1 k+1 = 1− λν q + 1 + abcdq − q ad 4q 3/2 ad k=0 by (1.3)–(1.4) and (2.4), where n = 1, 2, 3, . . . or ∞. Therefore u ν (x; a, b, c; d) is, really, a function of λν as well. The definition of the u ν in this paper is the same as in [32], but different by a constant multiple from the original one in [35] and [37]. This definition emphasizes the symmetry properties. Function u ν (x; a, b, c; d) in (2.7)–(2.8) is obviously symmetric in b and c. Applying (III.23) and Bailey’s transform (III.36) of [12] once again we obtain u ν (x; a, b, c; d) (abcq ν+z , q 1−ν+z /d; q)∞ = (abcq z , q 1+z /d; q)∞

q 1−z z−1 −ν ν−1 z z z × 8 W7 abcq ; q , abcdq , aq , bq , cq ; q, d ν ν ν ν−1 1−ν+z 1−ν−z (abq , acq , bcq , abcdq , q /d, q /d; q)∞ = 2ν−1 1+z 1−z (ab, ac, bc, abcdq , q /d, q /d; q)∞   −ν 1−ν 1−ν q , q /ad, q /bd, q 1−ν /cd × 4 ϕ3 2−2ν ; q, q q /abcd, q 1−ν+z /d, q 1−ν−z /d (q −ν , q 1−ν /ad, q 1−ν /bd, q 1−ν /cd, abcq ν+z , abcq ν−z ; q)∞ (ab, ac, bc, q 1−2ν /abcd, q 1+z /d, q 1−z /d; q)∞   abcdq ν−1 , abq ν , acq ν , bcq ν × 4 ϕ3 ; q, q abcdq 2ν , abcq ν+z , abcq ν−z +

=

(aq −z , bq −z , cq −z , abcq ν+z , q 1−ν+z /d; q)∞ (ab, ac, bc, q −2z , q 1+z /d; q)∞

(2.10)

190

SUSLOV

 × 4 ϕ3

aq z , bq z , cq z , q 1+z /d

q 1+2z , abcq ν+z , q 1−ν+z

 ; q, q

(aq z , bq z , cq z , abcq ν−z , q 1−ν−z /d; q)∞ (ab, ac, bc, q 2z , q 1−z /d; q)∞  −z  aq , bq −z , cq −z , q 1−z /d × 4 ϕ3 ; q, q . q 1−2z , abcq ν−z , q 1−ν−z +

(2.11)

These representations show that u ν (x; a, b, c; d) is actually symmetric in a, b, and c. The second 4 ϕ3 representation was found with the help of Krattenthaler’s MATHEMATICA package HYPQ [27]. Here we have used the standard notation, 2r +2 W2r +1 (a; a1 , a2 , . . . , a2r −1 ; q, t)



:=

2r +2 ϕ2r +1

 √ √ a, q a, −q a, a1 , a2 , . . . , a 2r −1 ; q, t √ √ a, − a, qa/a1 , qa/a2 , . . . , qa/a 2r −1

(2.12)

for a very-well-poised basic hypergeometric series. Let us discuss analyticity properties of the function u ν (z) defined in (2.7)–(2.8). One can easily see, that for integers ν = n = 0, 1, 2, . . . this function is just a multiple of the Askey–Wilson polynomial (1.1) of the n-th degree, u n (x; a, b, c; d) =

(−1)n q −n(n−1)/2 −n d pn (x; a, b, c, d), (ab, ac, bc; q)n

(2.13)

where x = (q z + q −z )/2. In this case the Askey–Wilson polynomials pn (x; a, b, c, d) are symmetric with respect to a permutation of all four parameters a, b, c, and d due to Sears’s transformation [6]. On the other hand, if ν is not an integer, the essential poles of the 8 ϕ7 -solution in (2.7) coincide with the simple poles of the infinite product (q 1+z /d, q 1−z /d; q)−1 ∞. Therefore, the function vν (z) = vν (x(z); a, b, c; d) := (q 1+z /d, q 1−z /d; q)∞ u ν (x(z); a, b, c; d),

(2.14)

where u ν (x; a, b, c; d) is defined in (2.7)–(2.8), is an entire function in the complex z-plane. Let us mention also that function u ν (x; a, b, c; d) satisfies the simple difference-differentiation formula δ (1 − q −ν )(1 − abcdq ν−1 ) 2q u ν (x(z); a, b, c; d) = δx(z) (1 − q)d (1 − ab)(1 − ac)(1 − bc) × u ν−1 (x(z); aq 1/2 , bq 1/2 , cq 1/2 ; dq 1/2 ), where δ f (z) = f (z + 1/2) − f (z − 1/2) and x(z) = (q z + q −z )/2.

(2.15)

SOME ORTHOGONAL 8 ϕ7 -FUNCTIONS

3.

191

Solutions of the Pearson equation

In order to rewrite Eq. (2.1) for the function (2.7)–(2.8) in the self-adjoint form (2.5), we have to find a solution of the Pearson-type equation (2.6). In the case of the q-quadratic grid x = (q z + q −z )/2 this equation can be rewritten in the form ρ(z + 1) σ (−z) = ρ(z) σ (z + 1) = q −4z−2 q 2z+1

(1 − aq z )(1 − bq z )(1 − cq z )(1 − q −z /d) . (1 − aq −z−1 )(1 − bq −z−1 )(1 − cq −z−1 )(1 − q z+1 /d)

(3.1)

It is easy to check that ρ0 (z + 1) = q −4z−2 ρ0 (z) ρα (z + 1) = q −2z−1 ρα (z) ρa (z + 1) 1 − aq −z−1 = ρa (z) 1 − aq z

(q 2z , q −2z ; q)∞ , q z − q −z

for

ρ0 (z) =

for

ρα (z) = (αq z , αq −z , q 1+z /α, q 1−z /α; q)∞ , (3.3)

for

ρa (z) = (aq z , aq −z ; q)∞ .

(3.2)

(3.4)

(See, for example, [29, 33, 34] for methods of solving the Pearson equation.) Therefore, one can choose the following solution of (3.1), ρ(z) =

(q z − q −z )−1 (q 2z , q −2z , q 1+z /d, q 1−z /d; q)∞ , (αq z , αq −z , q 1+z /α, q 1−z /α, aq z , aq −z , bq z , bq −z , cq z , cq −z ; q)∞

(3.5)

where α is an arbitrary additional parameter. It was shown in [37] that this solution satisfies the correct boundary conditions for the second order divided-difference Askey–Wilson operator (2.5) for certain values of the parameter α. Special cases α = d or α = q/d of (3.5) result in the weight function for the Askey– Wilson polynomials (cf. [6, 9]). One can rewrite (3.1) as ρ(z + 1) (1 − aq z )(1 − bq z )(1 − q −z /c)(1 − q −z /d) = . ρ(z) (1 − aq −z−1 )(1 − bq −z−1 )(1 − q z+1 /c)(1 − q z+1 /d)

(3.6)

We shall use the corresponding solution, ρ(z) = in Section 7.

(q 1+z /c, q 1−z /c, q 1+z /d, q 1−z /d; q)∞ , (aq z , aq −z , bq z , bq −z ; q)∞

(3.7)

192

SUSLOV

It is worth mentioning a multiparameter extension of (3.5), namely, ρ(z) =

(q z − q −z )−1 (q 2z , q −2z , q 1+z /d, q 1−z /d; q)∞ (αq z , αq −z , q 1+z /α, q 1−z /α, aq z , aq −z , bq z , bq −z , cq z , cq −z ; q)∞ n  (βk q z , βk q −z , q 1+z /βk , q 1−z /βk ; q)∞ × (αk q z , αk q −z , q 1+z /αk , q 1−z /αk ; q)∞ k=1

(3.8)

where αi = βk . 4.

Orthogonality property

Now we can prove the orthogonality relation of the 8 ϕ7 -functions (2.7) with respect to the weight function (3.5) established in [37]. Let us apply the following q-version of the Sturm– Liouville procedure (cf. [9, 10, 29]). Consider the difference equations for the functions u ν (z) = u ν (x(z); a, b, c; d) and u µ (z) = u µ (x(z); a, b, c; d) in self-adjoint form,

 ∇u µ (z) σ (z) ρ(z) + λµ ρ(z) u µ (z) = 0, ∇x1 (z) ∇x(z)

 ∇u ν (z) σ (z) ρ(z) + λν ρ(z) u ν (z) = 0, ∇x1 (z) ∇x(z)

(4.1) (4.2)

where the eigenvalues λ = λν and λ = λµ are defined by (2.4). Let us multiply the first equation by u ν (z), the second one by u µ (z), and subtract the second equality from the first one. As a result we get (λµ − λν )u µ (z) u ν (z)ρ(z) ∇x1 (z) = [σ (z) ρ(z)W (u µ (z), u ν (z))],

(4.3)

where    u µ (z) u ν (z)    W (u µ (z), u ν (z)) =  ∇u µ (z) ∇u ν (z)    ∇x(z) ∇x(z) ∇u ν (z) ∇u µ (z) = u µ (z) − u ν (z) ∇x(z) ∇x(z) u ν (z) u µ (z − 1) − u µ (z) u ν (z − 1) = x(z) − x(z − 1)

(4.4)

is the analog of the Wronskian [29]. We need to know the explicit pole structure of the analog of the Wronskian W (u µ , u ν ) in (4.4). Let us transform the u’s to the entire functions v’s by (2.14), u ε (z) = ϕ(z)vε (z),

(4.5)

SOME ORTHOGONAL 8 ϕ7 -FUNCTIONS

193

where ε = µ, ν and ϕ(z) = (q 1+z /d, q 1−z /d; q)−1 ∞.

(4.6)

W (u µ (z), u ν (z)) = ϕ(z)ϕ(z − 1)W (vµ (z), vν (z)),

(4.7)

Thus,

where the new “Wronskian,” W (vµ (z), vν (z)), is clearly an entire function in z. Integrating (4.3) over the contour C indicated in Figure 1; where the variable z is such that z = iθ/log q −1 and −π ≤ θ ≤ π (we shall assume that 0 < q < 1 throughout this work); gives  (λµ − λν ) u µ (z) u ν (z)ρ(z) ∇x1 (z) dz C  = [σ (z) ρ(z) ϕ(z) ϕ(z − 1)W (vµ (z), vν (z))] dz. (4.8) C

Figure 1.

The contour C in the complex z-plane.

194

SUSLOV

All poles of the integrand in the right side of (4.8) coincide with the simple poles of the function σ (z) ρ(z) ϕ(z) ϕ(z − 1) d (q 2z , q 1−2z ; q)∞ (αq z , αq −z , q 1+z /α, q 1−z /α; q)∞ × (aq z , aq 1−z , bq z , bq 1−z , cq z , cq 1−z , q 1+z /d, q 2−z /d; q)−1 ∞.

=−

(4.9)

The integrand in the right side of (4.8) has the natural purely imaginary period T = 2πi/ log q −1 when 0 < q < 1, so this integral is equal to  [σ (z) ϕ(z) ϕ(z − 1) ρ(z)W (vµ (z), vν (z))] dz, (4.10) D

where D is the boundary of the rectangle on the figure oriented counterclockwise. The analog of the Wronskian W (vµ , vν ) is an entire function. Thus, when max(|a|, |b|, |c|, |q/d|) < 1, the poles of the integrand in (4.10) inside the rectangle in the figure are just the simple poles of ρ(z) at z = α0 and z = 1 − α0 , where q α0 = α and 0 < Re α0 < 1/2. By Cauchy’s theorem,  1 [σ (z) ϕ(z) ϕ(z − 1) ρ(z)W (vµ (z), vν (z))] dz 2πi D = Res f (z) | z=α0 + Res f (z) | z=1−α0 , (4.11) where f (z) = σ (z) ρ(z) ϕ(z) ϕ(z − 1)W (vµ (z), vν (z)) =− ×

d (q 2z , q 1−2z ; q)∞ (αq z , αq −z , q 1+z /α, q 1−z /α; q)∞ W (vµ (z), vν (z)) z 1−z z 1−z (aq , aq , bq , bq , cq z , cq 1−z , q 1+z /d, q 2−z /d; q)

∞

.

(4.12)

Evaluation of the residues at these simple poles gives Res f (z) | z=α0 = lim (z − α0 ) f (z) z→α0

=−

d W (vµ (z), vν (z))| z=α0 log q (q, q, αa, qa/α, αb, qb/α, αc, qc/α, qα/d, q 2 /αd; q)∞

(4.13)

and Res f (z) | z=1−α0 =

lim (z − 1 + α0 ) f (z)

z→1−α0

= −

d W (vµ (z), vν (z)) | z=1−α0 . log q (q, q, αa, qa/α, αb, qb/α, αc, qc/α, qα/d, q 2 /αd; q)∞ (4.14)

SOME ORTHOGONAL 8 ϕ7 -FUNCTIONS

195

But W (vµ (z), vν (z)) | z=α0 = W (vµ (z), vν (z)) | z=1−α0

(4.15)

due to the last line in (4.4) and the symmetry vε (z) = vε (−z), x(z) = x(−z). Thus, the residues are equal and as a result we evaluate the integral in the right side of (4.8),  [σ (z) ϕ(z) ϕ(z − 1) ρ(z)W (vµ (z), vν (z))] dz C

4π idW (vµ (α0 ), vν (α0 )) log q (q, q, αa, qa/α, αb, qb/α, αc, qc/α, qα/d, q 2 /αd; q)∞ 4πi d(α/d, q/αd; q)∞ W (u µ (α0 ), u ν (α0 )) = − log q (q, q, αa, qa/α, αb, qb/α, αc, qc/α; q)∞ = −

(4.16)

by (4.7). Combining (4.8) and (4.16), we, finally, arrive at the main equation,  (λµ − λν ) u µ (z) u ν (z)ρ(z) δx(z) dz C

= −

4πi d(α/d, q/αd; q)∞ log q(q, q, αa, qa/α, αb, qb/α, αc, qc/α; q)∞

× W (u µ (α0 ), u ν (α0 )),

(4.17)

where max(|a|, |b|, |c|, |q/d|) < 1 and 0 < Re α0 < 1/2, q α0 = α. Equation (4.17) can be extended in a larger domain in the space of parameters by analytic continuation. It is worth mentioning an important special case first. If both of “degree” parameters µ and ν are nonnegative integers: µ = m = 0, 1, 2, . . . and ν = n = 0, 1, 2, . . .; Eqs. (4.17) and (2.13) result in the following real integral  π pm (cos θ ; a, b, c; d) pn (cos θ; a, b, c; d) iθ −iθ , b eiθ , b e−iθ , c eiθ , c e−iθ ; q) ∞ 0 (a e , a e (e2iθ , e−2iθ , q eiθ /d, q e−iθ /d; q)∞ dθ (α eiθ , α e−iθ , q eiθ /α, qe−iθ /α; q)∞ (α/d, q/αd; q)∞ = (q, q, αa, qa/α, αb, qb/α, αc, qc/α; q)∞ ×

×

−4πq 1/2 d W ( pm (η; a, b, c; d), pn (η; a, b, c; d)) 1−q λ m − λn

(4.18)

involving the Askey–Wilson polynomials. Here we use the notation η = x(α0 ) =

1 (α + α −1 ). 2

(4.19)

One can easily see that when α = d or α = q/d our Eq. (4.18) implies the orthogonality relation for the Askey–Wilson polynomials (1.7).

196

SUSLOV

The new orthogonality property for the 8 φ7 -function appears when both of the parameters µ and ν are not nonnegative integers. It is convenient in this case to rewrite (4.17) in terms of the entire functions v’s as follows  π vµ (cos θ ; a, b, c; d)vν (cos θ ; a, b, c; d) iθ , a e−iθ , b eiθ , b e−iθ , c eiθ , c e−iθ , q eiθ /d, q e−iθ /d; q) (a e ∞ 0 (e2iθ , e−2iθ ; q)∞ × dθ (α eiθ , α e−iθ , q eiθ /α, q e−iθ /α; q)∞ = (q, q, αa, qa/α, αb, qb/α, αc, qc/α, qα/d, q 2 /αd; q)−1 ∞ ×

−4πq 1/2 d W (vµ (η; a, b, c; d), vν (η; a, b, c; d)) . 1−q λ µ − λν

(4.20)

The limiting case µ → ν of (4.20) is also of interest. From (4.4), W (vµ (z), vν (z)) λ µ − λν ∇vν (z) ∇vµ (z) − vν (z) vµ (z) ∇x(z) ∇x(z) = lim µ→ν λ µ − λν

∂vµ (z) ∇vν (z) ∂ ∇vµ (z) − vν (z) ∂µ ∇x(z) ∂µ ∇x(z) = lim µ→ν ∂λµ /∂µ

∂vν (z) ∇vν (z) ∂ ∇vν (z) − vν (z) ∂ν ∇x(z) ∂ν ∇x(z) = . ∂λν /∂ν

lim

µ→ν

Therefore,  0

π

(a

(vν (cos θ ; a, b, c; d))2 iθ −iθ iθ e , a e , b e , b e−iθ , c eiθ , c e−iθ , q eiθ /d, q

e−iθ /d; q)∞

(e2iθ , e−2iθ ; q)∞ dθ (α eiθ , α e−iθ , q eiθ /α, q e−iθ /α; q)∞ = (q, q, αa, qa/α, αb, qb/α, αc, qc/α, qα/d, q 2/αd; q)−1 ∞

 1/2   −4πq d ∂ ∇ vν (x; a, b, c; d)  × vν (η; a, b, c; d) 1−q ∂λν ∇x x =η 

  ∇ ∂ . − vν (η; a, b, c; d) vν (x; a, b, c; d)  ∂λ ∇x ×

ν

(4.21)

x =η

Finally, choosing the parameters µ and ν as ε-solutions of the equation

1 −1 vε (α + α ); a, b, c; d = 0, 2

(4.22)

SOME ORTHOGONAL 8 ϕ7 -FUNCTIONS

197

we arrive from (4.20) and (4.21) at the orthogonality relation of the 8 ϕ7 -functions under consideration, 

π

(a

0

×

vµ (cos θ ; a, b, c; d)vν (cos θ ; a, b, c; d) iθ −iθ e , a e , b eiθ , b e−iθ , c eiθ , c e−iθ , q eiθ /d, q e−iθ /d; q)

∞

−2iθ

; q)∞ (e , e dθ = 0 (α eiθ , α e−iθ , q eiθ /α, q e−iθ /α; q)∞ 2iθ

(4.23)

if µ = ν, and  0

π

(vν (cos θ ; a, b, c; d))2 (a eiθ , a e−iθ , b eiθ , b e−iθ , c eiθ , c e−iθ , q eiθ /d, q e−iθ /d; q)∞ (e2iθ , e−2iθ ; q)∞ × dθ (α eiθ , α e−iθ , q eiθ /α, q e−iθ /α; q)∞

= (q, q, αa, qa/α, αb, qb/α, αc, qc/α, qα/d, q 2 /αd; q)−1 ∞

 1/2  −4πq d ∂ ∇ × vν (η; a, b, c; d) vν (x; a, b, c; d)  x = η 1−q ∂λν ∇x

(4.24)

if µ = ν, respectively. We remind the reader that max(|a|, |b|, |c|, |q/d|) < 1. This will be assumed throughout this work. One can easily see that vε (x; a, b, c; d) with ε = µ, ν in (4.23)–(4.24) are real-valued functions of x for real ε which are orthogonal with respect to a positive weight function when all parameters a, b, c, d, and α are real, or when any two of a, b, c are complex conjugate and all other parameters are real. We shall usually assume that as well. Remark 4.1. Taking into account the multiparameter solution (3.8) of (3.1) one can extend (4.17) in the following manner  (λµ − λν )

u µ (x(z); a, b, c; d) u ν (x(z); a, b, c; d)ρ(z) δx(z) dz C

=

4πid (α/d, q/αd; q)∞ log q −1 (q, q)2∞ (αa, qa/α, αb, qb/α, αc, qc/α; q)∞ n  (αβk , βk /α, qα/βk , q/αβk ; q)∞ × (αα k , αk /α, qα/αk , q/ααk ; q)∞ k=1 × W (u µ (η; a, b, c; d), u ν (η; a, b, c; d)) + idem (α; α1 , α2 , . . . , αn ),

(4.25)

where η = (α + α −1 )/2 and η p = (α p + α −1 p )/2, p = 1, 2, . . . , n. Here the symbol idem(α; α1 , α2 , . . . , αn ) after the first expression in the right side denotes the sum of n terms obtained from this expression by interchanging α with α1 , α2 , . . . , αn , respectively.

198 5.

SUSLOV

Properties of zeros and asymptotics

In the previous section we have proved that the 8 ϕ7 -functions (2.7)–(2.8) are orthogonal if the “boundary” condition (4.22) is satisfied. Let us discuss now some properties of ν-zeros of the corresponding “boundary” function,

1 −1 vν (α + α ); a, b, c; d 2 (qa/d, bcq ν , αq 1−ν /d, q 1−ν /αd; q)∞ = (q 1−ν a/d, bc; q)∞     aq −ν aq −ν aq −ν −ν q 1−ν q 1−ν   ,q , −q ,q , , , αa, a/α  d  d d bd cd ν × 8 ϕ7  ; q, bcq       aq −ν aq −ν aq q 1−ν α q 1−ν ,− , , ab, ac, , d d d d αd 1−ν ν (qα/d, q /αd, abcq /α; q)∞ = (abc/α; q)∞     abc q 1/2 abc q 1/2 abc −ν b c a   ,− , q , abcdq ν−1 , , ,  qα , α α α α α α . × 8 ϕ7  ; q, q    d ν 1−ν   abc abc q abc q ,− , , , ab, ac, bc qα qα α αd

(5.1)

(5.2)

We have used (III.23) of [12] to transform (5.1) to (5.2), compare also (2.10). Again, vν (η; a, b, c; d) is real-valued function of ν when all parameters a, b, c, d, and α are real, or when any two of a, b, c are complex conjugate and all other parameters are real. Main properties of zeros of the function (5.1)–(5.2) can be investigated by using the same methods as in [11, 15, 17, 18]. The first property is that under certain conditions the real-valued function vν (η; a, b, c; d) has an infinity of positive ν-zeros. In order to establish this fact, one can consider the large ν-asymptotics of the 8 ϕ7 -function in (5.2),     abc q 1/2 abc q 1/2 abc −ν a b c   , ,− , q , abcdq ν−1 , , ,  qα α α α α α α   ϕ ; q, q 8 7   d   ν 1−ν abc abc q abc q ,− , , , ab, ac, bc qα qα α αd     abc q 1/2 abc q 1/2 abc a b c   , ,− , , ,  qα α α α α α −→ ϕ  2  ; q, α  ν →∞ 6 5      abc abc ,− , ab, ac, bc qα qα

SOME ORTHOGONAL 8 ϕ7 -FUNCTIONS

199

(αa, αb, αc, abc/α; q)∞ (ab, ac, bc, α 2 ; q)∞

(5.3)

=

by (II.20) of [12]. Therefore,

1 (αa, αb, αc, qα/d; q)∞ vν (α + α −1 ); a, b, c; d = 2 (ab, ac, bc, α 2 ; q)∞ × (q 1−ν /αd; q)∞ [1 + o(1)],

(5.4)

as ν → ∞. But for the positive values of q and αd the function (q 1−ν /αd; q)∞ oscillates and has an infinity of real zeros as ν approaches infinity (see [15, 17]). Indeed, consider the points ν = ωn = ω0 + n, such that q ω0 = β, where n = 0, 1, 2, . . . and q < β < 1, as test points. Then, by using (I.8) of [12], vω0 +n (η; a, b, c; d) =

(αa, αb, αc, qα/d; q)∞ (ab, ac, bc, α 2 ; q)∞ (αβd; q)n × (−1)n q −n(n−1)/2 (q/αβd; q)∞ [1 + o(1)], (αβd)n

(5.5)

as n → ∞, and one can see that the right side of (5.5) changes sign infinitely many times at the test points ν = ωn as ν approaches infinity. Thus we have established the following theorem. Theorem 5.1. The real-valued function vν (η; a, b, c; d) defined by (5.2) has an infinity of positive ν-zeros when αq < d, α 2 < 1, αd > 0, and 0 < q < 1. Also, all parameters a, b, c are real, or when any of two of a, b, c are complex conjugate and the remainder parameter is real. Now we can prove the next result. Theorem 5.2. Function vν (η; a, b, c; d) defined in (5.2) has only real ν-zeros under the following conditions: (1) parameters α and d are real; parameters a, b, c are all real, or, if any two of them are complex conjugate, the third parameter is real; (2) the following inequalities holds: (a) max(|a|, |b|, |c|, |q/d|) < 1, q 1/2 < α < 1; (b) αd > q, αabc < q. Proof: Suppose that ν0 is a zero of function (5.2) which is not real. Let ν1 be the complex number conjugate to ν0 , so that ν1 is also a zero of (5.2) because this function is real under the hypotheses of the theorem.

200

SUSLOV

Consider Eq. (4.20) with µ = ν0 and ν = ν1 . The integral on the left does not equal zero due to the positivity of the integrand, but the analog of the Wronskian on the right is zero in view of (4.22). Therefore, λν0 = λν1 ,

(5.6)

the eigenvalues defined by (2.4) are real. The last equation can be rewritten as (1 − q ν1 −ν0 )(1 − abcdq ν0 +ν1 −1 ) = 0.

(5.7)

The first solution is ν0 = ν1 , so the roots are real in this case. The second solution of (5.7) is  q

ν0

=

q eiχ , abcd

 q

ν1

=

q e−iχ , abcd

(5.8)

where abc > 0 and χ is an arbitrary real number. But in this case our function can be represented as a multiple of a positive function. Indeed, when ν = ν0 or ν = ν1 , vν (η; a, b, c; d) (q/αd, q 1−ν α/d, αabcq ν ; q)∞ = (αabc; q)∞    αabc  1/2 −ν ν−1 1/2  q , q αabc, − q αabc, q , abcdq , αa, αb, αc q    × 8 ϕ7  ; q,      αd αabc αabc ν ,− , q αabc, q 1−ν α/d, ab, ac, bc q q 1−ν ν (q/αd, q α/d, αabcq ; q)∞ = (αabc; q)∞  ∞  (1 − αabcq 2n−1 )(αabcq −1 , αa, αb, αc; q)n × 1+ (1 − αabcq −1 )(q, ab, ac, bc; q)n n=1 n  q (q −ν , abcdq ν−1 ; q)n (5.9) × ν 1−ν (q αabc, q α/d; q)n αd by (5.2), (III.23) and (I.31) of [12]. One can easily see that under the hypotheses of the theorem all products in the last but one line of (5.9) are positive. From (1.3)–(1.4) and (5.8), (q

−ν

, abcdq

ν−1



abcd abcd 2k cos χq k + q q q k=0 

2

n−1  abcd abcd = 1− cos χq k + sin2 χq 2k q q k=0

; q)n =

n−1 

1−2

(5.10)

SOME ORTHOGONAL 8 ϕ7 -FUNCTIONS

201

and ν

(q αabc, q

1−ν

n−1 



abcq k 2 abcq 2k 1 − 2α α/d; q)n = cos χq + α q d d k=0 

2

n−1  abcq abcq = 1−α cos χq k + α 2 sin2 χq 2k , d d k=0 (5.11)

where n = 1, 2, 3, . . . , ∞. Thus, these products are positive too, which proves the positivity of the “boundary” function in (5.9). So we have obtained a contradiction and the complex zeros (5.8) cannot exist. This completes the of the theorem. ✷ Theorem 5.3. The real ν-zeros of vν (η; a, b, c; d) are simple under the hypotheses of Theorem 5.2. Proof: This follows directly from the relations (4.21) or (4.24). Indeed, the integral on ∂ the left side is positive under the conditions of the theorem, which means that ∂ν vν (η; a, b, c; d) = 0 when vν (η; a, b, c; d) = 0. It is also clear from (5.9)–(5.11) that vν (η; a, b, c; d) = 0 for q ν = ±(q/abcd)1/2 when ∂λ/∂ν = 0. ✷ Let us consider two functions, f (ν) = vν (η; a, b, c; d),   ∇ g(ν) = . vν (x; a, b, c; d)  ∇x x =η

(5.12) (5.13)

Our next property is that positive zeros of f (ν) and g(ν) are interlaced. Theorem 5.4. If ν1 , ν2 , ν3 , . . . are the positive zeros of f (ν) arranged in ascending order of magnitude, and µ1 , µ2 , µ3 , . . . are those of g(ν), then 0 < µ 1 < ν1 < µ2 < ν 2 < µ3 < ν3 < · · · ,

(5.14)

if the conditions of Theorem 5.2 are satisfied. Proof: Suppose that νk and νk+1 are two successive zeros of f (ν). Then the derivative ∂ f (ν) has different signs at ν = νk and ν = νk+1 . This means, in view of the positivity of ∂ν the integral on the left side of (4.21), that g(ν) changes its sign between νk and νk+1 and, therefore, has at least one zero on each interval (νk , νk+1 ). To complete the proof of the theorem, we have to show that g(ν) changes its sign on each interval (νk , νk+1 ) only once. Suppose that g(µk ) = g(µk+1 ) = 0 and νk < µk < µk+1 < νk+1 . Then, by (4.21), function f (ν) has different signs at ν = µk and ν = µk+1 and, therefore, this function has at least one more zero on (νk , νk+1 ). So, we have obtained a

202

SUSLOV

contradiction, and, therefore, function g(ν) has exactly one zero between any two successive zeros of f (ν). In order to finish the proof of our theorem one has to show that µ1 < ν1 . We would like to leave the proof of this fact as an exercise for the reader. ✷ Let us discuss asymptotic behavior of u ν (x; a, b, c; d) for large values of ν applying the methods used in [12, 24, 30] at the level of the Askey–Wilson polynomials. In a similar manner, the 8 ϕ7 -function in (2.10) can be transformed by (III.37) of [12],

q 1−z z−1 ; q −ν , abcdq ν−1 , aq z , bq z , cq z ; q, 8 W7 abcq d =

(aq −z , bq −z , cq −z , bq 1+ν+z , cq 1+ν+z , dq ν−z , abcq z , bcdq ν+z ; q)∞ (q −2z , ab, ac, bc, q 1+ν , bdq ν , cdq ν , bcq 1+ν+2z ; q)∞ × 8 W7 (bcq ν+2z ; bcq ν , bq z , cq z , q 1+z /a, q 1+z /d; q, adq ν ) +

(aq z , bq z , cq z , bq 1+ν−z , cq 1+ν−z , dq ν+z , abcq z , bcdq ν−z ; q)∞ (q 2z , ab, ac, bc, q 1+ν , bdq ν , cdq ν , bcq 1+ν−2z ; q)∞

×

(q 1−ν−z /d, q 1+z /d, abcq ν−z ; q)∞ (q 1−z /d, q 1−ν+z /d, abcq ν+z ; q)∞

× 8 W7 (bcq ν−2z ; bcq ν , bq −z , cq −z , q 1−z /a, q 1−z /d; q, adq ν ).

(5.15)

Therefore, u ν (x; a, b, c; d) (bq 1+ν+z , cq 1+ν+z , dq ν−z , abcq ν+z , bcdq ν+z ; q)∞ = (q 1+ν , bdq ν , cdq ν , bcq 1+ν+2z ; q)∞ ×

(aq −z , bq −z , cq −z , q 1−ν+z /d; q)∞ (ab, ac, bc, q −2z , q 1+z /d; q)∞

× 8 W7 (bcq ν+2z ; bcq ν , bq z , cq z , q 1+z /a, q 1+z /d; q, adq ν ) +

(bq 1+ν−z , cq 1+ν−z , dq ν+z , abcq ν−z , bcdq ν−z ; q)∞ (q 1+ν , bdq ν , cdq ν , bcq 1+ν−2z ; q)∞

×

(aq z , bq z , cq z , q 1−ν−z /d; q)∞ (ab, ac, bc, q 2z , q 1−z /d; q)∞

× 8 W7 (bcq ν−2z ; bcq ν , bq −z , cq −z , q 1−z /a, q 1−z /d; q, adq ν ).

(5.16)

We shall use the last expression to determine the large ν asymptotic of our orthogonal later. For the “boundary” function (5.2) Eq. (5.16) takes the form

1 −1 vν (α + α ); a, b, c; d 2

8 ϕ7 -function

SOME ORTHOGONAL 8 ϕ7 -FUNCTIONS

=

203

(αbq 1+ν , αcq 1+ν , dq ν /α, αabcq ν , αbcdq ν ; q)∞ (q 1+ν , bdq ν , cdq ν , α 2 bcq 1+ν ; q)∞ ×

(a/α, b/α, c/α, q/αd, αq 1−ν /d; q)∞ (ab, ac, bc, α −2 ; q)∞

× 8 W7 (α 2 bcq ν ; bcq ν , αb, αc, qα/a, qα/d; q, adq ν ) +

(bq 1+ν /α, cq 1+ν /α, αdq ν , abcq ν /α, bcdq ν /α; q)∞ (q 1+ν , bdq ν , cdq ν , bcq 1+ν /α 2 ; q)∞

×

(αa, αb, αc, qα/d, q 1−ν /αd; q)∞ (ab, ac, bc, α 2 ; q)∞

× 8 W7 (bcq ν /α 2 ; bcq ν , b/α, c/α, q/αa, q/αd; q, adq ν ).

(5.17)

The last term dominates here as ν approaches infinity (cf. (5.4)). The proof of Theorem 5.1 has strongly indicated that asymptotically the large positive ν-zeros of vν (η; a, b, c; d) are ν n = n + n ,

0 ≤ n < 1

(5.18)

as n → ∞. The same consideration as in [11, 15, 18] shows that this function changes its sign only once between any two successive test points ν = ωn and ν = ωn+1 defined in (5.5) for sufficiently large values of n. We include details of this proof in Section 9 to make this work as self-contained as possible. Our next conjecture provides a more accurate estimate for the distribution of the large positive zeros of the “boundary” function (5.2). Conjecture 5.5. If ν1 , ν2 , ν3 , . . . are the positive zeros of vν (η; a, b, c; d) arranged in ascending order of magnitude, then νn = n −

log(αd) + o(1), log q

(5.19)

as n → ∞. The following motivation holds. All zeros of our function (5.2) coincide with the zeros of a new function, wν (η; a, b, c; d) := (−q 1−ν /αd; q)−1 ∞ vν (η; a, b, c; d),

(5.20)

because (−q 1−ν /αd; q)∞ is positive for real ν. Equations (5.17) and (5.20) give us the large ν-asymptotic of wν (η; a, b, c; d). When ν = γn such that q γn = q n /αd and n = 1, 2, 3, . . . , the second term in (5.17) vanishes and we get wγn (η; a, b, c; d) =

(qα 2 , a/α, bc, qb/d, qc/d, abc/d; q)∞ (−q, ab, ac, bc, αbcq/d; q)∞

204

SUSLOV

(q/αd, b/α, c/α, αbcq/d; q)n (−α 2 )n (−1, bc, b/d, qc/d, abc/d; q)n

αbc n b n α α a × 8 W7 q ; q , αb, αc, q , q ; q, q n d α a d α ×

(5.21)

with the help of (I.9) of [12]. Thus, lim wγn (η; a, b, c; d) = 0,

n→∞

(5.22)

which gives a motivation of our conjecture. Let us discuss also the large positive ν-asymptotic of u ν (x; a, b, c; d) when x = cos θ belongs to the interval of orthogonality −1 < x < 1. It is clear from (5.16) that the leading terms in the asymptotic expansion of this function are given by u ν (cos θ ; a, b, c; d) ∼

(a eiθ , b eiθ , c eiθ , q 1−ν e−iθ /d; q)∞ (ab, ac, bc, e2iθ , q e−iθ /d; q)∞ +

(a e−iθ , b e−iθ , c e−iθ , q 1−ν eiθ /d; q)∞ . (ab, ac, bc, e−2iθ , q eiθ /d; q)∞

(5.23)

In particular, when ν = νn are large zeros of u ν (η; a, b, c; d), we can estimate u νn (cos θ; a, b, c; d) ∼ u γn (cos θ; a, b, c; d),

(5.24)

where q γn = q n /αd, due to (5.19) as n → ∞. Relations (5.23)–(5.24) lead to the following theorem. Theorem 5.6. For −1 < x = cos θ < 1 and |q| < 1 the leading term in the asymptotic expansion of u γn (cos θ; a, b, c; d) as n → ∞ is given by u γn (cos θ ; a, b, c; d) ∼

q −(n−1)(n−2)/2 (ab, ac, bc; q)∞ × |A(eiθ )| cos((n − 1)θ − χ ),

(5.25)

where A(eiθ ) = |A(eiθ )|−2 =

(a eiθ , b eiθ , c eiθ , α e−iθ , q eiθ /α; q)∞ , (e2iθ , q e−iθ /d; q)∞

(5.26)

(e2iθ , e−2iθ ; q)∞ (a eiθ , a e−iθ , b eiθ , b e−iθ , c eiθ , c e−iθ ; q)∞ ×

(q eiθ /d, q e−iθ /d; q)∞ , (α eiθ , α e−iθ , q eiθ /α, q e−iθ /α; q)∞

(5.27)

and χ = arg A(eiθ ).

(5.28)

SOME ORTHOGONAL 8 ϕ7 -FUNCTIONS

205

It is worth mentioning that (5.27) coincides with the weight function in the orthogonality relation for u ν (cos θ; a, b, c; d). In a similar fashion, one can consider some other properties of zeros of the 8 ϕ7 -function (5.1)–(5.2) close to those established in [11, 17, 18] at the level of the basic trigonometric functions and the q-Bessel function, respectively. 6.

Evaluation of some constants

In this section we shall find an explicit expression for the squared norm,  π u 2ν (cos θ ; a, b, c; d) 2 dν = iθ −iθ , b eiθ , b e−iθ , c eiθ , c e−iθ ; q) ∞ 0 (a e , a e 2iθ −2iθ , q eiθ /d, q e−iθ /d; q)∞ (e , e × dθ, (α eiθ , α e−iθ , q eiθ /α, q e−iθ /α; q)∞

(6.1)

on the right side of (4.21). It is more convenient to write the u ν ’s instead of vν ’s here, cf. (2.14), (4.21), and (6.1). Using (2.8) with a and c interchanged we get dν2 =

(abq ν , q 1−ν /cd; q)∞ (ab, q/cd; q)∞ ∞  (q −ν , abcdq ν−1 ; q)n × qn (q, ac, bc, cd; q)n n=0  π u ν (cos θ; a, b, c; d) × iθ , a e−iθ , b eiθ , b e−iθ , cq n eiθ , cq n e−iθ ; q) (a e ∞ 0 ×

(e2iθ , e−2iθ , q eiθ /d, q e−iθ /d; q)∞ dθ (α eiθ , α e−iθ , q eiθ /α, q e−iθ /α; q)∞

(q −ν , abcdq ν−1 , qa/d, qb/d; q)∞ (ab, ac, bc, cd/q; q)∞ ∞  (abq ν , q 1−ν /cd; q)n × qn (q, qa/d, qb/d, q 2 /cd; q)n n=0  π u ν (cos θ ; a, b, c; d) × iθ −iθ iθ , b e , b e−iθ , q 1+n eiθ /d, q 1+n e−iθ /d; q)∞ 0 (a e , a e +

×

(e2iθ , e−2iθ , q eiθ /d, q e−iθ /d; q)∞ dθ. (α eiθ , α e−iθ , q eiθ /α, q e−iθ /α; q)∞

(6.2)

Both integrals on the right side have the same structure and can be evaluated in a similar way. Let  π u ν (cos θ ; a, b, c; d) In (γ ) := iθ , a e−iθ , b eiθ , b e−iθ , γ q n eiθ , γ q n e−iθ ; q) (a e ∞ 0 ×

(e2iθ , e−2iθ , q eiθ /d, q e−iθ /d; q)∞ dθ (α eiθ , α e−iθ , q eiθ /α, q e−iθ /α; q)∞

206

SUSLOV

=

(qa/d, bcq ν ; q)∞ (q 1−ν a/d, bc; q)∞

 π a −ν −ν 1−ν × q ; q , q /bd, q 1−ν /cd, a eiθ , ae−iθ ; q, bcq ν 8 W7 d 0 (e2iθ , e−2iθ ; q)∞ × iθ −iθ (a e , a e , b eiθ , b e−iθ , γ q n eiθ , γ q n e−iθ ; q)∞ ×

(q 1−ν eiθ /d, q 1−ν e−iθ /d; q)∞ dθ (α eiθ , α e−iθ , q eiθ /α, q e−iθ /α; q)∞

(6.3)

by (2.7) with γ = c and γ = q/d. The last integral can be evaluated in terms of two balanced 5 ϕ4 ’s by (6.4) of [21], 

π

8 W7 (dg/q; h, r, g/ f, d

eiθ , d e−iθ ; q, g f / hr )

0

×

(a

(e2iθ , e−2iθ ; q)∞ iθ −iθ iθ e , a e , b e , b e−iθ , c eiθ , c e−iθ , d

eiθ , d e−iθ ; q)∞

(g eiθ , g e−iθ ; q)∞ dθ ( f eiθ , f e−iθ ; q)∞ 2π (abcd, dg, g/d; q)∞ = (q, ab, ac, ad, bc, bd, cd, d f, f /d; q)∞   ad, bd, cd, g/ f, dg/ hr ; q, q × 5 ϕ4 abcd, qd/ f, dg/ h, dg/r ×

2π (abc f, dg, g/ f ; q)∞ (q, ab, ac, a f, bc, b f, c f, df , d/ f ; q)∞ ( f g/ h, f g/r, dg/ hr ; q)∞ × (dg/ h, dg/r, f g/ hr ; q)∞   a f, b f, c f, g/d, g f / hr ; q, q , × 5 ϕ4 abc f, q f /d, g f / h, g f /r +

(6.4)

this formula appears in a straightforward manner if one uses Bailey’s transform (III.36) of [12] and the Askey–Wilson integral. Thus, In (γ ) =

2π(qa/d, q 1−ν /ad, bcq ν , aγ q; q)∞ (q, q, ab, bc, b/a, αa, qa/α, αγ , qγ /α, aγ ; q)∞ (αγ , qγ /α, aγ ; q)n (aγ q; q)n   aγ q n , acq ν , q 1−ν /bd, αa, qa/α ; q, q × 5 ϕ4 aγ q 1+n , ac, qa/b, qa/d ×

SOME ORTHOGONAL 8 ϕ7 -FUNCTIONS

207

2π(qb/d, q 1−ν /bd, acq ν , bγ q; q)∞ (q, q, ab, ac, a/b, αb, qb/α, αγ , qγ /α, bγ ; q)∞ (αγ , qγ /α, bγ ; q)n × (bγ q; q)n   bγ q n , bcq ν , q 1−ν /ad, αb, qb/α ; q, q , × 5 ϕ4 bγ q 1+n , bc, qb/a, qb/d +

(6.5)

where γ = c and γ = q/d. One can see that the second term here equals the first one with a and b interchanged. Combining (6.2), (6.3), and (6.5), we obtain  dν2

= 0

× =

π

vν2 (cos θ ; a, b, c; d) (a eiθ , a e−iθ , b eiθ , b e−iθ , c eiθ , c e−iθ , q eiθ /d, q e−iθ /d; q)∞

(e2iθ , e−2iθ ; q)∞ dθ (α eiθ , α e−iθ , q eiθ /α, q e−iθ /α; q)∞

2π (qa/d, abq ν , bcq ν , q 1−ν /ad, q 1−ν /cd; q)∞ 2 (1 − ac)(q, ab; q) (bc, b/a, q/cd, αa, qa/α, αc, qc/α; q)∞ ×

∞  n=0

× 5 ϕ4 + ×

 acq n , acq ν , q 1−ν /bd, αa, qa/α acq n+1 , ac, qa/b, qa/d

∞ 

qn





; q, q

(q −ν , abcdq ν−1 , αc, qc/α; q)n (q, qαc, ac, cd; q)n

bcq n , bcq ν , q 1−ν /ad, αb, qb/α

× 5 ϕ4

×

(q −ν , abcdq ν−1 , αc, qc/α; q)n (q, αcq, bc, cd; q)n

2π (qb/d, abq ν , acq ν , q 1−ν /bd, q 1−ν /cd; q)∞ 2 (1 − bc)(q, ab; q) (ac, a/b, q/cd, αb, qb/α, αc, qc/α; q)∞

n=0

+

qn

bcq n+1 , bc, qb/a, qb/d



; q, q

(q −ν , abcdq ν−1 , q 1−ν /ad, bcq ν , qb/d; q)∞ 2π(qa/d; q)2∞ (1 − qa/d)(q, ab, bc; q)2 (ac, b/a, cd/q, αa, qa/α, qα/d, q 2 /αd; q)∞ ∞  n=0

× 5 ϕ4

qn

(abq ν , q 1−ν /cd, qα/d, q 2 /αd; q)n (q, qb/d, q 2 a/d, q 2 /cd; q)n

 aq n+1 /d, acq ν , q 1−ν /bd, αa, qa/α aq n+2 /d, ac, qa/b, qa/d

; q, q



208

SUSLOV

+ ×

(q −ν , abcdq ν−1 , q 1−ν /bd, acq ν , qa/d; q)∞ 2π(qb/d; q)2∞ (1 − qb/d)(q, ab, ac; q)2 (bc, a/b, cd/q, αb, qb/α, qα/d, q 2 /αd; q)∞ ∞  n=0

qn



× 5 ϕ4

(abq ν , q 1−ν /cd, qα/d, q 2 /αd; q)n (q, qa/d, q 2 b/d, q 2 /cd; q)n

bq n+1 /d, bcq ν , q 1−ν /ad, αb, qb/α bq n+2 /d, bc, qb/a, qb/d



; q, q .

(6.6)

One can see again that the second and the fourth terms in this formula are equal to the first and the third ones, respectively, with a and b interchanged. When ν satisfies the boundary condition (4.22) the last integral gives the values of the normalization constants in the orthogonality relation (4.23)–(4.24). 7.

An identity

In this section we shall derive an interesting relation involving a determinant of four 8 ϕ7 functions of type (2.7). Let u(z) = u ν (x(z); a, b, c; d),

(7.1)

v(z) = u ν (x(z); a, b, d; c) = u(z) | c↔d

(7.2)

be two solutions of Eq. (2.1) corresponding to the same eigenvalue (2.4). Then, due to (4.3), [σ (z) ρ(z)W (u(z), v(z))] = 0,

(7.3)

where W (u, v) is the analog of the Wronskian defined in (4.4) and ρ(z) is the appropriate solution of the Pearson equation (3.7). Using the difference-differentiation formula (2.15) we can rewrite the “Wronskian” as W (u(z), v(z)) =

2q (1 − q −ν )(1 − abcdq ν−1 ) (1 − q)c (1 − ab)(1 − ad)(1 − bd)   × u ν−1 x(z − 1/2); aq 1/2 , bq 1/2 , dq 1/2 ; cq 1/2 × u ν (x(z); a, b, c; d) 2q (1 − q −ν )(1 − abcdq ν−1 ) − (1 − q)d (1 − ab)(1 − ac)(1 − bc)   × u ν−1 x(z − 1/2); aq 1/2 , bq 1/2 , cq 1/2 ; dq 1/2 × u ν (x(z); a, b, d; c).

One can easily see that the function g(z) = σ (z) ρ(z)W (u(z), v(z))

(7.4)

SOME ORTHOGONAL 8 ϕ7 -FUNCTIONS

(q z /c, q 1−z /c, q z /d, q 1−z /d; q)∞ (aq z , aq 1−z , bq z , bq 1−z ; q)∞ × W (u(z), v(z)

209

= cd

(7.5)

in (7.3) is doubly periodic function in z without poles in the rectangle on the Figure 1. Therefore, this function is just a constant by Liouville’s theorem, σ (z) ρ(z)W (u(z), v(z)) = C.

(7.6)

To find the value of this constant we can choose here z = z 0 such that q z0 = a. From (7.4) and (2.7) one gets W (u(z 0 ), v(z 0 )) 2q(1 − q −ν )(1 − abcdq ν−1 ) = (1 − q)  (q, qbc, qb/d, qc/d, bdq ν , q 1−ν /ac, q 1−ν a/d, a 2 bcq ν ; q)∞ × acd(ab, qac, bc, bd, 1/ac, q/ad, a/d, qabc/d; q)∞ × 8 W7 (abc/d; ab, ac, a/d, q 1−ν /ad, bcq ν ; q, q) (q, qbd, qb/c, qd/c, bcq ν , q 1−ν /ad, q 1−ν a/c, a 2 bdq ν ; q)∞ − acd(ab, qad, bd, bc, 1/ad, q/ac, a/c, qabd/c; q)∞  × 8 W7 (abd/c; ab, ad, a/c, q 1−ν /ac, bdq ν ; q, q) =

2q(q, a 2 , qb/a, d/c, qc/d, q −ν , abcdq ν−1 , abq ν , q 1−ν /cd; q)∞ (1 − q)acd(ab, qac, ad, bc, bd, 1/ac, q/ad, a/c, a/d; q)∞

(7.7)

by (III.24) and (II.25) of [12]. As a result, from (7.6) and (7.7) we find the value of the “Wronskian” of the 8 ϕ7 -functions (7.1) and (7.2), W (u(z), v(z)) =

2q(c/d, qd/c, q −ν , abcdq ν−1 , abq ν , q 1−ν /cd; q)∞ (1 − q)c(ab, ab, ac, ad, bc, bd; q)∞ (aq z , aq 1−z , bq z , bq 1−z ; q)∞ × z . (q /c, q 1−z /c, q z /d, q 1−z /d; q)∞

Due to (7.4) and (2.7) the last equation can also be rewritten in a more explicit form, d(ac, adq; q)∞ 2−ν (q a/c, q 1−ν a/d; q) ×

(q

∞

/c, q /c, q 1−ν+z /d, q 1−ν−z /d; q)∞ (q z /c, q 1−z /c, q 1+z /d, q 1−z /d; q)∞

1−ν+z

2−ν−z

× 8 W7 (q 1−ν a/c; q 1−ν , q 1−ν /bc, q 1−ν /cd, aq z , aq 1−z ; q, q) × 8 W7 (q −ν a/d; q −ν , q 1−ν /bd, q 1−ν /cd, aq z , aq −z ; q, q)

(7.8)

210

SUSLOV

− ×

c(ad, acq; q)∞ 2−ν (q a/d, q 1−ν a/c; q) (q

∞

/d, q /d, q 1−ν+z /c, q 1−ν−z /c; q)∞ (q z /d, q 1−z /d, q 1+z /c, q 1−z /c; q)∞

1−ν+z

2−ν−z

× 8 W7 (q 1−ν a/d; q 1−ν , q 1−ν /bd, q 1−ν /cd, aq z , aq 1−z ; q, q) × 8 W7 (q −ν a/c; q −ν , q 1−ν /bc, q 1−ν /cd, aq z , aq −z ; q, q) =

(c/d, qd/c, q 1−ν , abcdq ν , abq ν , q 1−ν /cd; q)∞ c(ab, abq, qa/c, qa/d, bcq ν , bdq ν ; q)∞

×

d(aq z , aq 1−z , bq z , bq 1−z ; q)∞ . (q z /c, q 1−z /c, q z /d, q 1−z /d; q)∞

(7.9)

The second term on the left side here is the same as the first one with c and d interchanged. One can see from (7.8) that two solutions u(z) and v(z) are linearly dependent when ν is an integer. In this case due to (2.13) both solutions are the Askey–Wilson polynomials, up to a factor, which are related by Sears’ transformation. On the other hand, Eq. (7.8) shows that there is no analog of Sears’s transformation at the level of very-well-poised 8 ϕ7 -functions.

8.

Some special and limiting cases

The Askey–Wilson polynomials are known as the most general system of classical orthogonal polynomials. They include all other classical orthogonal polynomials as special and/or limiting cases [1, 6, 12, 26, 29]. Let us discuss in a similar manner a few interesting special cases of the orthogonal 8 ϕ7 -functions (2.7). 8.1.

Extension of continuous dual q-Hahn polynomials

Letting c → 0 in (2.7) and then changing d by c we get (qa/c, q 1−ν+z /c, q 1−ν−z /c; q)∞ (q 1−ν a/c, q 1+z /c, q 1−z /c; q)∞     aq −ν aq −ν aq −ν −ν q 1−ν z −z ,q , −q ,q , , aq , aq  b  c  c c bc × 7 ϕ7   ; q, q    c −ν −ν 1−ν+z 1−ν−z aq aq aq q q ,− , , ab, 0, , c c c c c   q −ν , aq z , aq −z (q 1−ν /ac; q)∞ = ; q, q 3 ϕ2 (q/ac; q)∞ ab, ac

u ν (x; a, b; c) =

+

(q −ν , qb/c, aq z , aq −z ; q)∞ (ab, ac/q, q 1+z /c, q 1−z /c; q)∞

SOME ORTHOGONAL 8 ϕ7 -FUNCTIONS

× 3 ϕ2

 q 1−ν /ac, q 1+z /c, q 1−z /c qb/c, q 2 /ac

211  ; q, q .

(8.1)

The 7 ϕ7 -function here can also be transformed to a single 3 ϕ2 by (3.2.11) of [12], u ν (x; a, b; c) =

(abq ν , q 1−ν+z /c, q 1−ν−z /c; q)∞ (ab, q 1+z /c, q 1−z /c; q)∞   q −ν , q 1−ν /ac, q 1−ν /bc ν × 3 ϕ2 ; q, abq . q 1−ν+z /c, q 1−ν−z /c

(8.2)

One can easily see that for an integer ν our function (8.1)–(8.2) is just a multiple of the continuous dual q-Hahn polynomial [1, 6, 26, 29]. The orthogonality relation for the corresponding entire function,   vν (x; a, b; c) = q 1+z /c, q 1−z /c; q ∞ u ν (x; a, b; c),

(8.3)

takes the form 

π

vµ (cos θ ; a, b; c) vν (cos θ ; a, b; c) iθ , a e−iθ , b eiθ , b e−iθ , q eiθ /c, q e−iθ /c; q) (a e ∞ 0 (e2iθ , e−2iθ ; q)∞ × dθ = 0 (α eiθ , α e−iθ , q eiθ /α, q e−iθ /α; q)∞

(8.4)

if µ = ν, and  0

π

(a

(vν (cos θ ; a, b; c))2 iθ −iθ e , a e , b eiθ , b e−iθ , q eiθ /c, q

e−iθ /c; q)∞

(e2iθ , e−2iθ ; q)∞ dθ eiθ /α, q e−iθ /α; q)∞ = (q, q, αa, qa/α, αb, qb/α, qα/c, q 2 /αc; q)−1 ∞

  −4πq 1/2 c ∂ ∇ × vν (η; a, b; c) vν (x; a, b; c)  1 − q ∂λν ∇x x=η ×

(α eiθ , α e−iθ , q

(8.5)

if µ = ν, respectively. The “degree” parameters µ and ν here satisfy the “boundary” condition

1 −1 vε (α + α ); a, b; c 2   q −ε , a/α, b/α α 1−ε = (qα/c, q /αc; q)∞ 3 ϕ2 = 0. (8.6) ; q, q c q 1−ε /αc, ab Properties of zeros of this function follow as a special case of the results of Section 5.

212 8.2.

SUSLOV

Extension of Al-Salam and Chihara polynomials

Letting b → 0 in (8.1)–(8.2) and then changing c by b one gets (q 1−ν+z /b, q 1−ν−z /b; q)∞ (q 1+z /b, q 1−z /b; q)∞   q −ν , q 1−ν /ab a ; q, q × 2 ϕ2 1−ν+z b q /b, q 1−ν−z /b   q −ν , aq z , aq −z (q 1−ν /ab; q)∞ = ; q, q 3 ϕ2 (q/ab; q)∞ ab, 0

u ν (x; a; b) =

(q −ν , aq z , aq −z ; q)∞ (ab/q, q 1+z /b, q 1−z /b; q)∞   q 1−ν /ab, q 1+z /b, q 1−z /b × 3 ϕ2 ; q, q . q 2 /ab, 0 +

(8.7)

For an integer ν this function is a multiple of the Al-Salam and Chihara polynomial [1, 4, 6, 26]. The orthogonality relation of the entire function,   vν (x; a; b) = q 1+z /b, q 1−z /b; q ∞ u ν (x; a; b),

(8.8)

is 

π

vµ (cos θ; a; b)vν (cos θ ; a; b) iθ −iθ , q eiθ /b, q e−iθ /b; q) ∞ 0 (a e , a e (e2iθ , e−2iθ ; q)∞ × dθ = 0, (α eiθ , α e−iθ , q eiθ /α, q e−iθ /α; q)∞

(8.9)

where µ = ν are solutions of vε

1 (α + α −1 ); a; b 2



= (qα/b, q 1−ε /αb; q)∞ 2 ϕ1

For properties of zeros see Section 5.

 q −ε , a/α

α ; q, q 1−ε b q /αb

 = 0.

(8.10)

SOME ORTHOGONAL 8 ϕ7 -FUNCTIONS

8.3.

213

Extension of continuous big q-Hermite polynomials

Letting a → 0 in (8.7) and then changing b by a we have u ν (x; a) =

(q 1−ν+z /a, q 1−ν−z /a; q)∞ (q 1+z /a, q 1−z /a; q)∞   q −ν 2−ν 2 × 1 ϕ2 1−ν+z ; q, q /a . q /a, q 1−ν−z /a

(8.11)

For an integer ν this function is a multiple of the continuous big q-Hermite polynomial [26]. The orthogonality relation has the form  0

π

vµ (cos θ; a)vν (cos θ ; a)(e2iθ , e−2iθ ; q)∞ dθ = 0, (α eiθ , α e−iθ , q eiθ /α, q e−iθ /α, q eiθ /a, q e−iθ /a; q)∞

(8.12)

where µ = ν satisfy vε

1 (α + α −1 ); a 2



= (q 1−ε /αa; q)∞ 1 ϕ1



q/αa

q 1−ε /αa

 ; q, q 1−ε α/a

=0

(8.13)

and vε (x; a) = (q 1+z /a, q 1−z /a; q)∞ u ε (x; a). 8.4.

Extension of continuous q-Hermite polynomials

The continuous q-Hermite polynomials Hn (x | q) are the simplest special case a = b = c = d = 0 of the Askey–Wilson polynomials pn (x; a, b, c, d) or the special case a = 0 of the continuous big q-Hermite polynomials pn (x; a) [6, 12, 26, 29]. See [2, 4] for the proof of lim pn (x; a) = Hn (x | q)

a→0

directly from the series representation of the continuous big q-Hermite polynomials. Let us consider the difference Eq. (2.1) with a = b = c = d = 0 and let us choose the following solution, u ν (z) = Hν (x(z) | q), such that Hν (x | q) =

(−q 1−ν+2z , −q 1−ν−2z ; q 2 )∞ (−q 1+2z , −q 1−2z ; q 2 )∞   q −ν , q 1−ν 2 × 2 ϕ2 ;q ,q −q 1−ν+2z , −q 1−ν−2z

(8.14)

214

SUSLOV

as a nonterminating extension of the continuous q-Hermite polynomials. This solution differs from the corresponding one in [8] by a periodic factor. For an integer ν function (8.14) coincides with Hn (x | q) up to a constant. Comparing (8.14) with the Eq. (2.8) of [11] one can see that function Hν (x | q) is a multiple of the basic cosine function C(x; ω) for certain values of parameter ν. Therefore, this function satisfies the orthogonality relation (1.16) under the “boundary” conditions (1.20). We would like to leave the details to the reader. In a similar fashion one can consider some other special and limiting cases of the orthogonal 8 ϕ7 -functions.

Appendix: Estimate of number of zeros In this section we give an estimate for the number of zeros of the “boundary” function vν (η; a, b, c; d) on the basis of Jensen’s theorem (see, for example, [28]). We shall apply the method proposed by Mourad Ismail at the level of the third Jackson q-Bessel functions [15] (see also [18, 11] for an extension of his idea to q-Bessel functions on a q-quadratic grid and q-trigonometric functions, respectively). Let us consider the entire function

1 −1 f (ζ ) = vν (α + α ); a, b, c; d 2 (qα/d; q)∞ = (abc/α; q)∞ ∞  × (1 − ζ q k+1 /αd + abcq 2k+1 /α 2 d) ×

k=0 ∞  m=0

×

m−1  k=0

α q d

m

(1 − abcq 2m−1 /α)(abcq −1 /α, a/α, b/α, c/α; q)m (1 − abcq −1 /α)(q, ab, ac, bc; q)m

1 − ζ q k + abcdq 2k−1 1 − ζ q k+1 /αd + abcq 2k+1 /α 2 d

(9.1)

in a complex variable ζ = q −ν + abcdq ν−1 ,

(9.2)

where |qα/d| < 1 (cf. (5.2)). Let n f (r ) be the number of zeros of f (ζ ) in the circle |ζ | < r . Consider also circles of radius R = Rn = q −n /β + βabcdq n−1 , q < β < 1 with n = 1, 2, 3, . . . in the complex ζ -plane. Since n f (r ) is nondecreasing with r one can write n f (Rn ) ≤ n f (r ) ≤ n f (Rn+1 )

(9.3)

SOME ORTHOGONAL 8 ϕ7 -FUNCTIONS

215

if Rn ≤ r ≤ Rn+1 , and, therefore,  Rn+1  Rn+1  Rn+1 n f (r ) dr dr n f (Rn ) ≤ dr ≤ n f (Rn+1 ) . r r r Rn Rn Rn

(9.4)

But 

Rn+1 Rn

 Rn+1

−1  dr q + β 2 abcdq 2n  = log q −1 + o(1) = log r  = log r 1 + β 2 abcdq 2n−1 Rn

(9.5)

as n → ∞, and, finally, one gets log q

−1

 n f (Rn )