Q-Hermite Polynomials and Classical Orthogonal Polynomials ∗

Q-Hermite Polynomials and Classical

Orthogonal Polynomials ∗

Christian Berg and Mourad E. H. Ismail

February 17, 1995

Abstract

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

Running title: Classical Orthogonal Polynomials.

1990 Mathematics Subject Classiﬁcation: Primary 33D45, Secondary 33A65, 44A60. em Key words and phrases. Askey-Wilson polynomials, q-orthogonal polynomials, orthogonality relations, q-beta integrals, q-Hermite polynomials, biorthogonal rational functions.

1. Introduction and Preliminaries. The q-Hermite polynomials seem to be at the bottom of a hierarchy of the classical q-orthogonal polynomials, [6]. They contain no parameters, other than q, and one can get them as special or limiting cases of other orthogonal polynomials. The purpose of this work is to show how one can systematically build the classical q-orthogonal polynomials from the q-Hermite polynomials using a simple procedure of attaching generating functions to measures. Let p (x) be orthogonal polynomials with respect to a positive measure µ with moments { n } of any order and inﬁnite support such that

∗Research partially supported by NSF grant DMS 9203659

1 ∞ (1.1) pn(x)pm(x) dµ(x)= ζnδm,n. −∞ Assume that we know a generating function for p (x) , that is we have { n } ∞ n (1.2) pn(x)t /cn = G(x, t), n=0 for a suitable numerical sequence of nonzero elements c . This implies that the orthogonality { n} relation (1.1) is equivalent to

∞ ∞ n (t1t2) (1.3) G(x, t1)G(x, t2)dµ(x)= ζn 2 , −∞ cn 0 provided that we can justify the interchange of integration and sums. Our idea is to use

G(x, t1)G(x, t2) dµ(x) as a new measure, the total mass of which is given by (1.3), and then look for a system of functions (preferably polynomials) orthogonal or biorthogonal with respect to it. If such a system is found one can then repeat the process. The generating function in (1.2) is assumed to be an elementary function, that is a quotient of products of powers and infinite products. It it clear that we cannot indefinitely continue this process. The form of the generating function will become too complicated at a certain level, and the process will then terminate. The referee wondered whether there is a principal reason which forbids that nice explicit (bi)orthogonal systems can be found with respect to measures which are not elementary. We do not know the answer to this question especially since in the case of associated orthogonal polynomials [11], [19], [28] the weight function involves the reciprocal of the square of the absolute value of a transcendental function. Part of the difficulty is that we do not have direct proofs of the orthogonality of the associated polynomials. If µ has compact support it will often be the case that (1.2) converges uniformly for x in the support and t sufficiently small. In this case the justification is obvious. | | We mention the following general result with no assumptions about the support of µ. For 0 < ρ we denote by D(0, ρ) the set of z C with z < ρ. ≤∞ ∈ | |

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

∞ √ζ (1.4) n zn cn n=0 2 has a radius of convergence ρ with 0 < ρ . ≤∞ (i) Then there is a µ-null set N R such that (1.2) converges absolutely for t < ρ, x R N. ⊆ | | ∈ \ Furthermore (1.2) converges in L2(µ) for t < ρ, and (1.3) holds for t , t < ρ. | | | 1| | 2| (ii) If µ is indeterminate then (1.2) converges absolutely and uniformly on compact subsets of Ω= C D(0, ρ), and G is holomorphic in Ω. ×

Proof. For 0 < r 0 such that (√ζ / c )rn C for n 0, and we 0 n | n| ≤ ≥ ﬁnd

∞ n r0 2 pn(x) L (µ), | | cn ∈ n=0 | | and in particular the sum is ﬁnite for µ-almost all x. This implies that there is a µ-null set N R such that p (x)(tn/c ) is absolutely convergent for t < ρ and x R N. ⊆ n n | | ∈ \ The series (1.2) can be considered as a power series with values in L2(µ), and by assumption its radius of convergence is ρ. It follows that the series in (1.2) converges in L2(µ) to some G(x, t) for t < ρ, and (1.3) is a consequence of Parseval’s formula. | | If µ is indeterminate it is well known that p (x) 2/ζ converges uniformly on compact | n | n subsets of C, cf. [1], [26], and the assertion follows. 2

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

(1.5) 2xH (x q)= H (x q)+(1 qn)H (x q), n | n+1 | − n−1 |

(1.6) xH (x : q)= H (x : q)+ qn−1(1 qn)H (x : q), n n+1 − n−1 and the initial conditions

(1.7) H (x q)= H (x : q)=1, H (x q)=2x, H (x : q)= x. 0 | 0 1 | 1 3 We will describe the q-Hermite polynomials on the unit circle later in the Introduction. The discrete and continuous q-Hermite polynomials have generating functions

∞ H (x : q) (t, t; q) (1.8) n tn = − ∞ , (q; q)n (xt; q)∞ 0 and

∞ Hn(x q) n 1 (1.9) | t = iθ −iθ , x = cos θ, (q; q)n (te , te ; q)∞ 0 respectively, where we used the notation in [16] for the q-shifted factorials

n (1.10) (a; q) := 1, (a; q) := (1 aqk−1), n =1, 2, , or , 0 n − ∞ k=1 and the multiple q-shifted factorials

k (1.11) (a , a , , a ; q) := (a ; q) . 1 2 k n j n j=1 A basic hypergeometric series is

a , ... , a (1.12) φ 1 r q, z = φ (a , ... , a ; b , ... , b ; q, z) r s   r s 1 r 1 s b1, ... , bs   ∞ (a1, ... , ar; q)n = zn(( 1)nqn(n−1)/2)s+1−r. (q, b1, ... , bs; q)n − n=0 In (1.9) e±iθ is x √x2 1 and the square root is chosen so that √x2 1 x as x . This ± − − ≈ →∞ makes e−iθ eiθ . It is clear that the right hand sides of (1.8) and (1.9) are analytic functions | |≤| | of the complex variable t for t < 1/ x , t < e−iθ . | | | | | | | | In Section 2 we apply the procedure outlined at the beginning of the Introduction to the continuous q-Hermite polynomials for q < 1 and we reach the Al-Salam-Chihara polynomials | | in the ﬁrst step and the second step takes us to the Askey-Wilson polynomials. It is worth mentioning that the Askey-Wilson polynomials are the general classical orthogonal polynomials, [6]. As a byproduct we get a simple evaluation of the Askey-Wilson q-beta integral, [10]. This seems to be the end of the line in this direction. The case q > 1 will be studied in Section 5, see comments below. In Section 3 we apply the same procedure to the polynomials U (a)(x; q) and { n } V (a)(x; q) of Al-Salam and Carlitz [2]. They are generated by the recurrences { n }

(1.13) U (a) (x; q) = [x (1 + a)qn]U (a)(x; q)+ aqn−1(1 qn)U (a) (x; q), n> 0, n+1 − n − n−1

4 (1.14) V (a) (x; q) = [x (1 + a)q−n]V (a)(x; q) aq1−2n(1 qn)V (a) (x; q), n> 0, n+1 − n − − n−1 and the initial conditions

(1.15) U (a)(x; q)= V (a)(x; q)=1, U (a)(x; q)= V (a)(x; q)= x 1 a, 0 0 1 1 − −

(a) (a) [2], [14]. It is clear that Un (x;1/q) = Vn (x; q), so there is no loss of generality in assuming

n −1/2 k (q; q)n (q z) (1.16) n(z; q)= . H (q; q)k (q; q)n−k k=0 Szeg˝ointroduced these polynomials in [27] to illustrate his theory of polynomials orthogonal on the unit circle. Szeg˝oused the Jacobi triple product identity to prove the orthogonality relation

2π −n 1 iθ iθ 1/2 iθ 1/2 −iθ (q; q)n q (1.17) m(e ; q) n(e ; q)(q e , q e ; q)∞ dθ = δm,n. 2π 0 H H (q; q)∞

In Section 4 we show how generating functions transform (1.16) to a q-beta integral of Ramanujan.

This explains the origin of the biorthogonal polynomials of Pastro [24] and the 4φ3 biorthogonal rational functions of Al-Salam and Ismail [4]. In section 5 we consider the q-Hermite polynomials for q > 1. They are orthogonal on the imaginary axis. For 0 1, more precisely to the polynomials

(1.18) u (x; t , t )= v ( 2x; q, (t + t )/q,t t q−2, 1), n 1 2 n − − 1 2 1 2 −

5 cf. [9]. We prove that for any positive orthogonality measure µ for the q−1-Hermite polynomials

( t eξ, t e−ξ, t eξ, t e−ξ; q) (1.19) dν (sinh ξ; t , t ) := − 1 1 − 2 2 ∞ dµ(sinh ξ), µ 1 2 ( t t /q; q) − 1 2 ∞ is an orthogonality measure for u . { n} The attachment procedure applied to u leads to the biorthogonal rational functions { n}

−n n−2 q , t1t2q , t1t3/q, t1t4/q (1.20) ϕn(sinh ξ; t1, t2, t3, t4) := 4φ3 − − − q, q .  t eξ, t e−ξ, t t t t q−3  − 1 1 1 2 3 4   of Ismail and Masson [18] in the special case t3 = t4 = 0. The referee raised the question of what determines the starting point in our process. In each case we used the polynomials with fewest possible parameters. In the cases of the polynomials in Sections 2, 4 and 5 we started with polynomials with no parameters, other than q. In Section 3 we used the 1-parameter family of Al-Salam-Carlitz polynomials. We cannot set a = 0 in the Al-Salam-Carlitz polynomials and maintain their orthogonality, so it seems that the full Al- Salam-Carlitz polynomials are the correct starting point. We do not have a canonical answer. The referee also remarked that the Askey scheme in [22] contains many q-polynomials at the lowest level of the classiﬁcation and wondered if these q-polynomials are Hermite like and when we apply our procedure to such Hermite like polynomials we may obtain other results. We plan to investigate this point in a future work.

2. The Continuous q-Hermite Ladder. Here we assume 1

π 2iθ −2iθ 2π(q; q)n (2.1) Hm(cos θ q)Hn(cos θ q)(e , e ; q)∞dθ = δm,n, 0 | | (q; q)∞ and follows easily from the Jacobi triple product identity [16]. The series in (1.9) converges for t < 1 uniformly in θ [0, π]. The reason is that | | ∈ n (q; q)n i(n−2k)θ Hn(cos θ q)= e | (q; q)k(q; q)n−k k=0 implies H (x q) H (1 q) for x [ 1, 1] and (1.9) converges at x = 1. Thus (1.2), (1.3) and | n | | ≤ n | ∈ − the generating function (1.9) imply

π 2iθ −2iθ (e , e ; q)∞ 2π (2.2) iθ −iθ iθ −iθ dθ = , t1 , t2 < 1, 0 (t1e , t1e , t2e , t2e ; q)∞ (q, t1t2; q)∞ | | | |

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

∞ (a; q) (az; q) (2.3) n zn = ∞ , (q; q)n (z; q)∞ 0 with a = 0. The next step is to ﬁnd polynomials p (x) orthogonal with respect to the weight function { n } (e2iθ, e−2iθ; q) 1 (2.4) w (x; t , t ) := ∞ , x = cos θ, 1 1 2 iθ −iθ iθ −iθ 2 (t1e , t1e , t2e , t2e ; q)∞ √1 x − which is positive for t , t ( 1, 1). Here we follow a clever technique of attachment which was 1 2 ∈ − used by Askey and Andrews and by Askey and Wilson in [10]. Write p (x) in the form { n } n −n iθ −iθ (q , t1e , t1e ; q)k (2.5) pn(x)= an,k, (q; q)k k=0 iθ −iθ then determine an,k such that pn(x) is orthogonal to (t2e , t2e ; q)j, j = 0, 1, , n 1. Note iθ −iθ − that (ae , ae ; q)k is a polynomial in x of degree k, since k−1 (2.6) (aeiθ, ae−iθ; q) = (1 2axqj + a2q2j) k − j=0 = ( 2a)kqk(k−1)/2xk + lower order terms. − iθ −iθ iθ −iθ The reason for choosing the bases (t1e , t1e ; q)k and (t2e , t2e ; q)j is that they attach { } { iθ −iθ} iθ −iθ nicely to the weight function and (2.2) enables us to integrate (t1e , t1e ; q)k(t2e , t2e ; q)j against the weight function w1(x; t1, t2). Indeed

iθ −iθ iθ −iθ k j (t1e , t1e ; q)k(t2e , t2e ; q)jw1(x; t1, t2)= w1(x; t1q , t2q ). Therefore 1 iθ −iθ (t2e , t2e ; q)jpn(x)w1(x; t1, t2)dx −1

n −n π 2iθ −2iθ (q ; q)k (e , e ; q)∞ dθ = an,k k iθ k −iθ j iθ j −iθ (q; q)k 0 (t1q e , t1q e , t2q e , t2q e ; q)∞ k=0 n −n 2π (q ; q)kan,k = k+j (q; q)∞ (q; q)k(t1t2q ; q)∞ k=0 n −n j 2π (q , t1t2q ; q)k = j an,k. (q, t1t2q ; q)∞ (q; q)k k=0 At this stage we look for an,k as a quotient of products of q-shifted factorials in order to make the above sum vanish for 0 j < n. The q-Chu-Vandermonde sum [16, (II.6)] ≤ 7 −n (c/a; q)n n (2.7) 2φ1(q , a; c; q, q)= a (c; q)n suggests

k an,k = q /(t1t2; q)k.

Therefore

1 −j iθ −iθ 2π(q ; q)n j n (t2e , t2e ; q)jpn(x)w1(x; t1, t2)dx = j (t1t2q ) . −1 (q, t1t2q ; q)∞ (t1t2; q)n n It follows from (2.5) and (2.6) that the coeﬃcient of x in pn(x) is

(2.8) ( 2t )nqn(n+1)/2(q−n; q) /(q, t t ; q) = (2t )n/(t t ; q) . − 1 n 1 2 n 1 1 2 n

This leads to the orthogonality relation

1 2n 2π(q; q)nt1 (2.9) pm(x)pn(x)w1(x; t1, t2)dx = δm,n. −1 (q, t1t2; q)∞(t1t2; q)n

Furthermore the polynomials are given by

q−n, t eiθ, t e−iθ (2.10) p (x)= φ 1 1 q, q . n 3 2   t1t2, 0   The polynomials we have just found are the Al-Salam-Chihara polynomials and were ﬁrst iden- tiﬁed by W. Al-Salam and T. Chihara [3]. Their weight function was given in [9] and [10].

One might hope that it is possible to make other choices for the coeﬃcients an,k and possibly use summation theorems other than (2.7). We went through the summation theorems in [16] and found that (2.7) is the only summation theorem that works in the case at hand. It is also worth mentioning that the generating function (1.9) is the only elementary generating function for the q-Hermite polynomials known to us, [8]. Observe that the orthogonality relation (2.9) and the uniqueness of the polynomials orthog- −n onal with respect to a positive measure show that t1 pn(x) is symmetric in t1 and t2. This gives the known transformation

q−n, t eiθ, t e−iθ q−n, t eiθ, t e−iθ (2.11) φ 1 1 q, q =(t /t )n φ 2 2 q, q 3 2   1 2 3 2   t1t2, 0 t1t2, 0    

8 as a byproduct of our analysis. Our next task is to repeat the process with the Al-Salam-Chihara polynomials as our starting point. The representation (2.10) needs to be transformed to a form more amenable to generating functions. This can be done using an idea of Ismail and Wilson [21]. First write the 3φ2 as a sum over k then replace k by n k. Applying −

(a; q)n k −kn+k(k−1)/2 (2.12) (a; q)n−k = 1−n ( q/a) q (q /a; q)k − we obtain iθ −iθ n k −n 1−n (t1e , t1e ; q)n −n(n−1)/2 n ( t2/t1) (q , q /t1t2; q)k k(k+1)/2 (2.13) pn(x)= q ( 1) − 1−n iθ 1−n −iθ q . (t1t2; q)n − (q, q e /t1, q e /t1; q)k k=0 Applying the q-analogue of the Pfaﬀ-Kummer transformation [16, (III.4)]

∞ (A, C/B; q)n n(n−1)/2 n (z; q)∞ (2.14) q ( Bz) = 2φ1(A, B; C; q, z), (q, C, Az; q)n − (Az; q)∞ n=0 with

−n iθ 1−n iθ −iθ A = q , B = t2e , C = q e /t1, z = qe /t1 to (2.13), we obtain the representation

(t e−iθ; q) tneinθ q−n, t eiθ p (x)= 1 n 1 φ 2 q, qe−iθ/t . n 2 1  1−n iθ 1 (t1t2; q)n q e /t1   Using (2.12) we express a multiple of pn as a Cauchy product of two sequences. The result is

n n iθ −iθ (q; q)nt1 (t2e ; q)k −ikθ (t1e ; q)n−k i(n−k)θ pn(x)= e e . (t1t2; q)n (q; q)k (q; q)n−k k=0 This and the q-binomial theorem (2.3) establish the generating function

∞ (t1t2; q)n n (tt1, tt2; q)∞ (2.15) pn(x)(t/t1) = −iθ iθ . (q; q)n (te , te ; q)∞ n=0 The orthogonality relation (2.9), the q-binomial theorem and the generating function (2.15) imply the Askey-Wilson q-beta integral, [10], [16]

π 2iθ −2iθ (e , e ; q)∞ 2π (t1t2t3t4; q)∞ (2.16) 4 iθ −iθ dθ = . 0 j=1(tje , tje ; q)∞ (q; q)∞ 1≤j

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

q−n, t t t t qn−1, t eiθ, t e−iθ (2.17) p (x; t , t , t , t q)= t−n(t t , t t , t t ; q) φ 1 2 3 4 1 1 q, q . n 1 2 3 4 1 1 2 1 3 1 4 n 4 3   | t1t2, t1t3, t1t4   The orthogonality relation of the Askey-Wilson polynomials is [10, (2.3)-(2.5)]

π (2.18) pm(cos θ; t1, t2, t3, t4 q)pn(cos θ; t1, t2, t3, t4 q)w(cos θ; t1, t2, t3, t4)dθ 0 | |

2n n−1 2π (t1t2t3t4q ; q)∞ (t1t2t3t4q ; q)n = n+1 n δm,n, (q ; q)∞ 1≤j

2iθ −2iθ (e , e ; q)∞ (2.19) w(cos θ; t1, t2, t3, t4)= 4 iθ −iθ . j=1(tje , tje ; q)∞ Observe that the weight function in (2.19) and the right-hand side of (2.18) are symmetric functions of t , t , t , t . The weight function in (2.18) is positive when max t , t , t , t < 1, 1 2 3 4 {| 1| | 2 | 3| | 4|} and the uniqueness of the polynomials orthogonal with respect to a positive measure shows that the Askey-Wilson polynomials are symmetric in the four parameters t1, t2, t3, t4. This symmetry is the Sears transformation [16, (III.15)], a fundamental transformation in the theory of basic hypergeometric functions. The Sears transformation may be stated in the form

q−n, a, b, c bc n ( de , df ; q) q−n,a,d/b,d/c (2.20) φ q, q = bc bc n φ q, q , 4 3   4 3  de df  d, e, f d (e, f; q)n d, , bc bc     where abc = defqn−1. Ismail and Wilson [21] used the Sears transformation to establish the generating function ∞ p (cos θ; t , t , t , t q) (2.21) n 1 2 3 4| tn (q, t1t2, t3t4; q)n n=0 t eiθ, t eiθ t e−iθ, t e−iθ = φ 1 2 q, te−iθ φ 3 4 q, teiθ . 2 1   2 1   t1t2 t3t4     Thus (2.18) leads to the evaluation of the following integral

10 π 6 t eiθ, t eiθ t e−iθ, t e−iθ (2.22) φ 1 2 q, t e−iθ φ 3 4 q, t eiθ 2 1  j  2 1  j  0 j=5 t1t2 t3t4     2iθ −2iθ (e , e ; q)∞ 4 iθ −iθ dθ × j=1(tje , tje ; q)∞ 2π (t t t t ; q) = 1 2 3 4 ∞ (q; q)∞ 1≤j

Rogers’s linearization formula is the special case t4 = 0 of (2.16).

3. The Discrete q-Hermite Ladder. Here we assume 0

∞ (a) (a) (a) n n(n−1)/2 (3.2) Um (x; q)Un (x; q) dµ (x)=( a) q (q; q)nδm,n, −∞ − with µ(a) a discrete probability measure on [a, 1] given by

∞ n n (a) q q (3.3) µ = εqn + εaqn . (q,q/a; q)n(a; q)∞ (q, aq; q)n(1/a; q)∞ n=0

In (3.3) εy denotes a unit mass supported at y. The form of the orthogonality relation (3.2)-(3.3) given in [2] and [14] contained a complicated form of a normalization constant. The value of the constant was simpliﬁed in [17]. Since the radius of convergence of (1.4) is ρ = we can apply ∞ Proposition 1.1 and get ∞ ∞ n (a) ( at1t2) n(n−1)/2 (3.4) G(x; t1) G(x; t2) dµ (x)= − q =(at1t2; q)∞, t1, t2 C, −∞ (q; q)n ∈ n=0 where the last expression follows by Euler’s theorem [16, (II.2)]

11 ∞ (3.5) znqn(n−1)/2/(q; q) =( z; q) . n − ∞ n=0 This establishes the integral

∞ dµ(a)(x) (at t ; q) (3.6) = 1 2 ∞ . −∞ (xt1, xt2; q)∞ (t1, t2, at1, at2; q)∞

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

(Aq/C,Bq/C; q) A, B (A, B; q) Aq/C,Bq/C (3.7) ∞ φ q, q + ∞ φ q, q 2 1   2 1  2  (q/C; q)∞ C (C/q; q)∞ q /C     =(ABq/C ; q)∞.

We now restrict the attention to t , t (a−1, 1) in the case of which 1/(xt , xt ; q) is a 1 2 ∈ 1 2 ∞ positive weight function on [a, 1]. The next step is to ﬁnd polynomials orthogonal with respect (a) to dµ (x)/(xt1, xt2; q)∞. Deﬁne Pn(x) by

n −n (q , xt1; q)k k (3.8) Pn(x)= q an,k (q; q)k k=0 where an,k will be chosen later. Using (3.6) it is easy to see that

∞ n −n k+m (xt2; q)m (a) (q ; q)k k (at1t2q ; q)∞ Pn(x) dµ (x)= q an,k k k m m −∞ (xt1, xt2; q)∞ (q; q)k (t1q , at1q , t2q , at2q ; q)∞ k=0

m n −n (at1t2q ; q)∞ (q , t1, at1; q)k k = m m m an,kq . (t1, at1, t2q , at2q ; q)∞ (q, at1t2q ; q)k k=0

The choice an,k =(λ; q)k/(t1, at1; q)k allows us to apply the q-Chu-Vandermonde sum (2.7). The n−1 choice λ = at1t2q leads to

∞ m m+1−n n−1 n (xt2; q)m (a) (at1t2q ; q)∞ (q ; q)n (at1t2q ) (3.9) Pn(x) dµ (x)= m m m . −∞ (xt1, xt2; q)∞ (t1, at1, t2q , at2q ; q)∞ (at1t2q ; q)n

The right-hand side of (3.9) vanishes for 0 m < n. The coeﬃcient of xn in P (x) is ≤ n −n n−1 n−1 (q , at1t2q ; q)n n n(n+1)/2 (at1t2q ; q)n n ( t1) q = t1 . (q, t1, at1; q)n − (t1, at1; q)n

Therefore

12 q−n, at t qn−1, xt (3.10) P (x)= ϕ (x; a, t , t )= φ 1 2 1 q, q , n n 1 2 3 2   t1, at1   satisﬁes the orthogonality relation

∞ dµ(a)(x) (3.11) ϕm(x; a, t1, t2)ϕn(x; a, t1, t2) −∞ (xt1, xt2; q)∞

n−1 2n (q, t2, at2, at1t2q ; q)n (at1t2q ; q)∞ 2 n n(n−1)/2 = ( at1) q δm,n. (t1, at1, t2, at2; q)∞ (t1, at1; q)n −

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

q−n,αβqn+1, x (3.12) P (x; α,β,γ : q)= φ q, q . n 3 2   αq,γq   Note that we may rewrite the orthogonality relation (3.11) in the form

∞ (a) −m −n dµ (x) (3.13) t1 (t1, at1; q)mϕm(x; a, t1, t2) t1 (t1, at1; q)nϕn(x; a, t1, t2) −∞ (xt1, xt2; q)∞

n−1 2n (q, t1, at1, t2, at2, at1t2q ; q)n (at1t2q ; q)∞ n n(n−1)/2 = ( a) q δm,n. (t1, at1, t2, at2; q)∞ −

(a) Since dµ (x)/(xt1, xt2; q)∞ and the right-hand side of (3.13) are symmetric in t1 and t2, then

−n t1 (t1, at1; q)n ϕn(x; a, t1, t2) must be symmetric in t1 and t2. This gives the known 3φ2 transformation

q−n, at t qn−1, xt tn(t , at ; q) q−n, at t qn−1, xt (3.14) φ 1 2 1 q, q = 1 2 2 n φ 1 2 2 q, q . 3 2   n 3 2   t1, at1 t2 (t1, at1; q)n t2, at2     We now consider the polynomials V (a)(x; q) and restrict the parameters to 0 < a, 0

∞ n n2 (a) a q (3.15) m =(aq; q)∞ εq−n , (q, aq; q)n n=0

13 and in the second case it is

∞ −n n2 (a) a q (3.16) σ =(q/a; q)∞ εaq−n , (q,q/a; q)n n=0 cf. [12]. The total mass of these measures was evaluated to 1 in [17]. If q

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

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

∞ (a) (a) n −n2 (3.18) Vm (x; q)Vn (x; q)dµ(x)= a q (q; q)nδm,n. −∞ The polynomials have the generating function [2], [14]

∞ n(n−1)/2 (a) q n (xt; q)∞ (3.19) V (x; t) := Vn (x; q) ( t) = , t < min (1, 1/a), (q; q)n − (t, at; q)∞ | | n=0

−n2 n and ζn = q a (q; q)n. The power series (1.4) has the radius of convergence q/a, and therefore (1.3) becomes ∞ ∞ (xt1, xt2; q)∞ dµ(x) = V (x, t1) V (x, t2)dµ(x) −∞ (t1, at1, t2, at2; q)∞ −∞ ∞ (at t /q)n = 1 2 (q; q)n n=0 1 = , t , t < q/a. (at t /q; q) | 1| | 2| 1 2 ∞ This identity with µ = m(a) or µ = σ(a) is nothing but the q-analogue of the Gauss theorem,

(c/a, c/b; q)∞ (3.20) 2φ1(a, b; c; q, c/ab)= , c < ab , (c, c/ab; q)∞ | | | | [16, (II.8)]. Specializing to the density (3.17) we get

∞ (xt , xt ; q) dx πa(t , at , t , at ; q) (3.21) 1 2 ∞ = 1 1 2 2 ∞ , −∞ (x/a; q)2 + γ2(x; q)2 a 1 γ(q, aq, q/a, at t /q; q) ∞ ∞ | − | 1 2 ∞ 14 valid for q 0. Formula (3.21) seems to be new. We now seek polynomials or rational functions that are orthogonal with respect to the measure

(3.22) dν(x)=(xt1, xt2; q)∞ dµ(x),

where µ satisﬁes (3.18). It is clear that we can integrate 1/[(xt1; q)k(xt2; q)j] with respect to the measure ν. Set

n −n k (q ; q)k q an,k (3.23) ψn(x; a, t1, t2) := . (q; q)k (xt1; q)k k=0

The rest of the analysis is similar to our treatment of the Un’s. We get

∞ n −n ∞ ψn(x; a, t1, t2) (q ; q)k k k m dν(x)= q an,k (xt1q , xt2q ; q)∞ dµ(x), −∞ (xt2; q)m (q; q)k −∞ k=0

and if we choose an,k =(t1, at1; q)k/(at1t2/q; q)k the above expression is equal to

(t , at , t qm, at qm; q) q−n, at t qm−1 1 1 2 2 ∞ φ 1 2 q, q , m−1 2 1   (at1t2q ; q)∞ at1t2/q   m m −m (t1, at1, t2q , at2q ; q)∞(q ; q)n m−1 n = m−1 (at1t2q ) , (at1t2q ; q)∞(at1t2/q; q)n which is 0 for m < n. We have used the Chu-Vandermonde sum (2.7). Since ν is symmetric in t1, t2, this leads to the biorthogonality relation

∞ (t1, at1, t2, at2; q)∞(q; q)n n (3.24) ψm(x; a, t2, t1)ψn(x; a, t1, t2) dν(x)= (at1t2/q) δm,n. −∞ (at1t2/q; q)∞(at1t2/q; q)n

The ψn’s are given by

−n q , t1, at1 (3.25) ψn(x; a, t1, t2)= 3φ2 q, q .  xt1, at1t2/q    They are essentially the rational functions studied by Al-Salam and Verma in [5]. Al-Salam and Verma used the notation

β,αγ/δ,q−n (3.26) Rn(x; α,β,γ,δ; q)= 3φ2 q, q .  βγ/q,αqx    The translation between the two notations is

15 −1 (3.27) ψn(x; a, t1, t2)= Rn(βxq /α; α,β,γ,δ; q), with

(3.28) t1 = β, t2 = βδ/qα, a = αγ/βδ.

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

4. The Szeg˝oLadder. Here we assume 0

∞ n t −1/2 (4.1) (z, t) := n(z; q) =1/(t, tzq ; q)∞, H H (q; q)n n=0 for t < 1, tz < q1/2. ¿From (1.16) it follows that (z; q) ( z ; q). Furthermore | | | | |Hn | ≤ Hn | | Darboux’s method [23] and (4.1) give

(1; q) q−n/2/(q−1/2; q) . Hn ≈ ∞ Thus (1.17) and (4.1) imply the Ramanujan q-beta integral [16]

1/2 1/2 1 (q z, q /z; q)∞ dz (t1, t2; q)∞ (4.2) −1/2 −1/2 = , 2πi |z|=1 (t1q z, t2q /z; q)∞ z (q, t1t2/q; q)∞ for t < q1/2, t < q1/2, since (1.4) has radius of convergence q1/2 and the series in (4.1) | 1| | 2| converges uniformly in z for z on the unit circle. This can be proved by estimating (z; q) Hn directly from (1.16). Putting

1/2 1/2 (q, t1t2q, q z, q /z; q)∞ (4.3) Ω(z)= 1/2 1/2 (t1q, t2q, t1q z, t2q /z; q)∞ and applying the attachment technique to (4.2) we ﬁnd that the polynomials

q−n, t q1/2z, t q (4.4) p˜ (z, t , t ) := φ 1 1 q, q . n 1 2 3 2   0, t1t2q   satisfy the biorthogonality relation

16 1 dz (q; q)n n (4.5) p˜m(z, t1, t2)p˜n(z, t2, t1) Ω(z) = (t1t2q) δm,n. 2πi |z|=1 z (t1t2q; q)n Using the transformation [16, (III.7)] we see that

(q; q)n n (4.6) p˜n(z, t1, t2)= (t1q) pn(z, t1, t2), (t1t2q; q)n where

n (b; q)n −n 1−n 1/2 (aq; q)k (b; q)n−k −1/2 k (4.7) pn(z, a, b)= 2φ1(q , aq; q /b; q, q z/b)= (q z) , (q; q)n (q; q)k (q; q)n−k k=0 are the polynomials considered by Pastro [24] and for which the biorthogonality relation reads

1 dz (t1t2q; q)n −n (4.8) pm(z, t1, t2)pn(z, t2, t1) Ω(z) = q δm,n. 2πi |z|=1 z (q; q)n The special case when a and b are real was considered by Askey in his comments on [27] in Szeg˝o’s Collected Papers. Al-Salam and Ismail [4] used (4.8) and the generating function

∞ 1/2 n (atzq , bt; q)∞ (4.9) pn(z; a, b)t = −1/2 (tzq , t; q)∞ n=0 to establish a q-beta integral and found the rational functions biorthogonal to its integrand. The interested reader is referred to [4] for details.

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

(5.1) h (x q)=2xh (x q) q−n(1 qn)h (x q),n> 0, n+1 | n | − − n−1 |

(5.2) h (x q)=1, h (x q)=2x. 0 | 1 |

The polynomials h (x q) are called the q−1-Hermite polynomials, [18]. The corresponding { n | } moment problem is indeterminate. Let be the set of probability measures which solve the Vq problem. For any µ we have ∈Vq ∞ −n(n+1)/2 (5.3) hm(x q)hn(x q) dµ(x)= q (q; q)nδm,n. −∞ | | 17 ′ The hns have the generating function, [18],

∞ n t n(n−1)/2 ξ −ξ (5.4) hn(x q) q =( te , te ; q)∞, x = sinh ξ, t,ξ C. | (q; q)n − ∈ n=0 By Proposition 1.1 it is clear that (5.3) and (5.4) imply

∞ ξ −ξ ξ −ξ (5.5) ( t1e , t1e , t2e , t2e ; q)∞dµ(x)=( t1t2/q; q)∞, t1, t2 C,µ q −∞ − − − ∈ ∈V since the power series (1.4) has radius of convergence ρ = . Incidentally the function ∞

(5.6) χ (x)=( teξ, te−ξ; q) =( t(√x2 +1+ x), t(√x2 +1 x); q) t − ∞ − − ∞ belongs to L2(µ) for any µ and any t C. Therefore the complex measure ν (t , t ) deﬁned ∈Vq ∈ µ 1 2 by

χ (x)χ (x) (5.7) dν (x; t , t ) := t1 t2 dµ(x), µ , t , t C, t t = q1−k,k 0 µ 1 2 ( t t /q; q) ∈Vq 1 2 ∈ 1 2 − ≥ − 1 2 ∞ has total mass 1, and it is non-negative if t1 = t2. Note that ( teξ, te−ξ; q) is a polynomial of degree k in x = sinh ξ for each ﬁxed t = 0. Since − k

ξ k −ξ k ( te /q , te /q ; q) χ (x)= χ k (x), − k t t/q we see that the non-negative polynomial ( teξ/qk, te−ξ/qk; q) 2 of degree 2k is ν (t, t)-integrable. | − k| µ This implies that νµ(t, t) has moments of any order, and by the Cauchy-Schwarz inequality every polynomial is ν (t , t )-integrable for all t , t C,µ . µ 1 2 1 2 ∈ ∈Vq Introducing the orthonormal polynomials

˜ hn(x q) n(n+1)/4 (5.8) hn(x q)= | q | (q; q)n the q-Mehler formula, cf. [18], reads

∞ ξ+η −ξ−η ξ−η −ξ+η ˜ ˜ n ( zqe , zqe ,zqe ,zqe ; q)∞ (5.9) hn(sinh ξ q)hn(sinh η q)z = − − 2 | | (z q; q)∞ n=0

18 valid for ξ, η C, z < 1/√q. ∈ | | Applying the Darboux method [23] to (5.9) Ismail and Masson [18] found the asymptotic behavior of h˜ (sinh η q), and from their result it follows that (h˜ (sinh η q)zn) l2 for z < n | n | ∈ | | q−1/4, η C. In this case the right-hand side of (5.9) belongs to L2(µ) as a function of x = ∈ sinh ξ and the formula is its orthogonal expansion. Putting t = qzeη,s = qze−η, we have − z2 = stq−2, so if st < q3/2 we have z < q−1/4 and the right-hand side of (5.9) becomes − | | | | χ (sinh ξ)χ (sinh ξ)/( st/q; q) , which belongs to L2(µ). Using this observation we can give a t s − ∞ simple proof of the following formula from [18].

Proposition 5.1 Let µ and let t C, i = 1, , 4 satisfy t t , t t < q3/2. (This holds ∈ Vq i ∈ | 1 3| | 2 4| in particular if t < q3/4, i =1, , 4). Then 4 χ L1(µ) and | i| i=1 ti ∈ 4 1≤j

Proof. We write

qz eη1 = t , qz eη2 = t , qz e−η1 = t , qz e−η2 = t , 1 1 2 2 − 1 3 − 2 4 noting that z2 = t t q−2, z2 = t t q−2, so the equations have solutions z , η , i =1, 2 if t =0 1 − 1 3 2 − 2 4 i i i for i =1, , 4. We next apply Parseval’s formula to the two L2(µ)-functions χ χ , χ χ and t1 t3 t2 t4 get

4 ∞ χ dµ =( t t /q, t t /q; q) h˜ (sinh η q)h˜ (sinh η q)(z z )n, ti − 1 3 − 2 4 ∞ n 1| n 2| 1 2 i=1 n=0 which by the q-Mehler formula gives the right-hand side of (5.10). If t = 0 and t t t = 0 we apply Parseval’s formula to χ and χ χ , and if two of the 1 2 3 4 t3 t2 t4 parameters are zero the formula reduces to (5.5). 2

We shall now look at orthogonal polynomials for the measures νµ(t1, t2). When q > 1 the Al-Salam-Chihara polynomials are orthogonal on ( , ) and their moment problem may be −∞ ∞ indeterminate [9], [3], [15]. If one replaces q by 1/q in the Al-Salam-Chihara polynomials, they can be renormalized to polynomials v (x; q,a,b,c) satisfying { n }

(5.11) (1 qn+1)v (x; q,a,b,c)=(a xqn)v (x; q,a,b,c) (b cqn−1)v (x; q,a,b,c), − n+1 − n − − n−1 where 0

19 (5.12) u (x; t , t )= v ( 2x; q, (t + t )/q,t t q−2, 1), n 1 2 n − − 1 2 1 2 − where t1, t2 are complex parameters. The corresponding monic polynomialsu ˆn(x) satisfy the recurrence relation determined from (5.11)

1 1 (5.13) xuˆ (x)=û (x)+ (t + t )q−n−1uˆ (x)+ (t t q−2n−1 + q−n)(1 qn)û (x), n n+1 2 1 2 n 4 1 2 − n−1 so by Favard’s theorem, cf. [14], u (x; t , t ) are orthogonal with respect to a complex measure { n 1 2 } α(t , t ) if and only if t t = qn+1, n 1, and with respect to a probability measure α(t , t ) if 1 2 1 2 − ≥ 1 2 and only if t = t C R or t , t R and t t 0. In the affirmative case 2 1 ∈ \ 1 2 ∈ 1 2 ≥

∞ n(n−3)/2 q −n−1 (5.14) um(x; t1, t2)un(x; t1, t2) dα(x; t1, t2)= ( t1t2q ; q)nδm,n. −∞ (q; q)n −

It follows from (3.77) in [9] that the moment problem is indeterminate if t1 = t2. If t1, t2 are real, diﬀerent and t t 0, we can assume t < t without loss of generality, and in this case 1 2 ≥ | 1| | 2| the moment problem is indeterminate if and only if t /t > q. | 1 2| The generating function (3.70) in [9] takes the form (x = sinh ξ)

∞ ξ −ξ n ( te , te ; q)∞ (5.15) un(x; t1, t2)t = − , for t < min q/ t1 ,q/ t2 . ( t1t/q, t2t/q; q)∞ | | { | | | |} n=0 − −

By the method used in [9] we can derive formulas for un in the following way: Use the q- binomial theorem to write the right-hand side of (5.15) as a product of two power series in t and equate coeﬃcients of tn to get

n ξ −ξ (qe /t1; q)k k ( qe /t2; q)n−k n−k (5.16) un(x; t1, t2)= ( t1/q) − ( t2/q) . (q; q)k − (q; q)n−k − k=0 Application of the identity (I.11) in [16] gives the explicit representation

( qe−ξ/t ; q) q−n, qeξ/t (5.17) u (x; t , t )=( t /q)n − 2 n φ 1 q, t eξ , n 1 2 2 2 1  ξ n 1  − (q; q)n t2e /q − −   which by (III.8) in [16] can be transformed to

( q2/(t t ); q) q−n, qeξ/t , qe−ξ/t (5.18) u (x; t , t )= − 1 2 n ( t /q)n φ 1 − 1 q, (t /t )qn . n 1 2 2 3 1  2 1 2  (q; q)n − q /(t1t2) −   Writing the 3φ1 as a ﬁnite sum and applying the formula

20 (a; q) =(q1−k/a; q) ( a)kqk(k−1)/2, k k − we see that (5.18) can be transformed to

n+1 n −n ξ k −ξ k n ( t1t2/q ; q)n n(n+1)/2 (q , t1e /q , t1e /q ; q)k nk (5.19) un(x; t1, t2)=( 1/t1) − q − k+1 q . − (q; q)n (q, t1t2/q ; q)k k=0 −

By symmetry of t1, t2 a similar formula holds for t1 and t2 interchanged.

Theorem 5.2 For µ and t , t C such that t t = qn+1, n 1 the Al-Salam-Chihara ∈ Vq 1 2 ∈ 1 2 − ≥ polynomials u (x; t , t ) are orthogonal with respect to the complex measure ν (t , t ) given by { n 1 2 } µ 1 2 (5.7).

Proof. It follows by (5.13) thatu ˆ (x;0, 0)=2−nh (x q) so the assertion is clear for t = t = 0. n n | 1 2 Assume now that t = 0. By the three term recurrence relation it suﬃces to prove that 1

(5.20) un(x; t1, t2) dνµ(x; t1, t2)=0 for n 1. ≥ By (5.19) and (5.7) we get

un(x; t1, t2) dνµ(x; t1, t2)

n+1 n −n k n ( t1t2/q ; q)n n(n+1)/2 nk (q ; q)k χt1/q (x)χt2 (x) =( 1/t1) − q q k+1 dµ(x). − (q; q)n (q; q)k ( t1t2/q ; q)∞ k=0 − By (5.5) the integral is 1, and the sum is equal to φ (q−n; ; q, qn), which is equal to 0 for 1 0 − n 1 by (II.4) in [16]. 2 ≥ In particular, if t2 = t1 then νµ(t1, t1) is a positive measure and the Al-Salam-Chihara moment problem corresponding to u (x; t , t ) is indeterminate. The set { n 1 1 }

ν (t , t ) µ { µ 1 1 | ∈Vq} is a compact convex subset of the full set (t ) of solutions to the u (x; t , t ) -moment problem. C 1 { n 1 1 } If t = t R then 1 2 ∈

ν (t , t ) µ = (t ) { µ 1 1 | ∈Vq} C 1

21 since the measures on the left can have no mass at the zeros of χt1 (x). If t = t (q, 1) and t = 0 then the Al-Salam-Chihara moment problem is determinate and 1 ∈ 2 the set ν (t, 0) µ contains exactly one positive measure namely the one coming from { µ | ∈ Vq} µ being the N-extremal solution corresponding to the choice a = q/t in (6.27) and (6.30) ∈ Vq of [18], i.e. µ is the discrete measure with mass m at x for n Z, where n n ∈ 1 t qn+1 xn = 2 qn+1 − t and

(q/t)4nqn(2n−1)(1 + q2n+2/t2) m = . n ( q2/t2, t2/q,q; q) − − ∞

The function χt(x) vanishes for x = xn when n< 0 and we get

∞

νµ(t, 0) = cnεxn , n=0 where

q3n(n+1)/2(1 + q2n+2/t2)( q2/t2; q) c = − n . n t2n(q; q) ( q2/t2; q) n − ∞ We now go back to (5.5) and integrate 1/( t eξ, t e−ξ; q) against the integrand in (5.5). − 1 1 k Here again the attachment method works and we see that

q−n, t t qn−2, 0 (5.21) ϕ (sinh ξ; t , t ) := φ − 1 2 q, q . n 1 2 3 2  ξ −ξ  t1e , t1e −   satisﬁes the biorthogonality relation

∞ (5.22) ϕm(x; t1, t2)ϕn(x; t2, t1)χt1 (x)χt2 (x) dµ(x) −∞

n−2 1+ t1t2q n−1 n(n−3)/2 n = 2n−2 ( t1t2q ; q)∞(q; q)nq (t1t2) δm,n. 1+ t1t2q −

The biorthogonal rational functions (5.21) are the special case t3 = t4 = 0 of the biorthogonal rational functions

−n n−2 q , t1t2q , t1t3/q, t1t4/q (5.23) ϕn(sinh ξ; t1, t2, t3, t4) := 4φ3 − − − q, q .  t eξ, t e−ξ, t t t t q−3  − 1 1 1 2 3 4   22 of Ismail and Masson [18]. We have not been able to apply a generating function technique to (5.21) because we have not been able to ﬁnd a suitable generating function for the rational functions (5.21). We now return to the Al-Salam-Chihara polynomials u (x; t , t ) in the positive deﬁnite { n 1 2 } case and reconsider the generating function (5.15). The radius of convergence of (1.4) is ρ = 3/2 q /√t1t2, and we get by Proposition 1.1, (5.14) and the q-binomial theorem

∞ ( t1t3/q, t1t4/q, t2t3/q, t2t4/q, t3t4/q; q)∞ (5.24) χt3 (x)χt4 (x)dα(x; t1, t2)= − − − −3− − , −∞ (t1t2t3t4q ; q)∞ valid for t , t < ρ. | 3| | 4| Applying this to the measures α(t1, t2)= νµ(t1, t2), we get a new proof of (5.10), now under slightly diﬀerent assumptions on t , , t . 1 4 The attachment procedure works in this case, and we prove the biorthogonality relation of [18] under the same assumptions as in Proposition 5.1:

∞ 4 (5.25) ϕm(x; t1, t2, t3, t4) ϕn(x; t2, t1, t3, t4) χti (x) dµ(x) −∞ i=1

n−2 −3 n 2 n−1 1+ t1t2q (t1t2t3t4q ) (q, q /t3t4; q)n( t1t2q ; q)∞ 1≤j

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

References

[1] N. I. Akhiezer, The Classical Moment Problem and Some Related Questions in Analysis, English translation, Oliver and Boyd, Edinburgh, 1965.

[2] W. A. Al-Salam and L. Carlitz, Some orthogonal q-polynomials, Math. Nachr. 30 (1965), 47-61.

[3] W. A. Al-Salam and T. S. Chihara, Convolutions of orthogonal polynomials, SIAM J. Math. Anal. 7 (1976), 16-28.

23 [4] W. A. Al-Salam and M. E. H. Ismail, A q-beta integral on the unit circle and some biorthogonal rational functions, Proc. Amer. Math. Soc. 121 (1994), 553-561.

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Mathematics Institute, Copenhagen University, Universitetsparken 5, DK-2100 Copenhagen Ø, Denmark. Department of Mathematics, University of South Florida, Tampa, Florida 33620-5700. e-mail [email protected] and [email protected]