The Q-Laguerre Polynomials and Related Moment Problems

journal of mathematical analysis and applications 218, 155–174 (1998) article no. AY975771

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Mourad E. H. Ismail∗

Department of Mathematics, Imperial College, 180 Queen’s Gate, SW7 2BZ, London, England; Department of Mathematics, University of South Florida, Tampa, Florida 33620-5700

and

Mizan Rahman†

Department of Mathematics, Carleton University, Ottowa, Ontario, Canada K1S 5B6

Submitted by Bruce C. Berndt

Received January 14, 1997

Dedicated to A. W. Goodman on his 80th Birthday

We study two indeterminate Hamburger moment problems associated with q- Laguerre polynomials. The coefﬁcients in their recurrence relations are of exponential growth. This completes earlier work started by D. Moak. Some of the measures that the polynomials are orthogonal with respect to are found and lead to the evaluation of certain integrals. A symmetric set of orthogonal polynomials is also considered. © 1998 Academic Press Key Words: Hamburger moment problems; weight functions; Nevanlinna parameterization; q-Laguerre polynomials.

*Research partially supported by NSF Grant DMS 9625459 and a visiting fellowship from the Leverhulme Foundation. E-mail: [email protected]. †Research partially supported by NSERC Grant A6197. E-mail: mrahman@math. carleton.ca. 155

1. INTRODUCTION

α The q-Laguerre polynomials Ln x q satisfy the recurrence relation y 2n α 1 α x 1 q q + + Ln x q − − y 1 α α 1 n 1 n α q + Ln 1 x q q + qLn 1 x q = − + y + − − y n 1 n α 1 α 1 q q + q + + Ln x q (1.1) − + − − y and the initial conditions 1 α 1 α α α 1 q + L0 x q 1;L1xq q + x − : (1.2) y = y=− + 1 q − We shall always assume that

0 1: − It is known that the Hamburger moment problem associated with the q- Laguerre polynomials is indeterminate; that is, the measure with respect to α which Ln x q are orthogonal is not unique. Consider they three-term recurrence relation

(1.3) ωn 1 z z αn ωn z βnωn 1 z ; + = − − − with 0, 0, and real, 0. The difference equation (1.3) βn > n> αn n has two linearly independent solutions≥ and satisfying the Qn z Qn∗ z initial conditions

Q0 z 1;Q1zz α0;Q0∗z0;Q1∗z1: (1.4) x= x= − x= x= Both and are monic polynomials. Qn z Qn∗ z When the moment problem associated with (1.3) is indeterminate, the polynomials A z , B z , C z , and D z : n n n n 0 0 1 (1.5) An 1 z Qn∗ 1 z Qn∗ Qn∗ 1 Qn∗ z β1β2 βn − ; + x= + − + ··· 0 0 1 (1.6) Bn 1 z Qn 1 z Qn∗ Qn∗ 1 Qn z β1β2 βn − ; + x= + − + ··· 0 0 1 (1.7) Cn 1 z Qn∗ 1 z Qn Qn 1 Qn∗ z β1β2 βn − ; + x= + − + ··· 0 0 1 (1.8) Dn 1 z Qn 1 z Qn Qn 1 Qn z β1β2 βn − + x= + − + ··· converge uniformly on compact subsets of the complex plane to entire functions A z , B z , C z , and D z [17]. Furthermore, the measures with respect to which the ’s are orthogonal are parameterized by functions Qn σ z , analytic in the open upper and lower half planes, satisfy σ z σ z , = q-moment problems 157 and map the open upper (lower) half plane into z 0, z 0, respec- tively. The orthogonality measures ψ ;σ are related= ≤ to A=, B≥, C, D, and σ through · dψ t; σ A z σ z C z ∞ − ; z 0: (1.9) z t = B z σ z D z = 6= Z−∞ − − The cases σ are also included and in either case the right-hand side of (1.9) becomes=±∞ C z /D z . The zeros of A z , B z , C z , and D z are real and simple and the zeros of B z and D z interlace [17]. The orthogonality relation of the ’s is Qn n ∞ Q x Q x dψ x; σ β δ : (1.10) m n = k m; n −∞ k 1 Z Y= Among the measures ψ ;σ the measures corresponding to σ being a real constant, ﬁnite or inﬁnite,· are called extremal measures in [17] and N-extremal in [1]. They are purely discrete measures and are precisely the measures for which are complete in .Ifan - Qn x L2 ; ;dψ ;σ N extremal measure is supported at : −∞ ∞ , then· the mass at xn σ <σ x x σ , say m x , is given by −∞ ≤∞ n = n n n 1 ∞ 2 − m x Q x /β1β2 β ; m x 1: n n= m n ··· n n n= m 0

σ z a ib; z>0;σzσ z ;b>0;aR; = − = = ∈(1.12) gives rise to an absolutely continuous measure ψ with dψ x; σ b/π : (1.13) dx = B x aD x 2 b2D2 x − + This was also used by Ismail and Masson [14] and Berg and Valent [4]. We shall follow the notation for q-shifted factorials of Gasper and Rahman [10],

n k 1 a q 0 1; a q 1 aq − ; y x= y n x= − k 1 (1.14) m Y= 12 or a1; ;am q n aj q n;n;;; : ··· y x= j 1 y = ··· ∞ Y= The q-shifted factorials satisfy m n (1.15) a q m n a q m aq q n; a q n a q /aq q : y + = y y y = y ∞ y ∞ 158 ismail and rahman

The second formula in (1.15) enables us to extend the deﬁnition of the q-shifted factorials to the case n<0 by deﬁning it to be a q / aqn q . Moak [15] studied the moment problem associated with y the∞q-Laguerrey ∞ α polynomials Ln x q . He computed the functions B z and D z and showed that they arey related to q-Bessel functions. To be consistent with Moak’s notation, we use his

α 1 kk α k α q + q ∞ q + xq 1 L x q y ∞ 1 − ∞ α y x= q q k 0 q + ;q q k y ∞ X= y α/2 2 x1 q − J 2 x 1 q : (1.16) = − α − 2 q Here Jν x q is Jackson’s q-Bessel function; see [11, 10]. In this worky we ﬁnd the functions A z and C z and give a new derivation of B z and D z . This is done by constructing complete asymptotic expansions for the q-Laguerre polynomials and their numerator polynomials. We also provide a family of new orthogonality measures for the q- Laguerre polynomials. Finally, we study the Hamburger moment problem associated with symmetrized q-Laguerre polynomials. Since the zeros of the q-Laguerre polynomials lie in 0; , the Stieltjes moment problem is a more natural problem to study. On the∞ other hand, the symmetrized polynomials have their zeros in ; , so the Hamburger moment problem is the appropriate moment problem.−∞ ∞ Chihara [9] considered the relationship between the Hamburger moment problems associated with a set of polynomials with zeros in 0; and the corresponding symmetric set of orthogonal polynomials. We shall ∞ follow the notation and formulas in [6] where we start with birth-and-death-process polynomials F x satisfying n (1.17) xFn x µn 1Fn 1 x λn 1Fn 1 x λn µn Fn x − = + + + − − − + and

F0 x 1;F1x λ0 µ0 x/µ1: (1.18) x= x= + − The corresponding family of monic symmetric polynomials is gener- Fn x ated by 1 (1.19) F0 ; F1 x x; Fn 1 x xFn x βnFn 1 x ; x= x= + = − − where the β’s are given by (1.20) β2n 1 λn;β2n µn: + x= x= Here it assumed that 0 0 and 0 0 (1.21) µ0 ;λn > µn 1 > ;n : = + ≥ q-moment problems 159

It turns out that is a multiple of 2 . In Section 4 we will point out F2n x Fn x that the q-Laguerre polynomials are birth-and-death-process polynomials, up to a constant multiple, and we will study the moment problem associated with the corresponding . As by-products we evaluate several new Fn x integrals. Throughout this work we use the notation , ≈ a b if and only if a b 1 o 1 (1.22) n ≈ n n = n + as n . →∞

2. ASYMPTOTICS

The explicit form of the q-Laguerre polynomials is α 1 n n 1 k kk 1/2 kn α k α q + q n q− q k q q + q + x L x q : (2.1) n y y −α1 y = q q n k 0 q + ;q q k y X= y As Moak [15] pointed out we can let n inside the sum in (2.1) and obtain, as n , →∞ →∞ α 1 k k kk α α q + q ∞ q 1 x q + L x q ∞ : (2.2) n y − α 1 y → q q k 0 q; q + q k y ∞ X= y n α It is clear from (2.1) that the coefﬁcient of x in Ln x q is y n n n α q 1 q + / q q : − y n Hence,

n nn α α Q x q q q 1 − q− + Ln x q (2.3) n = y n − y and 1 q qn 1 qn 1 α α + − + − + + ;n0;1;:::; (2.4) n = 1 q q2n α 1 = − + + 1 qn 1 qn α β − − + ;n1;2;:::: (2.5) n = 1 q 2q4n 2α 1 = − + − Thus, as n , →∞ 1 n nn α α (2.6) Qn x q q q − q− + L x q : ≈ y ∞ − ∞ y On the other hand,

α 1 n nn α Q 0 q + q q 1 − q− + : (2.7) n = y n − The calculation of the functions A, B, C, and D requires knowing two α terms in the large n asymptotic expansion of Ln x q , so we ﬁrst give a α y complete asymptotic expansion of Ln x q . y 160 ismail and rahman

Theorem 2.1. For α a negative integer and as n , we have 6= →∞ m 1 j jn j j 1 /2 α q q + α j m 1 n Ln x q − L − x q O q + : (2.8) ∞ y =j 0 q q j y + X= y Proof. From (2.1), (1.15), and [10, (I.12)],

n q q n 1 k nk k k 1 /2 (2.9) q− q k y q− + − ; y = q q n k − y − and the q-binomial theorem, we get

α 1 n k k k α n k 1 α q + q xq 1 q + q − + q L x q ∞ ∞ n y −α 1 α n 1y y = q q k 0 q + ;q q k q + + q y ∞ = y y ∞ α 1 X j α n 1 q + q ∞ q + + y ∞ = q q j 0 q q j y ∞ X= y n x q 1 kqk k α + q k α q : −α 1 − − j ×k 0 q + ;q q k y X= y We now apply [10, (I.13)]

α α 1 k α q− q j q + q k jk q− − q y y q− (2.10) y j = qα 1 j q + − y k to simplify the above equation to

α 1 j α n 1 α α q + q ∞ q + + q− q j Ln x q y ∞ y y = q q j 0 q q j y ∞ X= y n xq 1 kqk k α j + − : (2.11) −α j 1 ×k 0 q − + ;q q k X= y Replace the -sum by and note that the second sum is k k 0 k n 1 2 ≥ − ≥ + O qn . Finally, we use (1.15) and [10, (I.10)] to write P P α 1 j j 1 /2 α j q + q 1 q − q− ∞ : α j y1 α j 1 α− q − + q = q − + q j = q− q j y ∞ y y These observations will reduce (2.11) to (2.8).

Corollary 2.2. For large n we have

n 1 α α q + α 1 2n Ln x q L x q L − x q O q : (2.12) y = ∞ y −1 q ∞ y + − This corollary readily follows from (2.8). q-moment problems 161

We now analyze the second solution to (1.1). Set

y x 1 q (2.13) = − and assume is a solution of the recursion (1.1) with replaced by pn y x y/ 1 q . Further assume the initial conditions − p0 y 0;p1yc/ 1 q : (2.14) = = − Set

∞ n (2.15) P y; t pn y t : x=n 0 X= Through the generating function technique the recursion (1.1) is trans- formed to the following functional equation for P y; t : 2 α 1 α 2 2 1 t 1 q qt P y; t 1 t q q + q + t P y; qt − + + − − + + α 1 2 ytq + P y; q t ct: (2.16) + = To solve for P y; t , we proceed as in the Ismail–Libis paper [13] by setting

P y; t f y; t / t q : (2.17) = y ∞ The result is that (2.16) becomes

α 1 α 1 2 2 f y; t f y; qt 1 q + t ytq + f y; q t ct q t q : (2.18) − − + = y ∞ In (2.18) we expand f y; t as ∞ n (2.19) f y; t fn y t ; =n 0 X= and after some simpliﬁcation we establish the recurrence relation

1 n 1 1 n n α 1 n 1 − c n 1 n 2 /2 (2.20) q fn y q + yq − fn 1 y − q − + : − =− + − + q q n 1 y − After setting

1 n q q g y − y n f y ; n = y q qαn n n 1 /2 n − y n + + we transform (2.20) to

c αn 1 gn y gn 1 y q− − ; − − =− y q − y n 162 ismail and rahman which, by telescopy and the initial conditions g0 y f0 y p0 y 0, gives = = =

n 1 αk qy q n 1 1 n 1 α n 1 n 1 n 2 /2 − q− (2.21) fn y − y − c − q − + − + : = q q n − k 0 qy q k y X= − y It is clear from (2.21) that by replacing k by n k 1 we have − − c 1 n 1qn n 1 /2 1 f y − − + as n : (2.22) n ≈ q q q 1 qα →∞ y ∞ − Therefore, f y; t is an entire function of t. Now apply (2.21) and the initial condition f0 y 0 to see that = ∞ n f y; t t fn 1 y t : = n 0 + = Hence, X tn qy q 1 nqα n k n n 3 /2 f y; t ct − y n− − + + : (2.23) = >n k 0 q q n 1 qy q k ∞ X≥ ≥ y + − y We shall show in the Appendix that

m 1 m α 1 ∞ tq + q y f y; t cq ∞ − − yα − = m 0 q− q m 1 = y + X n ∞ y cq α 1 tqα 1; y q − − + −α − − y ∞ n 0 q− q n 1 = y + m X ∞ y : (2.24) −α 1 × m 0 q; tq + q m = y In view of (2.17) we haveX established the generating relation

n ∞ y P y; t cq α 1 − − −α = n 0 t; q− q n 1 = y + X α 1 n α 1 tq + ; y q ∞ y cq ∞ − − − y −α − t q n 0 q− q n 1 y ∞ = y + m X ∞ y : (2.25) −α 1 × m 0 q; tq + q n 1 = y + We will need two termsX in the large asymptotics of . To do this, n pn y observe from (2.23) that f y; t /t is an entire function of tand that 1 f y; 1 qf y; q 1 P y; t − − 1 q tq2 q 1 t − 1 qt − y ∞ − − q-moment problems 163

2 is an analytic function of t when t

m 1 j j j 1 /2 1 − 1 q − + ; (2.27) 1 j t q m = j 0 q q j q q m 1 j tq y X= y y − − − which is a consequence of the q-Chu–Vandermonde sum [10, (II.7)]. There- fore,

1 m jn 1 j j j 1 /2 q q + j m 1 n p y − f y; q− O q + : (2.28) n = q q q q + j 0 j y ∞ X= y Clearly, (2.24) implies

α 1 cq− − α 1 f 0;t qt q tq + q : (2.29) = 1 q α y ∞ − y ∞ − −

3. THE NEVANLINNA PARAMETERIZATION

In this section we compute the A and C functions and give a new derivation of the B and D functions already computed by Moak [15]. From (2.3) and (2.12) it follows that

1 n nn α Qn x q q q − q− + ≈ y ∞ − n 1 n 1 q + α q + α 1 1 L x q L − x q × +1 q ∞ y −1 q ∞ y − − n nn α q q q 1 − q− + ≈ y ∞ − n 1 α q + α α 1 L x q L x q L − x q : (3.1) × ∞ y +1 q ∞ y − ∞ y − To identify , we appeal to the renormalization (2.13) and observe that Qn∗

n n n α q 1 − q− + q q p x 1 q − y n n − must be a constant multiple of Q x . We now choose n∗ α 1 c q 1q + (3.2) = − 164 ismail and rahman and get

n nn α Q∗ x q 1 − q− + q q p x1 q : (3.3) n = − y n n − The next step is to identify D z . From (2.5) we see that n α 1 1 2n n 2n 2α 1 (3.4) βk q + ;q q n q − q− + + : k 1 = y − Y= Hence, by (1.8), (2.3), and (2.7),

1 n α 1 D z lim q 1 − q− − − n = →∞ − n 1 α n α 1 α 1 q + L 1 z q 1 q + + Ln z q : × − n y − − y + Using (2.12), we get

α q− α α 1 D z 1 qα L z q L − z q ; (3.5) = 1 q − ∞ y − ∞ y − which is simply

α 1 D z zL + z q ; (3.6) = ∞ y and is in agreement with the lemma on page 43 of Moak’s work [15]. Moak’s derivation of (3.5) is completely different. 2 2 ν From the large z behavior of Jν z q in [7] it follows that z− Jν z q is an entire function of order zero; hence, y D z is of order zero. A theorem y of Berg and Pedersen [3] asserts that all the functions A z , B z , C z , and D z have the same order, type, and Phragm´en–Lindel of¨ indicator; hence, they all have order zero. When 0, simpliﬁes considerably. First note that (2.29) and (3.2) x Qn∗ give =

0 ∞ 0 n f ;t fn t =n 0 = X1 q α α 1 − q qt q q + t q : (3.7) = 1 qα y ∞ − y ∞ − So (2.17) gives

1 1 α 1 ∞ n q α q + t q P 0;t p 0t − q y ∞ ; (3.8) n 1 α 1 =n 0 = q t− t q = − − y ∞ which yields theX explicit form

1 q qα qα 1 q p 0 − 1 + y n : (3.9) n = 1 qα − q q − y n q-moment problems 165

Hence,

1 1 n q − nn α α α 1 Q∗ 0 − q− + + q + q q q : (3.10) n = 1 qα y n − y n − n o Formula (3.10) was also obtained by Moak [15] through a different derivation; see the formulation on the bottom of page 43 in [15]. Euler’s formula for the q-exponential when applied to (3.10) gives

1 1 n q − nn α α Q∗ 0 − q− + + n = 1 qα − α 1 q + q q q × y ∞ − y ∞ n 1 q + α α 1 2n q q q q + q O q : (3.11) −1 q y ∞ − y ∞ + − We now proceed to identify A, B, and C. From (1.6), (2.12), (2.3), and (3.11) one can easily establish

1 q q α 1 α B z y ∞ L − z q L z q : (3.12) = 1 qα − qα q ∞ y − ∞ y − y ∞ This is also in Moak [15] using a different approach. The three-term recur- 2 rence relation of the q-Bessel functions Jν z q , cast in terms of the L functions, is y ∞

α 1 α α 1 L − z q 1 qα L z q 1 q qα zL + z q : (3.13) ∞ y = − ∞ y − − ∞ y This leads to the following alternate form of (3.12):

q q α B z ∞ L z q αy1 =− q + q ∞ y y ∞ α 1 q q q q α 1 − 1 y ∞ zL + z q : (3.14) 1 α α 1 − q − q + q ∞ y − y ∞ By (1.5), (3.3), and (3.9) we have

n 1 q− − q q n 1 An 1 x 1 y + + = qα q qα 1 + y n − α 1 α 1 1 q + q n 1 q + q n 1 pn 1 y y pn y y + ; × + − q q n − − q q n 1 y y + (3.15) 166 ismail and rahman and, by (1.7), (2.7), and (3.3), we ﬁnd

n α 1 1 1 1 n 1 1 α n 1 Cn 1 x q− − − q − pn 1 y q + pn y q + + : + = − + − − − h (3.16)i

Using (2.29) and (2.26), letting n , and simplifying, we ﬁnally obtain →∞ 1 α α 1 1 A y q q f y; 1 q q q + q f y; q− =− qα;q q y ∞ + y ∞ − y ∞ y ∞ n n n 1 /2 1 q ∞ y q − + = qα; y q q q 1 qn α n 0 n − − y ∞ X= y − α α α 1 α 1 q q L x q q q q + q L − x q × y ∞ ∞ y + y ∞ − y ∞ ∞ y α q ;q α ∞ L− x q α − q− q ∞ y y ∞ α 1 q q 1 α q q q q ∞ L − x q (3.17) + yα − y ∞ − y ∞ q− q ∞ y y ∞ and α q− α 1 C x 1 q f y; 1 f y; q− = 1 q q q − − − y ∞ yf y; q 1 = + 1 q q q − y ∞ n n n 1 /2 y α 1 ∞ y q + L + x q ∞ 1 n α = y q y n 0 q q n q − − y ∞ X= y − q q α 1 ∞ L− − x q : (3.18) y α + y; q− q ∞ y − y ∞ From (2.24) and [10, Exercise 1.24] we ﬁnd that

2α 2 f y; 1 1 q q− − q q = − y ∞ n ∞ y y α/2J 2 2 y q − α −α × y q q 1 n 0 − n p X= y + q q α/2 2 ∞ y J 2 y q ; (3.19) y α α − y; q− q − y − y ∞ and f y; q can be computed from (3.15) and (3.16)p by using (2.18). The choice q 1 qα q q a − 1 y ∞ (3.20) 1 α α 1 = q − q + q − y ∞ q-moment problems 167 in the weight function (1.13) gives the orthogonality relation

α α Lm x q Ln x q dx ∞ y y 2 1 2 α α 1 2 2 α Z−∞ q q L x q / q + q b x L + x q y ∞ ∞ y y ∞ + ∞ y h α 1 i h i π q + q δ y n m; n : (3.21) = qn b q q y n The weight functions in (3.21) form a new one-parameter family of weight functions for the q-Laguerre polynomials. Since the moments of any nor- malized solution to the q-Laguerre moment problem are

α 1 q + q n αn n n 1 /2 µ y q− − + (3.22) n = 1 q n − [15, (2.13)], we have

n ∞ x dx 1 α α 1 2 2 2 α 2 −∞ q q L x q / q + q b x L + x q Z y ∞ ∞ y y ∞ + ∞ y α 1 π q + q n αn n n 1 /2 y q− − + (3.23) = b 1 q n − for b>0. The above integral is very curious and we do not see how to establish it directly using the existing theory of special functions.

4. THE SYMMETRIC POLYNOMIALS

Following the notation in [6], the normalizations in (1.1), (1.17), and (1.18) are

1 qn α 1 1qn λ − + + ;µ − : (4.1) n = 1 q q2n α 1 n=1qq2nα − + + −+ Hence,

n α 1 λk 1 n q + q n (4.2) πn − q y x= k 1 µk = q q n Y= y and

n α F x q Ln x q : (4.3) n = y Furthermore, F x is the solution to (1.17) with n∗ F0∗ x 0; and F1∗ x 1/µ1: (4.4) x= x=− 168 ismail and rahman

Theorem 4.2 in [6], which is due to Chihara [9], asserts that for the ’s, Fn the polynomials , , , and are given by A2n 1 z B2n 1 z C2n 1 z D2n 1 z + + + + 2 Fn∗ z (4.5) A2n 1 z z ; + =− πn 2 Fn z (4.6) B2n 1 z ; + =− πn 2 F z2 Fn∗ z n∗ 1 (4.7) C2n 1 z λnπn + ; + = πn − πn 1 + 2 2 λnπn Fn z Fn 1 z (4.8) D2n 1 z + : + = z πn − πn 1 + Here is Fn∗ n F ∗ z q p z 1 q ; (4.9) n = n − since both sides satisfy (1.17) and the initial conditions (4.4). Theorem 4.1. The entire functions A z , B z , C z , and D z are given by

q q α 2 α 1 2 B z ∞ L z q ;DzzL + z q ; αy1 (4.10) =− q + q ∞ y = ∞ y y ∞ 2 α 1 A z zf 1 q z ; 1 / q + q =− − y ∞ 2α 2 z q q q− − 1 q y ∞ − α 1 1 2 =− q + ; q z q − − y ∞ n 2n n n 1 /2 α 2 ∞ 1 q z q + q q α 2 L z q − y ∞ L− z q ; × ∞ y q q 1 qn α − q α q ∞ y n 0 n − − X= y − y ∞ (4.11) 1 qα f z2 1 q ;1 f z2 1 q ;q 1 C z − − − − − = qα 1 q q q − y ∞ 2 2α 2 n 2n n n 1 /2 z q q q− − 1 q α 1 2 ∞ 1 q z q + y ∞ 2 − L + z q − 1 ∞ 1 n α =− q z q y n 0 q q n q − − − y ∞ = y − 2α 2 X q− − q q a 1 α 2 α α 2 y ∞ q L − z q 1 q L z q : α 1 2 − − − q− ; q z q ∞ y + − ∞ y − − y ∞ (4.12) Proof. From (4.3) and (4.6) we get

q q n α 2 q q α 2 B2 1 z ;L z q ∞ L z q n αy1 n αy1 + =− q + q n y →− q + q ∞ y y y ∞ q-moment problems 169 as n , and hence, the evaluation of B z . Similarly, →∞ n α 1 q− − − 1 α 2 1 α 2 1 n α 1 n D2n 1 z q + + Ln z q q + Ln 1 z q ; + =z 1 q − y − − + y − which, following the steps leading to (3.5) and (3.6), will establish the second equality in (4.10). From (4.9) and (4.1) we conclude F z q q n∗ y n p z 1 q π = qα 1 q n − n + y n 1 α q q n 1 q 2n y ∞ 1 q − p z 1 q O q : (4.13) α 1 + 1 n = q + q − q − + y ∞ − The evaluation of A z follows by letting n in (4.5) after applying (2.26), (4.2), and (4.9). Similarly, we apply (4.13)→∞ and establish the expression for C z given (4.12) and the proof is complete. The referee has pointed out that Theorem 4.1 also follows from [9, (2.25)] since 0 1 α Qn∗ q q q q s0 lim − 1 y ∞ : 0 1 α α 1 = n Qn = q − q + q →∞ − y ∞ It follows from the asymptotic results in [7] and the work [3] that the entire functions in Theorem 4.1 all have order zero. To distinguish the speciﬁc under consideration, we shall denote it by Fn F z q . Thus, these polynomials satisfy (1.19), that is, n (4.14) Fn 1 z q zFn zq βnFn 1 zq ; + = − − with 1 qn β2 − ; n 1 2n α = q q + (4.15) −1 n α q + 0 β2n 1 − 2 1 ;n>; − = 1 q q n α − + − and F0 z q 1 and F1 z q z. The orthogonality relation is = = n ∞ F x q F x q dψ x σ δ β : (4.16) m n y = m;n k −∞ k 1 Z Y= We now consider the special measure (1.13) with a 0. Thus, = ∞ Fm x q Fn x q dx 1 α 2 α 1 2 2 2 α 2 2 −∞ q q L x q / q + q b x L + x q Z y ∞ ∞ y y ∞ + ∞ y π n β δ : (4.17) = b k m; n k 1 Y= 170 ismail and rahman

The weight function in (4.17) is a reminiscent of the integrand in the Gross- wald integral representation for 1/2 ; see [12]. z− Kν √z /Kν 1 √z The special case 12of is interesting+ and gives a - α / Fn xq q analogue of the Hermite=− polynomials. In this case,

1 qn/2 β − ;n>0; (4.18) n = 1 q qn 1/2 − − and (4.17) becomes

∞ Fm x q Fn x q dx 1 2 1 2 / 2 1/2 2 2 2 / 2 2 −∞ q q L− x q / q q b x L x q Z y ∞ ∞ y y ∞ + ∞ y 1/2 1/2 π q q n n2/2 y q− δ : (4.19) = b 1 q n m; n − We next simplify the denominator in the above equation for the special case q q b 1 q y ∞ : = − q1/2 q y ∞ Using p

1/2 1/2 1/2 3/2 1/2 q ;q q q q 2 and q ;q q q q 2 ; y k = y k y k = y k we get

1/2 2 1/2 2 L− x q i 1 qxL x q ∞ y + − ∞ y 1/2 n n 1 /4 n/2 n q q p∞ q − 1 q ix ∞ y 1/2 −1/2 = q q n 0 q q n y ∞ X= y 1/2 q q 1/2 y ∞ ix 1 q q : = q q − − y ∞ y ∞ p Therefore, the denominator in (4.19) is

ix 1 q; ix 1 q q1/2 x2 1 q q : − − − y ∞ =− − y ∞ Thus, (4.19) becomesp p

1/2 1/2 1/2 ∞ Fm x q Fn x q dx π q q n q q n2/2 y y ∞ q δ : (4.20) 2 1 1 n 1/2 − m; n x q q = q + q q Z−∞ − − y ∞ − y ∞ In (4.20) if we take m and n to be even, replace ∞ by 2 0∞, and then replace x by √x, we will get the continuous orthogonality−∞ relation for the R R q-Laguerre polynomials in [15]. q-moment problems 171

It is interesting to observe that the limit q 1 of (4.20) is the orthogonality relation of the q-Hermite polynomials.→ This is so since [10, Sect. 1.10]

1 x q q 1 q − y ∞ 0 x 0x as q 1− (4.21) − qx q = q → → y ∞ implies

lim 1 q q q / q1/2 q 0 1/2 √π: q 1 − y ∞ y ∞ = = → − p Furthermore,

lim 1/2 1 n 2 n and lim 2 n q q n/ q − Fn x q − Hn x; q 1 y − = q 1 = → → (4.22) where we used (4.14) and (4.18). Furthermore, in the special case α 1/2 the ’s have the generating function =− Fn 2 ∞ n n 1 /2 n Fn x q xt q (4.23) q − t 2 −1 y ∞ : n 0 q q n = t / q q X= y − y ∞ This indicates that the F ’s in this case are discrete q-Hermite polynomials and indeed a special case of the ’s of Al-Salam and Carlitz; see [8]. Vn Although Berg and Valent [5] studied the Al-Salam–Carlitz moment problem, the weight function in (4.20) in not included in their work but is in [15], after a change of variable.

APPENDIX

We ﬁrst rewrite (2.23) in the form

α n n n 3 /2 n αk ∞ tq q qy q q f y; t ct − + − y n − : (5.1) = n 0 q q n 1 k 0 qy q k X= y + X= − y Since n zk

k 0 a q k X= y n b q zk lim y k = b 0 a q → k 0 k = y X n 1 q; b b q n 1 n 1 q; bq + lim + 2φ1 q; z y z + 2 φ1 n 1 q; z = b 0 a − a q 1 aq → n + y +

172 ismail and rahman

b; qz q z; a/b n 1 z; a/b n 1 lim y ∞ 2φ1 q; b z + 2 φ1 q; bq + = b 0 a; z q qz − zq → y ∞ 1 k k k 1 /2 n 1 k k k 1 /2 ∞ a q − n 1 ∞ aq + q − − z + − ; (5.2) = a q q q 1 zqk − q q 1 zqk k 0 k k 0 k y ∞ X= y − X= y − we have

α n n n 3 /2 ∞ tq q f y; t ct + − n 1 = n 0 q q n 1 yq + q = y + − y ∞ X k k k 1 /2 k k n 1 k k 1 /2 ∞ y q ∞ y q + q α n 1 + + + : 1 k α − + 1 k α × k 0 q q k q − − k 0 q q k q − = y − = y − X X (5.3)

We now evaluate the n sums in (5.3). To sum the ﬁrst series, note that

α n n n 3 /2 ∞ tq q + − n 1 n 0 q q n 1 yq + q X= y + − y ∞ α n n n 3 /2 n 1 j ∞ tq q ∞ yq − + − + = n 0 q q n 1 j 0 q q j X= y + X= y j α n 1 j n 1 n n 3 /2 α 1 ∞ y ∞ tq + q + + + tq − − − =− j 0 q q j n 0 q q n 1 X= y X= y + j α n jn n 1 n 2 /2 α 1 ∞ y ∞ tq q + − + 1 tq − − − q− =− q q q q − j 0 j n 0 n X= y X= y j α 1 1 ∞ y α j 1 tq − q− − tq + + q 1 =− q q y ∞ − j 0 j X= y α 1 j α 1 1 tq + q ∞ y tq + − ∞ : (5.4) αy 1 −α 1 = tq + j 0 q; tq + q j + y q − X= y − y ∞ To evaluate the second n sum in (5.3), we observe that

n k n 1 n n 3 /2 ∞ t q q α + + + − − n 1 n 0 q q n 1 yq + q X= y + − y ∞ j α 1 1 qt q ∞ y tq + − ∞ : (5.5) yα 1 − =− tq + j 0 q q j qt q j k + y q X= y y + − y ∞ q-moment problems 173

Substituting (5.4) and (5.5) in (5.3), we get 1 j j k j k 1 k k 1 /2 α 1 ∞ y + tq + + q q + f y; t cq − y ∞ − − 1 k α = j; k 0 q q j q q k q − X= y y − 1 j j k α j 1 k k 1 /2 α 1 ∞ y + tq + + q q + cq − y ∞ : (5.6) − − 1 k α − j; k 0 q q j q q k q − X= y y − Setting j k n and replacing j by n k in the ﬁrst double series on the right-hand+ side= of (5.6), we ﬁnd that −

j j k j k 1 k k 1 /2 ∞ 1 y + tq + + q q + − y ∞ 1 k α j; k 0 q q j q q k q − X= y y − n n 1 n α ∞ y tq + q q− ;q− n 1 − y ∞ 2φ1 q; q + = q q 1 q α q1 α n 0 n − − = y − X n n 1 ∞ y tq + q q q n − y ∞ y 1 α 1 α = n 0 q q n q− q − q n = y − y X n ∞ y t q : (5.7) −α =y ∞n 0 t; q− q n 1 = y + Therefore, we have establishedX the following expression for f : n ∞ y f y; t cq α 1 t q − − −α = y ∞ n 0 t; q− q n 1 = y + X j α 1 α 1 ∞ y cq− − tq + q − 1 − y ∞ q; tqα q j 0 + j X= y k k k 1 /2 ∞ y q + : (5.8) 1 k α × k 0 q q k q − = y − Finally, we obtain X k k k 1 /2 ∞ y q + 1 k α k 0 q q k q − = y − X 1 α 1 α lim 2φ1 q− ;b q − q; yq/b = 1 q α b y y − − − →∞ 1 y q 1 α 1 α lim − y ∞ 2φ1 q; q /b q q; y 1 α − − = q− b yq/b q y y − − →∞ − y ∞ n ∞ y y q (5.9) −α =− y ∞n 0 q− q n 1 X= y + and f y; q can be computed from (3.15) and (3.16) by using (2.18). 174 ismail and rahman

ACKNOWLEDGMENTS

Part of this work was done while the ﬁrst-named author (M.I) was visiting Imperial College, London. He gratefully acknowledges the hospitality of Imperial College and support from both Imperial College and the Uni- versity of South Florida. Last but not least, we thank a very conscientious referee, whose eyes never seemed to miss an error or a misprint.

REFERENCES

1. N. I. Akhiezer, “The Classical Moment Problem and Some Related Questions in Analy- sis,” Oliver & Boyd, Edinburgh, 1965. 2. C. Berg and J. P. R. Christensen, Density questions in the classical theory of moments, Ann. Inst. Fourier (Grenoble) 31 (1981), 99–114. 3. C. Berg and H. L. Pedersen, On the order and type of the entire functions associated with an indeterminate Hamburger moment problem, Ark. Mat. 32 (1994), 1–11. 4. C. Berg and G. Valent, The Nevanlinna parameterization for some indeterminate Stieltjes moment problems associated with birth and death processes, Methods Appl. Analy. 1 (1994), 169–209. 5. C. Berg and G. Valent, Nevanlinna extremal measures for some orthogonal polynomials related to birth and death processes, J. Comput. Appl. Math. 57 (1995), 29–43. 6. Y. Chen and M. E. H. Ismail, Some indeterminate moment problems and freud-like weights, Constr. Approx. 14 (1998). 7. Y. Chen, M. E. H. Ismail, and K. A. Muttalib, Asymptotics of basic Bessel functions and q-Laguerre polynomials, J. Comput. Appl. Math. 54 (1995), 263–273. 8. T. S. Chihara, “An Introduction to Orthogonal Polynomials,” Gordon & Breach, New York, 1978. 9. T. S. Chihara, Indeterminate symmetric moment problems, J. Math. Anal. Appl. 85 (1982), 331–346. 10. G. Gasper and M. Rahman, “Basic Hypergeometric Series,” Cambridge Univ. Press, Cambridge, 1990. 11. M. E. H. Ismail, The zeros of basic Bessel functions, the functions and associated Jν ax orthogonal polynomials, J. Math. Anal. Appl. 86 (1982), 1–19. + 12. M. E. H. Ismail and D. H. Kelker, Special functions, Stieltjes transforms and inﬁnite divisibility, SIAM J. Math. Anal. 10 (1979), 884–901. 13. M. E. H. Ismail and C. Libis, Contiguous relations, basic hypergeometric functions and orthogonal polynomials I, J. Math. Anal. Appl. 141 (1989), 349–372. 14. M. E. H. Ismail and D. R. Masson, q-Hermite polynomials, biorthogonal rational functions and q-beta integrals, Trans. Amer. Math. Soc. 346 (1994), 61–116. 15. D. Moak, The q-analogue of the Laguerre polynomials, J. Math. Anal. Appl. 81 (1981), 20–47. 16. F. W. J. Olver, “Asymptotics and Special Functions,” Academic Press, New York, 1974. 17. J. Shohat and J. D. Tamarkin, “The Problem of Moments,” rev. ed., Amer. Math. Soc., Providence, 1950.