journal of mathematical analysis and applications 218, 155–174 (1998) article no. AY975771 The q-Laguerre Polynomials and Related Moment View metadata, citation and similar papers at core.ac.uk brought to you by CORE Problems provided by Elsevier - Publisher Connector 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 expo- nential growth. This completes earlier work started by D. Moak. Some of the mea- sures 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 param- eterization; 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 0022-247X/98 $25.00 Copyright © 1998 by Academic Press All rights of reproduction in any form reserved. 156 ismail and rahman 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 <q<1;α>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 func- tions 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 <n< X= −∞X ∞ (1.11) Here it is assumed that . xn σ <xn 1 σ Berg and Christensen [2] proved + that 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 deriva- tion of B z and D z . This is done by constructing complete asymptotic expansions for the q-Laguerre polynomials and their numerator polyno- mials. 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 poly- nomials 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 .