q -Classical Orthogonal Polynomials: A General Difference Calculus Approach
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q-Classical Orthogonal Polynomials: A General Difference Calculus Approach R.S. Costas-Santos · F. Marcellán
Received: 21 July 2007 / Accepted: 23 June 2009 / Published online: 4 July 2009 © Springer Science+Business Media B.V. 2009
Abstract It is well known that the classical families of orthogonal polynomials are characterized as the polynomial eigenfunctions of a second order homogeneous linear differential/difference hypergeometric operator with polynomial coefficients. In this paper we present a study of the classical orthogonal polynomials sequences, in short classical OPS, in a more general framework by using the differential (or difference) calculus and Operator Theory. The Hahn’s Theorem and a characterization theorem for the q-polynomials which belongs to the q-Askey and Hahn tableaux are proved. Finally, we illustrate our results applying them to some known families of orthogonal q-polynomials. Keywords Classical orthogonal polynomials · Discrete orthogonal polynomials · q-Polynomials · Characterization theorems · Rodrigues operator Mathematics Subject Classification (2000) 33C45 · 33D45 1 Introduction Classical orthogonal polynomials constitute a very important and interesting family of special functions. They are mathematical objects which have attracted attention not only because of their mathematical value but also because of their connections with physical problems. In fact, they are also related, among others, to continued fractions, Eulerian series, elliptic functions [7, 13], and quantum algebras [17, 18, 27]. They also satisfy a three-term recurrence relation (TTRR) [25] x(s)pn (x(s)) = αn pn+1 (x(s)) + βn pn (x(s)) + γn pn−1 (x(s)),
n ≥ 0,
R.S. Costas-Santos () Department of Mathematics, University of California, Santa Barbara, CA 93106, USA e-mail: [email protected] F. Marcellán Departamento de Matemáticas, Universidad Carlos III de Madrid, Avenida Universidad 30, 28911 Leganés, Spain e-mail: [email protected]
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R.S. Costas-Santos, F. Marcellán
where γn = 0, n ≥ 1, as well as their derivatives (or differences or q-differences) also constitute a sequence of orthogonal family (see e.g. [3, 4, 11, 25] for a more recent review). Indeed, a fundamental role is played by the so-called characterization theorems, i.e. the theorems which collect those properties that completely define and characterize the classical orthogonal polynomials. One of the many ways to characterize a family (pn ) of classical polynomials (Hermite, Laguerre, Jacobi, and Bessel), which was first posed by R. Askey and proved by W.A. AlSalam and T.S. Chihara [2] (see also [21]), is the structure relation φ(x)pn (x) = a˜ n pn+1 (x) + b˜n pn (x) + c˜n pn−1 (x),
n ≥ 0,
(1)
where φ is a fixed polynomial of degree at most 2 and c˜n = 0, n ≥ 1. A.G. Garcia, F. Marcellán, and L. Salto [14] proved that the relation (1) also characterizes the discrete classical orthogonal polynomials (Hahn, Krawtchouk, Meixner, and Charlier polynomials) when the derivative is replaced by the forward difference operator defined as f (x)
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