Pollaczek polynomials and Padé approximants: some closed-form expressions

摘要

Recently, some work has been devoted to the task of obtaining closed-form expressions for important classes of orthogonal polynomials. This has been done successfully for the associated Jacobi polynomials and their special cases, including the associated Laguerre and Hermite polynomials. Such expressions allow one to write closed-form expressions for the [n − 1/n] Padé approximants to important transcendental functions, for instance, expressions for the associated Jacobi polynomials provide in closed form the truncates of Gauss' continued fraction.In this paper, we continue this work by developing closed-form expressions for the Pollaczek polynomials Pnλ(x; a, b,c). These expressions involve a finite number of terms, each algebraic in character. This is to be contr with the traditional representation, which is a ratio of cross products of Gaussian hypergeometric functions, an expression not algebraic in the variable and not identifiably a polynomial. We examine some special cases of this formula, including the associated ultraspherical polynomials and the Chihara-Ismail polynomials. Finally, we determine the [n − 1/n] Padé approximants to a certain ratio of Gaussian hypergeometric functions.