PROCEEDINGS OF THE AMERICAN MATHEMATICAL SOCIETY Volume 52, October 1975 AN EQUIVALENCE THEOREMON GENERATING FUNCTIONS1 H. M. SRIVASTAVA ABSTRACT. The present paper establishes an equivalence theorem on generating relations for a sequence of functions \f (z)j defined by the Rodrigues formula (1) below. It is also shown how this theorem may be applied to a fairly large variety of special functions including, for in- stance, the classical orthogonal polynomials. 1. The main result. We prove the following generalization of certain re- sults on generating functions due to F. Brafman [l]: Theorem. Let the function F(z) be holomorphic on a domain D of the complex z-plane, and define (1) /„(*) = («!)" lDnz\(az + b)nF(z)\, Dz = d/dz, for n = 0, 1, 2, . . . , where a and b are complex constants, not both zero. Also let !A j be any sequence of complex numbers for which (2) l/P = hmsup|Aj1/" n — 9* is finite, or for which (3) R= lim |A A | n—•<» exists and is positive. Suppose further that (4) A*~ hVrk* n-0'1'2'-' for some positive integer N and some complex number w. Then Received by the editors May 8, 1974 and, in revised form, August 15, 1974. AMS (MOS) subject classifications (1970). Primary 33A65, 33A45; Secondary 42A52, 30A86. Key words and phrases. Equivalence theorem, generating relations, Rodrigues' formula, special functions, classical orthogonal polynomials, holomorphic functions, the Cauchy-Hadamard theorem, d'Alembert's ratio test, Cauchy's integral formula, uniform convergence, binomial theorem, analytic continuation, Hermite polynomials, Laguerre polynomials, Jacobi polynomials, Bessel polynomials, ultraspherical poly- nomials, Legendre polynomials, Tchebycheff polynomials, Charlier polynomials, gen- erating-function equivalence, Meixner polynomials. 1 This work was supported in part by the National Research Council of Canada under grant A-7353. Copyright © 1975, American Mathematical Society 159 License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use