Agnes Scott College

Sister Mary Celine Fasenmyer

Some Generalized Hypergeometric Polynomials
Bulletin of the American Mathematical Society, Vol. 53, No. 8 (Aug. 1947), 806-812

Introduction

We shall obtain some basic formal properties of the hypergeometric polynomials

equation 1

(n a non-negative integer) in an attempt to unify and to extend the study of certain sets of polynomials which have attracted considerable attention. Some special cases of the fn(ai; bj; x) are:

(a) fn(1/2; —; x) = Pn(1–2x) (Legendre)
(b) fn(1; —; x) = [n!/(1/2)n]Pn(-1/2,1/2)(1–2x) (Jacobi)
(c) fn(1, 1/2; b; x) = [n!/(b)n[Pn(b-1,1-b)(1–2x) (Jacobi)
(d) fn(1/2, ζ; ρ; ν) = Hn(ζ, ρ, ν)
(e) fn[1/2, (1+z)/2; 1; 1] = Fn(z)
(f) fn(1/2; 1; t) = Zn(t)
(g) fn[1/2, (z+m+1)/2; m+1; 1] = Fnm(z)

A dash indicates the absence of parameters.