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

(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 f_n(a_i; b_j; x) are:

(a) f_n(1/2; —; x) = P_n(1–2x) (Legendre)
(b) f_n(1; —; x) = [n!/(1/2)_n]P_n^(-1/2,1/2)(1–2x) (Jacobi)
(c) f_n(1, 1/2; b; x) = [n!/(b)_n[P_n^(b-1,1-b)(1–2x) (Jacobi)
(d) f_n(1/2, ζ; ρ; ν) = H_n(ζ, ρ, ν)
(e) f_n[1/2, (1+z)/2; 1; 1] = F_n(z)
(f) f_n(1/2; 1; t) = Z_n(t)
(g) f_n[1/2, (z+m+1)/2; m+1; 1] = F_n^m(z)

A dash indicates the absence of parameters.

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