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.