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 \[ \begin{align} {f_n}({a_i};{b_j};x) &\equiv {f_n}({a_1},{a_2}, \cdots ,{a_p};{b_1},{b_2}, \cdots ,{b_q};x) \\ &\equiv {}_{p + 2}{F_{q + 2}}\left[ \begin{array}{ccc} -n, & n+1, & a_1,a_2,\dots,a_p; \\ 1/2, & 1, & b_1,b_2,\dots,b_p; \end{array} \; x\right] \end{align} \]

(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) \(f_n(1/2,\zeta;\rho;\nu) = H_n(\zeta,\rho,\nu)\).
(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.