Applications of quadratic and cubic hypergeometric transformations

Abstract

The purpose of this paper is to consider five classes of quadratic and cubic hypergeometric transformations in the spirit of Bailey and Whipple. We shall successfully evaluate several hypergeometric functions, of the types 2F1(x), 3F2(x), and 4F3(x), with each function having one or more free parameters, and with the argument $x$ chosen to equal such unusual values as x=±1,−8,14,−18, (these four values having been explored initially by Gessel and Stanton). In each case, companion identities and/or inverse transformations are given, which are sometimes proved by a limiting process for a divergent hypergeometric series. Some of the proofs use the Clausen quadratic formula, Euler reflection formula, Legendre duplication, Gauss multiplication formula, Euler transformation, hypergeometric reversion formula and known hypergeometric summation formulas. The proofs in the terminating case are simpler and can lead to mixed summation formulas, which depend on values of a negative integer. Some of the formulas use the Digamma function and a dimension formula is referred to.