The purpose of this note is to provide an alternative proof of two transformation formulas contiguous to that of Kummers second transformation for the confluent hypergeometric function ${}_1F_1$ using a differential equation approach.
In this note we state (with minor corrections) and give an alternative proof of a very general hypergeometric transformation formula due to Slater. As an application, we obtain a new hypergeometric transformation formula for a ${}_5F_4(-1)$ series wi
th one pair of parameters differing by unity expressed as a linear combination of two ${}_3F_2(1)$ series.
The aim in this note is to provide a generalization of an interesting entry in Ramanujans Notebooks that relate sums involving the derivatives of a function Phi(t) evaluated at 0 and 1. The generalization obtained is derived with the help of expressi
ons for the sum of terminating 3F2 hypergeometric functions of argument equal to 2, recently obtained in Kim et al. [Two results for the terminating 3F2(2) with applications, Bull. Korean Math. Soc. 49 (2012) pp. 621{633]. Several special cases are given. In addition we generalize a summation formula to include integral parameter differences.
Expressions for the summation of a new series involving the Laguerre polynomials are obtained in terms of generalized hypergeometric functions. These results provide alternative, and in some cases simpler, expressions to those recently obtained in the literature.
