Extension of Some Classical Summation Theorems for the Generalized Hypergeometric Series with Integral Parameter Differences

We derive extensions of the classical summation theorems of Kummer and Watson for the generalized hypergeometric series where r pairs of numeratorial and denominatorial parameters differ by positive integers. The results are obtained with the help of a generalization of Kummer’s second summation theorem for the 2F1 series given recently by Rakha and Rathie [Integral Transforms and Special Functions, 22, 823–840 (2011)] together with generalizations of the Euler transformations for the r+2Fr+1(z) function. A few interesting special cases are also presented. Mathematics subject classification (2010): Primary 33C20, Secondary 33C15.