Euler-type transformations for the generalized hypergeometric function r+2Fr+1(x)

We provide generalizations of two of Euler’s classical transformation formulas for the Gauss hypergeometric function extended to the case of the generalized hypergeometric function r+2 F r+1(x) when there are additional numeratorial and denominatorial parameters differing by unity. The method employed to deduce the latter is also implemented to obtain a Kummer-type transformation formula for r+1 F r+1 (x) that was recently derived in a different way.