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.
Miller, A. R., & Paris, R. B. (2011). Euler-type transformations for the generalized hypergeometric function r+2Fr+1(x). Zeitschrift fur Angewandte Mathematik und Physik, 62(1), 31–45. https://doi.org/10.1007/s00033-010-0085-0