Extensions of certain classical integrals of Erdélyi for Gauss hypergeometric functions

摘要

It is shown how series manipulation technique and certain classical summation theorems for hypergeometric series can be used to prove Erdélyi's integral representations for 2F1(z), originally proved using fractional calculus. The method not only leads to generalizations but also leads to new integrals of Erdélyi type for certain q+1Fq(z) and corresponding Pochhammer contour integrals. The technique outlined here, compared to the method of fractional calculus, seems to be more effective as it not only provides transparent elementary proofs of Erdélyi's integrals but even leads to various generalizations.