New transformation formulas for the fourth Lauricella function

Abstract

The purpose of this article is to find summation, transformation and reduction formulas for Lauricella functions of n variables and for their special cases Appell functions ($n=2$) and $_{2}F _{1}$ functions. The method of proof is always to start with the Eulerian integral for the Lauricella function or its special cases, and then make an integral substitution, which maps (0, 1) to itself, and finally rewrite the new integral in terms of another Lauricella function or its special cases. In this way, quite interesting formulas are obtained, and the number of variables in the new Lauricella function can change. Some of the formulas generalize formulas for Appell functions, and some formulas are of general type, which are then specialized to concrete formulas. The important object reduced roots for a polynomial $f(\tau )$ is introduced in order to factorize $f(\tau )$. When the arguments of the double functions are $>1$, the results are only formal. Often these formulas are specialized to new summation formulas for Gaussian hypergeometric functions. Several of the new formulas have been checked by Mathematica.