Certain bilateral generating relations for generalized hypergeometric functions
In this short note the authors have given bilateral generating relations (in the form of a single result) for the generalized hypergeometric functions \(I^{\alpha; (a_p)}_{n; (b_q)} (x,w)\), defined by \[ I^{\alpha; (a_p)}_{n; (b_q)} (x,w) = {1 \over n! (x - w)^{[\alpha_w]}} \Delta^n_{x,w} \left[ (x - w)^{\bigl[ (\alpha + n) w \bigr]} _{p + 1} F_q \left( (a_p), - {x \over w}; (b_q); w \right) \right] \] where, \[ \Delta_{x,w} f(x) = {f(x + w) - f(x) \over w}. \] To establish their result they used series manipulation technique. The result involves generalized Lauricella hypergeometric functions of four variables and is too lengthy to be reproduced here.
Lauricella hypergeometric function of four variables
scientific article; zbMATH DE number 903365