Abstract
The identity F-2(1)(alpha, beta; 2 alpha; z) = (1 - z/2)F--beta(2)1(beta 2, beta+1 2; alpha+1 2;(z 2-z)(2)) given, either by I.S. Gradshteyn and I.M. Ryzhik in Table of integrals series and products named 9.134 or in the handbook "mathematical functions with formulas, graphs and mathematical tables" done by Abramowitz-Stegun named 15.3.20 or in the book "special functions" done by G. Andrews, R. Askey and R. Roy named 3.1.7 page 127 with a slight modification is true provided that {2 alpha+1, alpha+3/2} are not natural numbers and alpha-beta is not an integer (see Gradshteyn, Ryzhik, 9.130). In this manuscript we consider a case where alpha-beta is an integer by taking beta=2a, alpha=-n+1. We give and prove the right identity for any positive integer a and for any any positive integer n.