Another method for a new two-term relation for the hypergeometric function \(_{3}F_{2}\) due to Exton
In this note the authors give a very short proof of the following two-term relation for the hypergeometric function \(_{3}F_{2}\) of unit argument: \[ _{3}F_{2}\Bigg[\begin{matrix} a,\,b,\,c\\ d,\,e\\ \end{matrix} ; 1 \Bigg]=\frac{\Gamma(e)\Gamma(d+e-a-b-c)} {\Gamma(e-c)\Gamma(d+e-a-b)}\,_{3}F_{2}\Bigg[\begin{matrix} d-a,\,d-b,\,c\\ d,\,d+e-a-b\\ \end{matrix} ; 1 \Bigg]\,. \] This formula had been earlier obtained by \textit{H. Exton} [J. Comput. Appl. Math. 106, No.2, 395-397 (1999; Zbl 0930.33002)], by a different method.
scientific article; zbMATH DE number 2085553