Extended linear and bilinear generating relations for a class of generalized hypergeometric functions
The authors have given some linear and bilinear generating functions for the generalized hypergeometric function \(I_{n;(b_ q)}^{\alpha;(a_ p)} (x,w)\), defined by \[ I_{n; (b_ q)}^{\alpha; (a_ p)} (x,w)= {{(1+\alpha_ n)} \over {n!}} F_{q;1;0}^{p;2;1} \left[ \begin{matrix} (a_ p) : -n, {x\over w}; {x\over w};\\ (b_ q) : 1+ \alpha; -;w,w \end{matrix} \right]. \] To get their results they use manipulation techniques. One of their results is as follows: \[ \begin{multlined} \sum_{n=0}^ \infty \left( \begin{matrix} n+m\\ n\end{matrix} \right) {{(\lambda)_ n} \over {(1+ \alpha+m)_ n}} I_{n+m; (b_ q)}^{\alpha; (a_ p)} (x,w) t^ n=\\ (1-t)^{- \lambda} \left( \begin{matrix} \alpha+m\\ m\end{matrix} \right) F^{(3)} \left[ \begin{matrix} (a_ p) :: {x\over w}; -;-; \lambda;-m; - {x\over w};\\ (b_ q) :: 1+\alpha; -;-:-;-;-; {{wt} \over {t-1}}, w,w\end{matrix} \right].\end{multlined} \] For more generalized results of this type one is advised to see the paper of \textit{H. M. Srivastava} [Glas. Math., III. Ser. 4(24), 67-73 (1969; Zbl 0176.018)].
scientific article; zbMATH DE number 597307