Telescopic generalizations for two \(_3F_2\)-series identities (Q2905567)

Two nonterminating \(_3F_2(1)\)-series identities are derived: NEWLINE\[NEWLINE _3F_2\left [ \begin{alignedat}{2} 2,\;& 1+\alpha , & 1+\beta \gamma \\ & 2+\alpha \gamma , & 2+\beta \end{alignedat} \Big | 1\right ] = \frac {(1+\alpha \gamma)(1+\beta)}{(\alpha -\beta)(\gamma -1)} NEWLINE\]NEWLINE for \(\Re \{(\alpha -\beta)(\gamma -1)\}>0\) and NEWLINE\[NEWLINE _3F_2\left [\begin{alignedat}{2} 1+\alpha , \alpha \gamma , 1+\alpha \gamma -\beta \gamma \\ 2+\alpha \gamma , 1+\alpha \gamma -\beta \gamma +\beta \end{alignedat} \Big | 1 \right ]=(1+\alpha \gamma)\frac {\Gamma (1-\alpha +\beta)\Gamma (1+\alpha \gamma -\beta \gamma +\beta)}{\Gamma (1+\beta)\Gamma (1+(\alpha -\beta)(\gamma -1))} NEWLINE\]NEWLINE for \(\Re (1-\alpha +\beta)>0\), where \(\Gamma \) is Euler's gamma function.NEWLINENEWLINENEWLINENEWLINE The first of them confirms a conjecture of M. Milgram and the second one extends a couple of identities due to \textit{I. Gessel} and \textit{D. Stanton} [SIAM J. Math. Anal. 13, No. 2, 295--308 (1982; Zbl 0486.33003)].