Products of two restricted hypergeometric functions
Using decomposition of hypergeometric series into even an odd parts, the authors obtain a Cayley-Orr-type theorem: If \[ {}_p F_q [(2\alpha_p);(2\beta_q);z]_r F_s [(2a_r); (2b_s); \lambda z]= \sum^\infty_{n=0} c_n z^n, \] where \(\lambda\) equals 1 or \(-1\), then \dots (two formulas follow that are too long to reproduce here). Moreover, using special cases of this theorem together with suitable classical product theorems, they establish a number of identities involving products or squares of hypergeometric functions. These formulas are rather long, so only one of the particular cases is quoted: \[ {}_0 F_3 [ ;\mu+ {\textstyle {1\over 2}}, \mu+1, {\textstyle {1\over 2}}; z] {}_0 F_3 [ ;\mu+ 1, \mu+ {\textstyle {{3\over 2},{3\over 2}}}; z]= {}_2 F_5 \Biggl[\begin{aligned} \mu+{\textstyle {3\over 4}}, &\mu+{\textstyle {5\over 4}}\\ \mu+1, & \mu+ {\textstyle {3\over 2}},\;2\mu+1,\;2\mu+ {\textstyle {3\over 2}, {3\over 2}} \end{aligned} \Biggl|16z \Biggr] \] (a misprint in the paper is corrected). Finally, they establish the sum of a special terminating \({}_8 F_7 [1]\) with two free parameters.
scientific article; zbMATH DE number 906457