Miller, Allen R. A summation formula for Clausen’s series \(_3F_2(1)\) with an application to Goursat’s function \(_2F_2(x)\). (English) Zbl 1068.33009 J. Phys. A, Math. Gen. 38, No. 16, 3541-3545 (2005). The author proves the following summation formula for a terminating \(_{3}F_{2}\) generalized hypergeometric series of unit argument \[ _{3}F_{2}\biggl(\begin{matrix} -n,\,a,\,c+1\\ b,\,c \\ \end{matrix},1 \biggr)= \frac{(b-a-1)_{n}\,(\alpha+1)_{n}}{(b)_{n}\,(\alpha)_{n}}, \] where \(\alpha=\frac{c(1+a-b)}{a-c}\). Two different proofs of this formula are given. Then using this formula the author obtains a reduction formula for the Kampé de Fériet (double generalized hypergeometric) function \(F_{q:2;0}^{p:2;0}[-x,x]\), which generalizes some earlier results dealing with transformation formulas for a \(_{2}F_{2}\) generalized hypergeometric function. Reviewer: Stamatis Koumandos (Nicosia) Cited in 2 ReviewsCited in 17 Documents MSC: 33C20 Generalized hypergeometric series, \({}_pF_q\) 33C50 Orthogonal polynomials and functions in several variables expressible in terms of special functions in one variable 33C15 Confluent hypergeometric functions, Whittaker functions, \({}_1F_1\) Keywords:generalized hypergeometric series; summation formulae; transformation formulae; Kampé de Fériet function PDFBibTeX XMLCite \textit{A. R. Miller}, J. Phys. A, Math. Gen. 38, No. 16, 3541--3545 (2005; Zbl 1068.33009) Full Text: DOI