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Generalized hypergeometric functions at unit argument. (English) Zbl 0754.33003
The behaviour of the Gaussian hypergeometric series (for $p=1$) $$\sb{p+1}F\sb p\left(\left.{{a\sb 1,a\sb 2,\dots,a\sb{p+1}}\atop {b\sb 1,\dots,b\sb p}}\right\vert z\right)=\sum\sb{n=0}\sp \infty {{(a\sb 1)\sb n(a\sb 2)\sb n\dots(a\sb{p+1})\sb n} \over {(b\sb 1)\sb n\dots(b\sb p)\sb n n!}}z\sp n,\qquad \vert z\vert<1,$$ near the unit argument is given by a known analytic continuation formula when $s=\sum\sb{j=1}\sp p b\sb j- \sum\sb{j=1}\sp{p+1}a\sb j$ is not an integer. {\it R. J. Evans} and {\it D. Stanton} [SIAM J. Math. Anal. 15, 1010-1020 (1984; Zbl 0547.33001)] obtained continuation formulas near $z=1$ for $\sb{p+1}F\sb p$, $p=2$, when $s=0$. The author of this paper obtains a continuation formula near $z=1$ for $p=2$ and unrestricted $s$ and also for $p=3$ or 4 when $s$ is not an integer. {\it M. Saigo} and {\it H. M. Srivastava} [Proc. Am. Math. Soc. 110, No. 1, 71-76 (1990; Zbl 0706.33004)] obtained a continuation formula near $z=1$ for arbitrary $p$ when $s=0$. The author obtains here a continuation formula near $z=1$ for the series $\sb{p+1}F\sb p$ for arbitrary integral $p$ and unrestricted $s$.

33C20Generalized hypergeometric series, ${}_pF_q$
33C05Classical hypergeometric functions, ${}_2F_1$
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