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Generalized hypergeometric functions at unit argument. (English) Zbl 0754.33003

The behaviour of the Gaussian hypergeometric series (for p=1)

p+1 F p a 1 ,a 2 ,,a p+1 b 1 ,,b p z= n=0 (a 1 ) n (a 2 ) n (a p+1 ) n (b 1 ) n (b p ) n n!z n ,|z|<1,

near the unit argument is given by a known analytic continuation formula when s= j=1 p b j - j=1 p+1 a j is not an integer. R. J. Evans and D. Stanton [SIAM J. Math. Anal. 15, 1010-1020 (1984; Zbl 0547.33001)] obtained continuation formulas near z=1 for p+1 F 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. M. Saigo and 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 p+1 F p for arbitrary integral p and unrestricted s.


MSC:
33C20Generalized hypergeometric series, p F q
33C05Classical hypergeometric functions, 2 F 1