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Padé approximation to the logarithmic derivative of the Gauss hypergeometric function. (English) Zbl 1114.33002
Jia, Chaohua (ed.) et al., Analytic number theory. Proceedings of the 1st China-Japan seminar on number theory, Beijing, China, September 13–17, 1999 and the annual conference on analytic number theory, Kyoto, Japan, November 29–December 3, 1999. Dordrecht: Kluwer Academic Publishers (ISBN 1-4020-0545-8/hbk). Dev. Math. 6, 157-172 (2002).
The purpose of this paper is to give the explicit $$(n,n-1)$$-Padé approximation to the ratio $$H(a,b,c;z)=F(a+1,b+1,c+1;z)/F(a,b,c;z)$$ for any parameters $$a,b,c\in\mathbb C$$ satisfying $$c\not\in-\mathbb N_0$$, which is equal to the logarithmic derivative of Gauss hypergeometric function $$F(a,b,c;z)$$ multiplied by $$c/ab$$ ($$ab\neq 0$$). This result is shown by the simple combinatorial method used by W. Maier [J. Reine Angew. Math. 156, 93–148 (1927; JFM 53.0340.02)] and G. V. Chudnovsky [J. Math. Pures Appl. (9) 58, 445–476 (1979; Zbl 0434.10023)].
For the entire collection see [Zbl 0990.00042].

##### MSC:
 33C05 Classical hypergeometric functions, $${}_2F_1$$ 41A21 Padé approximation