MR3012577 Akihito EBISU Akihito EBISU Funkcialaj Ekvacioj. Serio Internacia 55 2012 255--283 http://fe.math.kobe-u.ac.jp/FE/FullPapers/55-2/55_255.pdf http://www.ams.org/mathscinet-getitem?mr=MR3012577 Three hypergeometric series $F(a,b,c;x)$ with the same parameters $(a,b,c)$ up to additive integers are linearly related over rational functions in $x$. This paper makes this linear relation explicit: the coefficients are given from sums of products of hypergeometric series. The hypergeometric series, Contiguity relation, Three term relation. 33C05. 55-255 2012 Akihito EBISU Akihito EBISU 1 Ebisu, A., Three term relations for the hypergeometric series (in Japanese), Master thesis at Kyushu University, February 2011 2 Erdélyi, A.; Magnus, W.; Oberhettinger, F.; Tricomi, F. G. Higher transcendental functions, vol. 1 McGraw-Hill Book Company, Inc., New York-Toronto-London 1953 MR0058756 3 Iwasaki, K.; Kimura, H.; Shimomura, S.; Yoshida, M. From Gauss to Painlevé -A modern theory of special functions Aspects of Mathematics, E16, Friedr. Vieweg & Sohn, Braunschweig 1991 MR1118604 4 Poole, E. G. C. Introduction to the theory of linear differential equations Oxford Univ. Press, London 1936 MR0111886