Some Transformation Formulas for Lauricella's Hypergeometric Functions $F

MR2547102 Keiji MATSUMOTO and Katsuyoshi OHARA Keiji MATSUMOTO and Katsuyoshi OHARA Some Transformation Formulas for Lauricella's Hypergeometric Functions $F_D$ Funkcialaj Ekvacioj. Serio Internacia 52 2009 203--212 http://fe.math.kobe-u.ac.jp/FE/FullPapers/52-2/52_203.pdf http://www.ams.org/mathscinet-getitem?mr=MR2547102 In this paper, we give some functional equations with a parameter $c$ for Lauricella's hypergeometric functions; they can be regarded as multivariable versions of the Gauss quadratic transformation formula for the hypergeometric function. These functional equations for $c=1$ are utilized for the study of arithmetic-geometric means of several terms. Hypergeometric functions, Transformation formulas. 33C65. 52-203 2009 Some Transformation Formulas for Lauricella's Hypergeometric Functions $F_D$ Keiji MATSUMOTO and Katsuyoshi OHARA Keiji MATSUMOTO and Katsuyoshi OHARA

1 Borwein, J. M.; Borwein, P. B. A cubic counterpart of Jacobi's identity and the AGM Trans. Amer. Math. Soc. 323 1991 691-701 MR1010408

2 Berndt, B. C.; Bhargava, S.; Garvan, F. G. Ramanujan's theories of elliptic functions to alternative bases Trans. Amer. Math. Soc. 347 1995 4163-4244 MR1311903

3 Erdélyi, A. Higher Transcendental Functions Robert E. Krieger Publishing Co., Inc., Melbourne, Florida 1981 MR0698779

4 Goursat, E. M. Sur l'Équation Différentielle Linéaire qui Admet pour Intégrale la Série Hypergéométrique Ann. Sci. École Norm. Sup. (2) 10 1881 3-142 MR1508709

5 Hattori, R.; Kato, T.; Matsumoto, K., Mean iterations derived from transformation formulas for the hypergeometric function, to appear in Hokkaido Math. J. 6 Iwasaki, K.; Kimura, H.; Shimomura, S.; Yoshida, M. From Gauss to Painlevé Vieweg, Braunschweig, Wiesbaden 1991 MR1118604 7 Kato, T.; Matsumoto, K., The common limit of a quadruple sequence and the hypergeometric function $F_D$ of three variables, to appear in Nagoya Math. J.

8 Koike, K.; Shiga, H. Isogeny formulas for the Picard modular form and a three terms arithmetic geometric mean J. Number Theory 124 2007 123-141 MR2320994

9 Koike, K.; Shiga, H. Extended Gauss AGM and corresponding Picard modular forms J. Number Theory 128 2008 2097-2126 MR2423753

10 Ohara, K., yang--a package for computation in the ring of differential-difference operators, http://www.openxm.org, 2007 11 Ohara, K., Calculations of some transformation for hypergeometric functions, http://www.math.kobe-u.ac.jp/OpenXM/Math/AGM, 2008