In this paper, we give a transformation formula for Appell's hypergeometric function $F_1$. As applications of this formula, we show that some common limits of triple sequences given by mean iterations of 3-terms can be expressed by $F_1$.
References
J. M. Borwein and P. B. Borwein, Pi and the AGM, Canad. Math. Soc. Ser. Monogr. Adv. Texts 4, A Wiley-Interscience Publication, Jhon Wiley & Sons, Inc., New York, 1998. MR1641658J. M. Borwein and P. B. Borwein, Pi and the AGM, Canad. Math. Soc. Ser. Monogr. Adv. Texts 4, A Wiley-Interscience Publication, Jhon Wiley & Sons, Inc., New York, 1998. MR1641658
B. C. Carlson, Algorithms involving arithmetic and geometric means, Amer. Math. Monthly 78 (1971), 496--505. MR283246 10.2307/2317754B. C. Carlson, Algorithms involving arithmetic and geometric means, Amer. Math. Monthly 78 (1971), 496--505. MR283246 10.2307/2317754
E. M. Goursat, Sur l'équation différentielle linéaire qui admet pour intégrale la série hypergéométrique, Ann. Sci. École Norm. Sup. (2) 10 (1881), 3--142. MR1508709E. M. Goursat, Sur l'équation différentielle linéaire qui admet pour intégrale la série hypergéométrique, Ann. Sci. École Norm. Sup. (2) 10 (1881), 3--142. MR1508709
R. Hattori, T. Kato and K. Matsumoto, Mean iterations derived from transformation formulas for the hypergeometric function, Hokkaido Math. J. 38 (2009), 563--586. MR2548236 05606281 euclid.hokmj/1258553977
R. Hattori, T. Kato and K. Matsumoto, Mean iterations derived from transformation formulas for the hypergeometric function, Hokkaido Math. J. 38 (2009), 563--586. MR2548236 05606281 euclid.hokmj/1258553977
K. Iwasaki, H. Kimura, S. Shimomura and M. Yoshida, From Gauss to Painlevé, Aspects Math. E16, Friedr, Vieweg & Sohn, Braunschweig, Wiesbaden, 1991. MR1118604K. Iwasaki, H. Kimura, S. Shimomura and M. Yoshida, From Gauss to Painlevé, Aspects Math. E16, Friedr, Vieweg & Sohn, Braunschweig, Wiesbaden, 1991. MR1118604
T. Kato and K. Matsumoto, A quadruple sequence and the hypergeometric function $F_D$ of three variables, Nagoya Math. J. 195 (2009), 113--124. MR2552956 1176.33018 euclid.nmj/1252934374
T. Kato and K. Matsumoto, A quadruple sequence and the hypergeometric function $F_D$ of three variables, Nagoya Math. J. 195 (2009), 113--124. MR2552956 1176.33018 euclid.nmj/1252934374
K. Koike and H. Shiga, Isogeny formulas for the Picard modular form and a three terms arithmetic geometric mean, J. Number Theory 124 (2007), 123--141. MR2320994 1128.11026 10.1016/j.jnt.2006.08.002K. Koike and H. Shiga, Isogeny formulas for the Picard modular form and a three terms arithmetic geometric mean, J. Number Theory 124 (2007), 123--141. MR2320994 1128.11026 10.1016/j.jnt.2006.08.002
K. Koike and H. Shiga, Extended Gauss AGM and corresponding Picard modular forms, J. Number Theory 128 (2008), 2097--2126. MR2423753 05379238 10.1016/j.jnt.2007.12.001K. Koike and H. Shiga, Extended Gauss AGM and corresponding Picard modular forms, J. Number Theory 128 (2008), 2097--2126. MR2423753 05379238 10.1016/j.jnt.2007.12.001
K. Matsumoto and K. Ohara, Some transformation formulas for Lauricella's hypergeometric functions $F_D$, Funkcial. Ekvac. 52 (2009), 203--212. MR2547102 05609719 10.1619/fesi.52.203K. Matsumoto and K. Ohara, Some transformation formulas for Lauricella's hypergeometric functions $F_D$, Funkcial. Ekvac. 52 (2009), 203--212. MR2547102 05609719 10.1619/fesi.52.203