The hypergeometric equation and Ramanujan functions. (English) Zbl 1072.11052

There are 4 sections in this paper. The author gives the history of the Ramanujan functions and related topics. Ramanujan functions satisfy a system of nonlinear differential equations. This paper deals with both nonlinear systems satisfied by modular functions and identities between these functions and hypergeometric functions for a special choice of parameters. The author gives some examples related to these functions and analogues of the Ramanujan functions and nonlinear differential equations for them. He investigates a modular structure of solutions for nonlinear differential systems. He finds identities between the Ramanujan functions and hypergeometric functions. He also gives a solution of transcendence problems concerning nonlinear systems, which is given by the table explicitly.


11J91 Transcendence theory of other special functions
11F03 Modular and automorphic functions
33C05 Classical hypergeometric functions, \({}_2F_1\)
37F05 Dynamical systems involving relations and correspondences in one complex variable
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