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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.

MSC:
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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