Nature Of The Gauss Hypergeometric Function And Related Special Functions

The Gaussian hypergeometric function F(a,b; c; z) is one of the most important classesof special functions. Many important special functions are only special cases of F(a,b; c; z).It is wellknown that many properties of the hypergeometric function F(a,b; c; z) are closelyassociated with the gamma functionΓ(x), its logarithmic derivativeΨ(x), and the betafunction B(x,y). The Ramanujan constant R(a) plays an important role in the study ofproperties of such functions as the hypergeometric function F(a,b; c; z), the generalizedelliptic integralsΚ_a(r),ε_a(r), and the solutionφK(a,r) to the generalized Ramanujan'smodular equations, etc. Therefore, it is significant to study properties of the Gaussianhypergeometric function F(a,b; c; z) and its related special functions.This thesis is divided into four chapters.In the first chapter, we introduce some concepts, notation and the developing historyof special functions and Ramanujan's modular equations.In the second chapter, we present some properties of the gamma functionΓ(x), itslogarithmic derivativeΨ(x), and the beta function B(x,y), and show some monotonicityand concavity properties of certain functions defined in terms of the Ramanujan constantR(a) and some elementary functions. Some new properties of R(a) are obtained, andsome known inequalities satisfied by R(a) are improved.In the third chapter, we obtain some inequalities and properties of the Gaussian hy-pergeometric function F(a,b; c; z), including its dependence on the parameters.In the fourth chapter, we study some analytic properties of the complete elliptic in-tegrals of the first kindΚ(r), the generalized Gr(?)tzsch ring functionμ_a(r), the solution ofthe generalized Ramanujan's modular equations ?K(a,r), and of the generalized distor-tion functionηΚ(a,t).