Series Expansions Of The Ramanujan Constant With Applications

It is well known that the Gaussian hypergeometric function F(a,b;c;x)is extremely important in the special functions.Some other important special functions are only special cases of F(a,b;c;x).Many properties of the hypergeometric function F(a,b;c;x)are related to the gamma function ?(x)?its logarithmic derivative ?(x)and the Beta function B(x,y).On the other hand,the Ramanujan constant R(a)plays an important role not only in the study of properties of the hypergeometric function F(a,b;c;x),but also in the study of some other special functions.For example,R(a)is indispensable in the study of the generalized elliptic integral ?a(r),?a(r),the solution of the Ramanujan modular equations ?K(a,r)and the generlized deviation functions ?(a,K)and ?K(a,t).However,the known properties of R(a)are still cannot meet the needs of the application,and one of the main obstacle to reveal the properties of R(a)is lack of effective research tool.A lot of researches show that the series expansion of R(a)is an important and effective tool.The main purpose of this thesis is to establish different series expansions of R(a),to reveal the close relationship between the Ramanujan constant and Beta function and to obtain some important properties of R(a)by studying the properties of certain combinations in terms of of R(a)and some elementary functions.This thesis is divided into three chapters:In chapter 1,we introduc some concepts,notions and some known results involved in this thesis.In chapter 2,we present different series expansions of the Ramanujan constant R(a),B(a)=B(a,1-a),and power series expansion of R(a)-B(a),and apply these results to obtain some identities for the Rieman zeta function.In chapter 3,we present several asymptotic sharp inequalities of R(a),by studying the the analysis properties of certain combinations defined in terms of Ramanujan constant R(a)and some elementary functions.