Endowing novel modular equations to substantiate the inquisitions of ramanujan

Abstract

This doctoral dissertation investigates computational formulas of newlinevarious topics related to the work of Ramanujan. We constructed several newlinenew P and#1048576; Q modular equations of various degrees involving Ramanujan s newlinetheta-functions. And, we showed how these modular equations are connected newlineto derive explicit formulas of Ramanujan s product of theta-functions, newlineRamanujan s class invariant, singular moduli and continued fractions. newlineRamanujan defined a remarkable product of theta-functions am;n in newlinehis first notebook and offered a list of 18 particular values for both odd values newlineof m and n. We established several general explicit formulas for evaluating am;n. newlineThen, we classified those formulas into two types based on n, namely n is even newlineand n is odd. In addition, we calculated several explicit values of am;n. newlineThe definition of the class invariant ln was given by Ramanujan in newlinehis lost notebook and he has listed 21 explicit values of ln for odd values of newlinen. Those values were proved by Berndt and his collaborators. We established newlineseveral novel explicit formulas of ln for even and odd values of n and evaluated newlineseveral explicit values. Moreover, we established explicit formulas of and#956;n, which newlineis closely related to ln as defined by Ramanathan. Further, we computed several newlinevalues of and#956;n. newline