Ramanujan's identities and monotonicity properties of quotients of theta functions

Abstract

In this work, we prove monotonicity results associated with certain quotients of theta functions. The monotonicity result for the second theta function has its application in the extremal problems on harmonic measure and steady-state heat distributions in special configurations. Along with these results, we include the generalization of theta functions in signature 3 and illustrate a beautiful identity of Ramanujan associated with the inversion of an analogue of an elliptic integral in signature 3.