A Product of Theta-Functions Analogous to Ramanujan's Remarkable Product of Theta-Functions and Applications

Abstract EN

We define a product lk,n for any positive real numbers k and n involving Ramanujan's theta-functions ϕ(q) and ψ(q) which is analogous to Ramanujan's remarkable product of theta-functions recorded by Ramanujan (1957) and study its several properties.

We prove general theorems for the explicit evaluations of lk,n and find some explicit values.

As application of the product lk,n, we also offer explicit formulas for explicit values of Ramanujan's continued fraction V(q) in terms of lk,n and give examples.