Circular Summations of Theta Functions in Ramanujan's Lost Notebook

Abstract

Let f(a, b) denote Ramanujan's symmetric theta function. In his Lost Notebook, Ramanujan claimed that the “circular” summation of n-th powers of f satisfies a factorization of the form f(a, b)F(ab). He listed elegant identities for n = 2, 3, 4, 5 and 7. We present alternative proofs of his claims.