Kronecker theta function and a decomposition theorem for theta functions I

Abstract

The Kronecker theta function is a quotient of the Jacobi theta functions, which is also a special case of Ramanujan's 1 ψ 1 summation. Using the Kronecker theta function as building blocks, we prove a decomposition theorem for theta functions. This decomposition theorem is the common source of a large number of theta function identities. Many striking theta function identities, both classical and new, are derived from this decomposition theorem with ease. A new addition formula for theta functions is established. Several known results in the theory of elliptic theta functions due to Ramanujan, Weierstrass, Kiepert, Winquist and Shen among others are revisited. A curious trigonometric identities is proved.Unless otherwise stated, it is assumed throughout that q=exp(2πiτ ), where Imτ > 0.