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 11 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.
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