Additive decompositions of θ-functions of multiple arguments

A generalized circular summation of theta function and its application

“…From the definition of θ 1 (see, for example, [5]), we find that Other applications of Theorem 2 can be found in [1] and [2].…”

“…Glasser, J. Zar and I.J. Zucker [2], but they did not realize its connection with Ramanujan's circular summation formulas. The rest of the paper is organized as follows.…”

On a new circular summation of theta functions

“…With a( y 1 , y 2 |τ ) defined as in Definition 1 and y 3 = −y 1 − y 2 , we have Taking y 1 = π 3 and y 2 = − π 3 in Proposition 1 and combining the second equation in (2.15), we have b(τ )θ 3 (3z|3τ ) = θ 3 (z|τ )θ 3 z + π 3 τ θ 3 z − π 3 τ .…”

“…In particular, with the theory of elliptic functions, the authors of [6] proved the following theta function identity, which is a dual form of the Ramanujan circular summation formula. (1.2) Motivated by [6] and [3], one former student of the second author, X.-F. Zeng [14], recently proved the following theorem, which unifies (1.1) and (1.2). Theorem 3.…”

“…Boon et al[3] proved the following additive decomposition of θ 3 .Theorem 2.For any positive integer n, we have the identity = nθ 3 (nz|nτ ).…”