A generalized circular summation of theta function and its application

Abstract

In this paper we prove a generalized Ramanujan's circular summation of theta function using the theory of elliptic and theta functions. This identity also includes a result of M. Boon, M.L. Glasser, J. Zar, I.J. Zucker and an addition formula of theta functions. Besides, we can get some new trigonometric identities.

A theta function identity and its applications

“…When n = 4, taking y = π 8 , we also find sin 8 From Proposition 2.1, we can derive the following identity on circular summation of theta functions.…”

“…3, using propositions in Sects. 1 and 2, we derive some new and nontrivial identities on circular summations of theta functions (see [1,8] …”

An alternate circular summation formula of theta functions and its applications

“…The first proof of the entire Theorem 1.1 was given by H. H. Chan, Z.-G. Liu and S. T. Ng [4], i.e., (12) and (13). By applying the Jacobi imaginary transformation to (12) Inspired by [4] and [3], X.-F. Zeng [17] proved the following important formula unifying Theorem 1.2 and 1.3 . Theorem 1.5.…”

On a new circular summation of theta functions

“…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.…”

“…Different studies on the Ramanujan circular summation can be found in [1,[6][7][8]10,[12][13][14]. 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.…”