A generalized circular summation of theta function and its application

In this paper we prove a generalized Ramanujanʹs 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.