Abstract
In this paper, we give two Ramanujan-type circular summation formulas by applying the way of elliptic functions and the properties of theta functions. As applications, we obtain the corresponding imaginary transformation formulas for Ramanujan-type circular summations and some theta function identities.
Highlights
1 Introduction, preparation, and motivation The classical Jacobi four theta functions θi(z|τ ), i = 1, 2, 3, 4, with the notation of Tannery and Molk, are defined as follows
Liu, and Ng [10] showed that Theorem 1.8 is an equivalent of the theorem below by applying the Jacobi imaginary transformation formulas [25, p. 475]
Motivated by [10, 11], and [15, 17, 28], by applying the theory of elliptic functions, we further investigate other two Ramanujan-type circular summations for theta functions θ1(z|τ ) and θ2(z|τ ), which are two variations of Ramanujan’s circular summations
Summary
Theorem 1.8 (Ramanujan’s circular summation) For each positive integer n, n–1 Theorem 1.9 (Ramanujan’s circular summation) For any positive integer n, there exists a quantity Gn(τ ) such that n–1 Liu and Luo [15] obtained the alternating circular summation formulas of theta function θ3(z|τ ).
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