On the Jacobi Elliptic functions and Applications

TL;DR

This paper explores developments of Jacobi elliptic functions, including trigonometric expansions of theta functions, differential systems derived from the heat equation, and various applications and expansions for elliptic and Zeta functions.

Contribution

It introduces a new trigonometric expansion of theta functions and derives a differential system from the heat equation related to Jacobi elliptic functions.

Findings

Established a differential system satisfied by theta function coefficients.

Derived new expansions for Jacobi elliptic functions.

Explored applications and alternative expansions for Zeta functions.

Abstract

In this paper we are interested in developments of elliptic functions of Jacobi. In particular a trigonometric expansion of the classical theta functions introduced by the author (Algebraic methods and q-special functions, Editors: C.R.M. Proceedings and Lectures Notes, A.M.S., vol 22, Providence, 1999, 53-57) permits one establish a differential system. This system is derived from the heat equation and is satisfied by their coefficients. Several applications may be deduced. Other types of expansions for the Jacobi elliptic functions as well as for the Zeta function are examined.