On the unimodality of the Taylor expansion coefficients of Jacobian elliptic functions

TL;DR

This paper investigates the shape properties of the Taylor expansion coefficients of Jacobian elliptic functions, showing they are symmetric, unimodal, and alternatingly increasing, using gamma-positivity theory.

Contribution

It establishes the unimodality and symmetry of these coefficients, providing new insights into their combinatorial and analytical structure.

Findings

Coefficients of sn(u,k) are symmetric and unimodal.

Coefficients of cn(u,k) are unimodal and alternatingly increasing.

Gamma-positivity is used to prove these properties.

Abstract

The Jacobian elliptic functions are standard forms of elliptic functions, and they were independently introduced by C.G.J. Jacobi and N.H. Abel. In this paper, we study the unimodality of Taylor expansion coefficients of the Jacobian elliptic functions sn(u,k) and cn(u,k). By using the theory of gamma-positivity, we obtain that the Taylor expansion coefficients of sn(u,k) are symmetric and unimodal, and that of cn(u,k) are unimodal and alternatingly increasing.