Jacobian elliptic functions and a family of bivariate peak polynomials

Abstract

The Jacobian elliptic function sn(u,k) is the inverse of the elliptic integral of the first kind and cn(u,k)=1−sn2(u,k). In this paper, we study coefficient polynomials in the Taylor series expansions of sn(u,k) and cn(u,k). We first provide a combinatorial expansion for a family of bivariate peak polynomials, which count permutations by their odd and even cycle peaks. A special case of this combinatorial expansion says that the coefficient polynomials of sn(u,k) are γ-positive. We then show that the coefficient polynomials of cn(u,k) are bi-γ-positive, which implies that these coefficient polynomials are unimodal with modes in the middle. Furthermore, by using context-free grammars, we find combinatorial interpretations of two associated coefficients in terms of increasing trees.