The Jacobian elliptic function $\mathrm{sn}(u,k)$ is the inverse of the elliptic integral of the first kind and $\mathrm{cn}(u,k)=\sqrt{1-{\mathrm{sn}}^2(u,k)}$. In this paper, we study coefficient polynomials in the Taylor series expansions of $\mathrm{sn}(u,k)$ and $\mathrm{cn}(u,k)$. 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 $\mathrm{sn}(u,k)$ are $\gamma$-positive. We then show that the coefficient polynomials of $\mathrm{cn}(u,k)$ are bi-$\gamma$-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.

雅可比椭圆函数 $\mathrm{锡}(你,克)$ 是第一类椭圆积分的逆 $\mathrm{cn}(你,克)=\sqrt{1-{\mathrm{锡}}^2(你,克)}$. 在本文中，我们研究了泰勒级数展开式中的系数多项式$\mathrm{锡}(你,克)$ 和 $\mathrm{cn}(你,克)$. 我们首先为一系列双变量峰值多项式提供组合扩展，这些多项式通过奇数和偶数周期峰值来计算排列。这种组合展开的一个特例表明，系数多项式为$\mathrm{锡}(你,克)$ 是 $\gamma$-积极的。然后我们证明系数多项式为$\mathrm{cn}(你,克)$ 是双$\gamma$-positive，这意味着这些系数多项式是单峰的，中间有模式。此外，通过使用上下文无关文法，我们根据递增树找到了两个相关系数的组合解释。