Ramanujan defined the polynomials $\psi_{k}(r,x)$ in his study of power series inversion. Berndt, Evans and Wilson obtained a recurrence relation for $\psi_{k}(r,x)$. In a different context, Shor introduced the polynomials $Q(i,j,k)$ related to improper edges of a rooted tree, leading to a refinement of Cayley's formula. He also proved a recurrence relation and raised the question of finding a combinatorial proof. Zeng realized that the polynomials of Ramanujan coincide with the polynomials of Shor, and that the recurrence relation of Shor coincides with the recurrence relation of Berndt, Evans and Wilson. So we call these polynomials the Ramanujan-Shor polynomials, and call the recurrence relation the Berndt-Evans-Wilson-Shor recursion. A combinatorial proof of this recursion was obtained by Chen and Guo, and a simpler proof was recently given by Guo. From another perspective, Dumont and Ramamonjisoa found a context-free grammar $G$ to generate the number of rooted trees on $n$ vertices with $k$ improper edges. Based on the grammar $G$, we find a grammar $H$ for the Ramanujan-Shor polynomials. This leads to a formal calculus for the Ramanujan-Shor polynomials. In particular, we obtain a grammatical derivation of the Berndt-Evans-Wilson-Shor recursion. We also provide a grammatical approach to the Abel identities and a grammatical explanation of the Lacasse identity.

中文翻译：

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Ramanujan-Shor 多项式的上下文无关文法

Ramanujan 在他的幂级数反演研究中定义了多项式 $\psi_{k}(r,x)$。Berndt、Evans 和 Wilson 获得了 $\psi_{k}(r,x)$ 的递推关系。在不同的上下文中，Shor 引入了与有根树的不正确边相关的多项式 $Q(i,j,k)$，从而改进了 Cayley 公式。他还证明了递推关系，并提出了寻找组合证明的问题。曾先生意识到拉马努金多项式与肖尔多项式重合，肖尔的递推关系与伯恩特、埃文斯和威尔逊的递推关系重合。所以我们称这些多项式为 Ramanujan-Shor 多项式，并将递推关系称为 Berndt-Evans-Wilson-Shor 递归。Chen 和 Guo 获得了这个递归的组合证明，郭最近给出了一个更简单的证明。从另一个角度来看，Dumont 和 Ramamonjisoa 发现了一种上下文无关文法 $G$ 来生成 $n$ 顶点上具有 $k$ 不正确边的有根树的数量。基于语法 $G$，我们找到了 Ramanujan-Shor 多项式的语法 $H$。这导致了 Ramanujan-Shor 多项式的正式微积分。特别是，我们获得了 Berndt-Evans-Wilson-Shor 递归的语法推导。我们还提供了 Abel 恒等式的语法方法和 Lacasse 恒等式的语法解释。这导致了 Ramanujan-Shor 多项式的正式微积分。特别是，我们获得了 Berndt-Evans-Wilson-Shor 递归的语法推导。我们还提供了 Abel 恒等式的语法方法和 Lacasse 恒等式的语法解释。这导致了 Ramanujan-Shor 多项式的正式微积分。特别是，我们获得了 Berndt-Evans-Wilson-Shor 递归的语法推导。我们还提供了 Abel 恒等式的语法方法和 Lacasse 恒等式的语法解释。