A new algorithm for computing the multivariate Faà di Bruno's formula

Faà di Bruno's formula and inversion of power series

Faà di Bruno's formula gives an expression for the derivatives of the composition of two real-valued functions. In this paper we prove a multivariate and synthesized version of Faà di Bruno's formula in higher dimensions, providing a combinatorial expression for the derivatives of chain compositions $F^{(1)} \circ \ldots \circ F^{(m)}$ of functions $F^{(l)} : \mathbb{R}^N \to \mathbb{R}^N$ in terms of sums over labelled trees. We give several applications of this formula, including a new involution formula for the inversion of multivariate power series. We use this framework to outline a combi…

The symmetric function theorem via the Faà di Bruno formula

The symmetric function theorem states that a polynomial that is invariant under permutation of variables, is a polynomial in the elementary symmetric polynomials. We deduce this classical result, in the analytic setting, from the multivariate Faà di Bruno formula. In two variables, this allows us to completely determine all coefficients that occur in the inductive equations.

Four explicit formulas for the prolongations of an infinitesimal Lie symmetry and multivariate Faa di Bruno formulas

In 1979, building on S. Lie's theory of symmetries of (partial) differrential equations, P.J. Olver formulated inductive formulas which are appropriate for the computation of the prolongations of an infinitesimal Lie symmetry to jet spaces, for an arbitrary number n\geq 1 of independent variables (x^1, ..., x^n) and for an arbitrary number m \geq 1 of dependent variables (y^1, ..., y^m). This paper is devoted to elaborate a formalism based on multiple Kronecker symbols which enables one to handle these ``unmanageable'' prolongations and to discover the underlying complicated combinatorics. Pro…

一个超几何Faà di Bruno公式

A Supergeometric Faà di Bruno Formula

本研究将多元Faà di Bruno公式推广至含反交换奇坐标的超几何情形，推导出具非平凡符号的对应公式，并借此得到Fan和Hon定义的广义超Bell多项式的显式组合公式。

通过莫比乌斯反演得到的离散法阿·迪·布鲁诺公式

Discrete Faà di Bruno via Möbius Inversion

该研究从莫比乌斯反演角度探讨离散和微分法阿·迪·布鲁诺公式，通过组合泰勒展开与反演得到离散公式，经分组、迭代得出相关版本及公式，还展示了覆盖与划分关系，最后将代数恒等式提升到分析设置，恢复并扩展了多元法阿·迪·布鲁诺公式。

A new method for computing asymptotics of diagonal coefficients of multivariate generating functions

Let \sum_{n\in N^d} f_{n_1, ..., n_d} x_1^{n_1}... x_d^{n_d} be a multivariate generating function that converges in a neighborhood of the origin of C^d. We present a new, multivariate method for computing the asymptotics of the diagonal coefficients f_{a_1n,...,a_dn} and show its superiority over the standard, univariate diagonal method.