A symbolic method for k-statistics

A Symbolic Treatment of Riordan Arrays

We approach Riordan arrays and their generalizations via umbral symbolic methods. This new approach allows us to derive fundamental aspects of the theory of Riordan arrays as immediate consequences of the umbral version of the classical Abel's identity for polynomials. In particular, we obtain a novel non-recursive formula for Riordan arrays and derive, from this new formula, some known recurrences and a new recurrence relation for Riordan arrays.

幂体重新审视——一种古老的符号方法

Poweroids revisited - an old symbolic approach

本文重审Jeffery 1861年的有限差分计算，扩展至一般delta算子，用于简洁证明多项式二项类型理论中的定理，提供紧凑的符号复合、二项性质和连接常数处理方法。

海森堡群上协变变换的嬗变与扩展伞形原理

Transmutations from the Covariant Transform on the Heisenberg Group and an Extended Umbral Principle

本文以海森堡群Weyl代数的升降算符表示为统一框架，研究伞形演算、广义平移与嬗变及算符符号演算之间的联系，回顾经典结果并揭示新机遇。

A new method for fast computing unbiased estimators of cumulants

We propose new algorithms for generating $k$-statistics, multivariate $k$-statistics, polykays and multivariate polykays. The resulting computational times are very fast compared with procedures existing in the literature. Such speeding up is obtained by means of a symbolic method arising from the classical umbral calculus. The classical umbral calculus is a light syntax that involves only elementary rules to managing sequences of numbers or polynomials. The cornerstone of the procedures here introduced is the connection between cumulants of a random variable and a suitable compound Poisson ra…

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

A new algorithm for computing the multivariate Faà di Bruno's formula is provided. We use a symbolic approach based on the classical umbral calculus that turns the computation of the multivariate Faà di Bruno's formula into a suitable multinomial expansion. We propose a MAPLE procedure whose computational times are faster compared with the ones existing in the literature. Some illustrative applications are also provided.

A unifying framework for $k$-statistics, polykays and their multivariate generalizations

Through the classical umbral calculus, we provide a unifying syntax for single and multivariate $k$-statistics, polykays and multivariate polykays. From a combinatorial point of view, we revisit the theory as exposed by Stuart and Ord, taking into account the Doubilet approach to symmetric functions. Moreover, by using exponential polynomials rather than set partitions, we provide a new formula for $k$-statistics that results in a very fast algorithm to generate such estimators.