Related papers: Some remarks about the derivative operator and gen…

We consider generalized Stirling numbers of the second kind $% S_{a,b,r}^{\alpha_{s},\beta_{s},r_{s},p_{s}}\left( p,k\right) $, $% k=0,1,\ldots .rp+\sum_{s=2}^{L}r_{s}p_{s}$, where $a,b,\alpha_{s},\beta_{s} $ are complex numbers, and…

This work introduces the families of generalized energy operators $([[.]^p]_k^+)_{k\in\mathbb{Z}}$ and $([[.]^p]_k^-)_{k\in\mathbb{Z}}$ ($p$ in $\mathbb{Z}^+$). One shows that with Lemma 1, the successive derivatives of $\big…

We study two generalizations of the gamma-expansion of Eulerian polynomials from the viewpoint of the decompositions of statistics. We first present an expansion formula of the trivariate Eulerian polynomials, which are the enumerators for…

Let $w$ be a word in alphabet $\{x,D\}$ with $m$ $x$'s and $n$ $D$'s. Interpreting "$x$" as multiplication by $x$, and "$D$" as differentiation with respect to $x$, the identity $wf(x) = x^{m-n}\sum_k S_w(k) x^k D^k f(x)$, valid for any…

Factorial series played a major role in Stirling's classic book "Methodus Differentialis" (1730), but now only a few specialists still use them. This article wants to show that this neglect is unjustified, and that factorial series are…

We define a new family of generalized Stirling permutations that can be interpreted in terms of ordered trees and forests. We prove that the number of generalized Stirling permutations with a fixed number of ascents is given by a natural…

Here presented is a unified approach to Stirling numbers and their generalizations as well as generalized Stirling functions by using generalized factorial functions, $k$-Gamma functions, and generalized divided difference. Previous…

We consider the generating polynomial of the number of rooted trees on the set $\{1,2,\dots,n\}$ counted by the number of descending edges (a parent with a greater label than a child). This polynomial is an extension of the descent…

In this article we generalize the $q$-difference operator due to Carlitz in order to derive explicit sum formulae for several extensions of Stirling numbers of the second kind, including complete homogeneous symmetric functions,…

We introduce a generalization of the Stirling numbers via symmetric functions involving two weight functions. The resulting extension unifies previously known Stirling-type sequences with known symmetric function forms, as well as other…

This paper studies the generalizations of the Stirling numbers of both kinds and the Lah numbers in association with the normal order problem in the Weyl algebra $W=\langle x,D|Dx-xD=1\rangle$. Any word $\omega\in W$ with $m$ $x$'s and $n$…

We characterize those linear operators that can be expressed as a sum over k of terms of the form f_k(D) x^k and give several examples.

This paper presents a family of rapidly convergent summation formulas for various finite sums of the form $\sum_{k=0}^{\lfloor x\rfloor}f(k)$, where $x$ is a positive real number.

By applying the Newton-Gregory expansion to the polynomial associated with the sum of powers of integers $S_k(n) = 1^k + 2^k + \cdots + n^k$, we derive a couple of infinite families of explicit formulas for $S_k(n)$. One of the families…

We define some generalizations of the classical descent and inversion statistics on signed permutations that arise from the work of Sack and Ulfarsson [20] and called after width-k descents and width-k inversionsof type A in Davis's work…

We introduce a class of $f(t)$-factorials, or $f(t)$-Pochhammer symbols, that includes many, if not most, well-known factorial and multiple factorial function variants as special cases. We consider the combinatorial properties of the…

We show that for each positive integer $a$ there exist only finitely many prime numbers $p$ such that $a$ appears an odd number of times in the period of continued fraction of $\sqrt{p}$ or $\sqrt{2p}$. We also prove that if $p$ is a prime…

We define a new class of generating function transformations related to polylogarithm functions, Dirichlet series, and Euler sums. These transformations are given by an infinite sum over the $j^{th}$ derivatives of a sequence generating…

Associated to each random variable $Y$ having a finite moment generating function, we introduce a different generalization of the Stirling numbers of the second kind. Some characterizations and specific examples of such generalized numbers…

We apply the Euler--Maclaurin formula to find the asymptotic expansion of the sums $\sum_{k=1}^n (\log k)^p / k^q$, ~$\sum k^q (\log k)^p$, ~$\sum (\log k)^p /(n-k)^q$, ~$\sum 1/k^q (\log k)^p $ in closed form to arbitrary order ($p,q…