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Traditionally, fractional counting processes, such as the fractional Poisson process, etc. have been defined using fractional differential and integral operators. Recently, Laskin (2024) introduced a generalized fractional counting process (FCP) by c
Externí odkaz:
http://arxiv.org/abs/2412.04334
Autor:
Kim, Taekyun, Kim, Dae san
The aim of this paper is to study degenerate Eulerian polynomials and degenerate Eulerian numbers, respectively as degenerate versions of the Eulerian polynomials and the Eulerian numbers, and to derive some of their properties. Specifically, we deri
Externí odkaz:
http://arxiv.org/abs/2412.03119
Autor:
Kim, Taekyun, Kim, Dae San
In combinatorics, a derangement is a permutation of the elements of a set, such that no element appears in its original position. The number of derangement of an n-element set is called the nth derangement number. Recently, the degenerate derangement
Externí odkaz:
http://arxiv.org/abs/2410.09394
Autor:
Farhi, Bakir
This paper is devoted to establishing several new formulas relating Bernoulli and Stirling numbers of both kinds.
Comment: 13 pages
Comment: 13 pages
Externí odkaz:
http://arxiv.org/abs/2410.05207
Autor:
Boualam, Rezig, Ahmia, Moussa
In this paper, we provide criteria for the log-concavity of rows and the strong $q$-log-convexity of the generating functions of rows in more generalized triangles. Additionally, we prove that the bi$^s$nomial transformation not only preserves the st
Externí odkaz:
http://arxiv.org/abs/2409.18886
Autor:
Grunwald, Jerzy, Serafin, Grzegorz
In this article, we provide a comprehensive analysis of the asymptotic behavior of Bell numbers, enhancing and unifying various results previously dispersed in the literature. We establish several explicit lower and upper bounds. The main results cor
Externí odkaz:
http://arxiv.org/abs/2408.14182
Autor:
Beauduin, Kei
In this paper, we explore the effectiveness of almost purely operational methods in the study of umbral calculus. To accomplish this goal, we systematically reconstruct the theory operationally, offering new proofs and results throughout. Our approac
Externí odkaz:
http://arxiv.org/abs/2407.16348
The generalized Stirling numbers of the second kind together with the Stirling numbers of the first kind are used to present a novel method to approximate the Riemann zeta function at integer values by rationals. We show that the error committed in s
Externí odkaz:
http://arxiv.org/abs/2407.07920
Based on the combinatorial interpretation of the ordered Bell numbers, which count all the ordered partitions of the set $[n]=\{1,2,\dots,n\}$, we introduce the Fibonacci partition as a Fibonacci permutation of its blocks. Then we define the Fibonacc
Externí odkaz:
http://arxiv.org/abs/2407.04409
Autor:
Kim, Taekyun, Kim, Dae san
Assume that the moment generating function of the random vari able Y exists in a neighborhood of the origin. We introduce the probabilistic multi-Stirling numbers of the second kind associated with Y and the proba bilistic multi-Lah numbers associate
Externí odkaz:
http://arxiv.org/abs/2407.00061