Related papers: Stirling numbers with higher level and records
A new family of fractional counting processes based on a three-parameter generalized Mittag-Leffler function was introduced and studied. As applications we develop a fractional generalized compound process, introduce and develop fractional…
By means of the generating function method, a linear recurrence relation is explicitly resolved. The solution is expressed in terms of the Stirling numbers of both the first and the second kind. Two remarkable pairs of combinatorial…
We consider two types of polynomials $F_n (x) = \sum_{\nu=1}^n \nu! S_2(n,\nu) x^\nu$ and $\hat{F}_n (x) = \sum_{\nu=1}^n \nu! S_2(n,\nu) H_\nu x^\nu$, where $S_2(n,\nu)$ are the Stirling numbers of the second kind and $H_\nu$ are the…
In this paper ideas of different types of convergence of a sequence of random variables in probability, namely, statistical convergence of order $\alpha$ in probability, strong $p$-Ces$\grave{\mbox{a}}$ro summability of order $\alpha$ in…
Physical and mathematical applications of fractional Poisson probability distribution have been presented. As a physical application, a new family of quantum coherent states has been introduced and studied. As mathematical applications, we…
The development of the theories of the second-order Eulerian polynomials began with the works of Buckholtz and Carlitz in their studies of an asymptotic expansion. Gessel-Stanley introduced Stirling permutations and presented combinatorial…
Recently, the $\lambda$-analogues of $r$-Stirling numbers of the first kind were studied by Kim-Kim. The aim of this paper is to introduce the $\lambda$-analogues of $r$-Stirling numbers of the second kind and to investigate some…
Scattered polynomials over finite fields attracted an increasing attention in the last years. One of the reasons is their deep connection with Maximum Rank Distance (MRD) codes. Known classification results for exceptional scattered…
This paper deals with the problem of extracting the activity of individual neurons from multi-electrode recordings. Important aspects of this work are: 1) the sorting is done in two stages - a statistical model of the spikes from different…
A record is an entry in a time series that is larger or smaller than all previous entries. If the time series consists of independent, identically distributed random variables with a superimposed linear trend, record events are positively…
Although the prime numbers are deterministic, they can be viewed, by some measures, as pseudo-random numbers. In this article, we numerically study the pair statistics of the primes using statistical-mechanical methods, especially the…
In this lecture notes I concentrate on the classic and widely applicable characterization of higher order statistics by joint moments, a.k.a. higher order correlation functions, and directly related statistics. I put special emphasis on…
In this brief note, we give two explicit formulas for the Bernoulli Numbers in terms of the Stirling numbers of the second kind, and the Eulerian Numbers. To the best of our knowledge, these formulas are new. We also derive two more…
It is known that Bernoulli scheme of independent trials with two outcomes is connected with the binomial coefficients. The aim of this paper is to indicate stochastic processes which are connected with the $q$-polynomial coefficients (in…
We provide a method that enables the simple calculation of the maximal correlation coefficient of a bivariate distribution, under suitable conditions. In particular, the method readily applies to known results on order statistics and…
New exceptional (i.e. non-repeating) prime number multiplets are given and formulated in terms of arithmetic progressions, along with laws governing them. Accompanying repeating prime number multiplets are pointed out. Prime number…
In this paper we study random orderings of the integers with a certain invariance property. We describe all such orders in a simple way. We define and represent random shuffles of a countable set of labels and then give an interpretation of…
We introduce a novel generalization of deranged Bell numbers by defining the partial deranged Bell numbers $w_{n,r}$, which count the number of set partitions of $\left[ n\right] $ with exactly $r$ fixed blocks, while the remaining blocks…
As properties of poly-Bernoulli numbers, a number of formulas such as the duality formula, explicit formula using the Stirling numbers of the second kind and periodicity for negative upper-index have been established. For the multi-indexed…
The asymptotic results that underlie applications of extreme random fields often assume that the variables are located on a regular discrete grid, identified with $\mathbb{Z}^2$, and that they satisfy stationarity and isotropy conditions.…