72 Results Found

The new type degenerate of Bell polynomials and numbers were recently introduced, which are a degenerate version of Bell polynomials and numbers and are different from the previously introduced partially degenerate Bell polynomials and numbers. Sever...

In this paper, we introduce central complete and incomplete Bell polynomials which can be viewed as generalizations of central Bell polynomials and central factorial numbers of the second kind, and also as ’central’ analogues for complete...

Here we would like to introduce the extended r-central incomplete and complete Bell polynomials, as multivariate versions of the recently studied extended r-central factorial numbers of the second kind and the extended r-central Bell polynomials, and...

In this paper, we introduce a new class of special polynomials called the generalized Bell polynomials, constructed by combining two-variable general polynomials with two-variable Bell polynomials. The concept of the monomiality principle was employe...

In this paper, we consider Bell-based Stirling polynomials of the second kind and derive some useful relations and properties including some summation formulas related to the Bell polynomials and Stirling numbers of the second kind. Then, we introduc...

Recently, the partially degenerate Bell polynomials and numbers, which are a degenerate version of Bell polynomials and numbers, were introduced. In this paper, we consider the new type degenerate Bell polynomials and numbers, and obtain several expr...

This paper aims to establish a new hybrid class of special polynomials, namely, the Fubini–Bell-based Appell polynomials. The monomiality principle is used to derive the generating function for these polynomials. Several related identities and...

This study uses a fractional operator technique to analyze a novel class of special polynomials. These polynomials are designated as fractional Gould–Hopper–Bell–Apostol-type polynomials. We first define the operational expression o...

This study introduces a novel generalized class of special polynomials using a fractional operator approach. These polynomials are referred to as the generalized Gould–Hopper–Bell-based Appell polynomials. In view of the operational metho...

This paper introduces a new family of q-special polynomials, termed q-general Bell polynomials, and systematically explores their structural and analytical properties. We establish their generating functions, derive explicit series representations, a...

In this paper, we study differential equations arising from the generating function of the $(r,\beta )$ -Bell polynomials. We give explicit identities for the $(r,\beta )$ -Bell polynomials. Finally, we find the zeros of the $(r$...

The problem of computing the Laplace transform of composed functions has not found its way into the literature because it was customarily believed that there were no suitable formula to solve it. Actually, it has been shown in previous work that by m...

In this paper, we introduce new class of Bell-based Apostol-type Frobenius–Euler polynomials and investigate some properties of these polynomials. We derive some explicit and implicit summation formulas and their symmetric identities by using d...

This paper introduces a new family of extended degenerate Bell-based Appell polynomials by applying Euler’s integral as a fractional operator to the Appell-type degenerate Bell polynomials. Beginning with the exponential operational rule and em...

After recalling the most important properties of the Bell polynomials, we show how to approximate a positive compact operator by a suitable matrix. Then, we derive a representation formula for functions of the obtained matrix, which can be considered...

In this paper, we introduce the extended degenerate r-central factorial numbers of the second kind and the extended degenerate r-central Bell polynomials. They are extended versions of the degenerate central factorial numbers of the second kind and t...

In the real world there are many applications that find the Bell distribution to be a useful and relevant model. One of these is the normal distribution. In this paper, we develop a new subclass of analytic bi-univalent functions by making use of the...

Using Bell’s polynomials it is possible to approximate the Laplace Transform of composite functions. The same methodology can be adopted for the evaluation of the Laplace Transform of higher-order nested functions. In this case, a suitable exte...

Several different subclasses of the bi-univalent function class $\mathsf{\Sigma }$ were introduced and studied by many authors using distribution series like Pascal distribution, Poisson distribution, Borel distribution, the Mittag-Leffler-type Borel distributi...

Bell’s polynomials are used in many different fields of mathematics, ranging from number theory to operator theory. This paper shows a relevant application in probability theory aimed at computing the moments of generalized Gaussian distributio...

In this paper, by means of the Faà di Bruno formula, with the help of explicit formulas for partial Bell polynomials at specific arguments of two specific sequences generated by derivatives at the origin of the inverse sine and inverse cosine...

A new class of matrices defined in terms of r-Stirling numbers is introduced. These r-Stirling matrices are totally positive and determine the linear transformation between monomial and r-Bell polynomial bases. An efficient algorithm for the computat...

The aim of this paper is to construct generating functions for new families of combinatorial numbers and polynomials. By using these generating functions with their functional and differential equations, we not only investigate properties of these ne...

In this paper, we introduce a new family of Stirling polynomials of the second kind, Bell polynomials, bivariate Bell polynomials, Bernoulli polynomials of higher order, and Euler polynomials of higher order arising from the Kaniadakis calculus viewp...

In this paper, we focus on the higher-order derivatives of the hyperharmonic polynomials, which are a generalization of the ordinary harmonic numbers. We determine the hyperharmonic polynomials and their successive derivatives in terms of the r-Stirl...

In this paper, we consider a new type of degenerate derangement polynomial and number, which shall be called the degenerate derangement polynomials and numbers of the second kind. These concepts are motivated by Kim et al.’s work on degenerate derang...

In the paper, by virtue of the Faà di Bruno formula, with the aid of some properties of the Bell polynomials of the second kind, and by means of a general formula for derivatives of the ratio between two differentiable functions, the authors establis...

In this article, we derive representation formulas for a class of r-associated Stirling numbers of the second kind and examine their connections with a class of generalized Bernoulli polynomials. Herein, we use the Blissard umbral approach and the fa...

The main aim of this paper is to introduce the degenerate truncated forms of multifarious special polynomials and numbers and is to investigate their various properties and relationships by using the series manipulation method and diverse special pro...

The derivatives of the inverse of a one-to-one function are needed in a range of applied contexts, from random variate generation and molecular simulation to nuclear physics and bias reduction for maximum likelihood estimates, yet existing treatments...

In this paper, we perform a further investigation for the Catalan numbers. By making use of the method of derivatives and some properties of the Bell polynomials, we establish two new expressions for sums of products of arbitrary number of the Catala...

In recent papers, new sets of Sheffer and Brenke polynomials based on higher order Bell numbers, and several integer sequences related to them, have been studied. The method used in previous articles, and even in the present one, traces back to prece...

In this paper, the authors give a simple review of closed-form, explicit, and recursive formulas and related results for the nth derivative of the power-exponential function $x^x$, establish two closed-form and explicit formulas for partial Bell polynom...

In the paper, the authors find a sufficient and necessary condition for the power-exponential function ${\left(1+\frac{1}{x}\right)}^{\alpha x}$ to be a Bernstein function, derive closed-form formulas for the nth derivatives of the power-exponential functions ${\left(1+\frac{1}{x}\right)}^{\alpha x}$ and ${(1}^{}$...

This paper presents an extensive investigation into several interrelated topics in mathematical analysis and number theory. The author revisits and builds upon known results regarding the Maclaurin power series expansions for a variety of functions a...

Gegenbauer polynomials constitute a numerical tool that has attracted the interest of many function theorists in recent times mainly due to their real-life applications in many areas of the sciences and engineering. Their applications in geometric fu...

We study inverse-binomial double series involving ordinary and alternating harmonic numbers. Using iterated integrals associated with admissible 3-posets, we establish twenty-four explicit evaluations in terms of multiple zeta values and alternating...

This paper presents analytical formulas for pricing generalized swaps, including the moment swap, gamma swap, entropy swap and self-quantoed variance swap. The formulas are based on closed-form formulas for the conditional expectations of the product...

Consider a multivariate function f that maps an open region of $R^p$ into $R^p$, together with its inverse function. This paper presents a general method for obtaining the partial derivatives of the inverse function, of any order, expressed in terms of the...

With the aim of improving the reconstruction of stochastic evolution equations from empirical time-series data, we derive a full representation of the generator of the Kramers–Moyal operator via a power-series expansion of the exponential operator. T...

We considered the time-inhomogeneous linear birth–death processes with immigration. For these processes closed form expressions for the transition probabilities were obtained in terms of the complete Bell polynomials. The conditional mean and t...

Recently, Kim-Kim (J. Math. Anal. Appl. (2021), Vol. 493(1), 124521) introduced the $\lambda$-Sheffer sequence and the degenerate Sheffer sequence. In addition, Kim et al. (arXiv:2011.08535v1 17 November 2020) studied the degenerate derangement polynomials a...

One of the main motivations of this paper is to construct generating functions for generalization of the Touchard polynomials (or generalization exponential functions) and certain special numbers. Many novel formulas and relations for these polynomia...

In the paper, the authors present a brief overview and survey of the scientific work by Chinese mathematician Feng Qi and his coauthors.

The Gould–Hopper–Laguerre–Sheffer matrix polynomials were initially studied using operational methods, but in this paper, we investigate them using matrix techniques. By leveraging properties of Pascal functionals and Wronskian matr...

We study the formulas for binomial sums of harmonic numbers of higher order ${\displaystyle \sum _{k=0}^n}{H}_k^{\left(r\right)}\left({\displaystyle \genfrac{}{}{0pt}{}{n}{k}}\right){(1-q)}^k{q}^{n-k}=H_n^{\left(r\right)}-{\displaystyle \sum _{j=1}^n}{\mathcal{D}}_r(n,j){\displaystyle \frac{q^j}{j}}.$ Recently, Mneimneh proved that ${\mathcal{D}}_1(n,j)=1$. In this paper, we find several different expressions of ${\mathcal{D}}_r(n$...

Explicit formulas for graphical Stirling and Bell numbers are known for relatively few graph families. We derive exact expressions for three classes whose independence structure admits a complete combinatorial description: complete multipartite graph...

The inverse problem under consideration concerns a one-dimensional parabolic equation whose thermal diffusivity takes the quadratic-in-space form $a_s\left(\tau \right){\kappa }^2+b_s\left(\tau \right)\kappa +c_s\left(\tau \right)$. The unknowns are three time-dependent coefficients $a_s(\mathrm{\&tau}$...

This paper considers polynomial characteristics useful for a better understanding of the behaviour of these functions. Taylor series for the polynomials are described by the items with even and odd derivatives and powered changes in the argument, whi...

This paper introduces fully Dowling polynomials of the first and second kinds, which are degenerate versions of the ordinary Dowling polynomials. Then, several important identities for these degenerate polynomials are derived. The relationship betwee...