A novel matrix technique for multi-order pantograph differential equations of fractional order

The main purpose of this article is to investigate a novel set of (orthogonal) basis functions for treating a class of multi-order fractional pantograph differential equations (MOFPDEs) computationally. These polynomials, denoted by S n ( x ) and called special polynomials , were first discovered in a study of a certain family of isotropic turbulence fields. They are expressible in terms of the generalized Laguerre polynomials and are related to the Bessel and Srivastava–Singhal polynomials. Unlike the Laguerre polynomials, all coefficients of the special polynomials are positive. We further introduce the fractional order of the special polynomials and use them along with some suitable collocation points in a special matrix technique to treat fractional-order MOFPDEs. Moreover, the convergence analysis of these polynomials is established. Through five example applications, the utility and efficiency of the present matrix approach are demonstrated and comparisons with some existing numerical schemes have been performed in this class.

Degenerate poly-Bell polynomials and numbers

Numerous mathematicians have studied ‘poly’ as one of the generalizations to special polynomials, such as Bernoulli, Euler, Cauchy, and Genocchi polynomials. In relation to this, in this paper, we introduce the degenerate poly-Bell polynomials emanating from the degenerate polyexponential functions which are called the poly-Bell polynomials when $\lambda \rightarrow 0$ λ → 0 . Specifically, we demonstrate that they are reduced to the degenerate Bell polynomials if $k = 1$ k = 1 . We also provide explicit representations and combinatorial identities for these polynomials, including Dobinski-like formulas, recurrence relationships, etc.

Degenerate Derangement Polynomials and Numbers

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 derangement polynomials and numbers. We investigate some properties of these new degenerate derangement polynomials and numbers and explore their connections with the degenerate gamma distributions for the case λ∈(−1,0). In more detail, we derive their explicit expressions, recurrence relations, and some identities involving our degenerate derangement polynomials and numbers and other special polynomials and numbers, which include the fully degenerate Bell polynomials, the degenerate Fubini polynomials, and the degenerate Stirling numbers of the first and the second kinds. We also show that those polynomials and numbers are connected with the moments of some variants of the degenerate gamma distributions. Moreover, we compare the degenerate derangement polynomials and numbers of the second kind to those of Kim et al.

The Mean Values of Character Sums and Their Applications

In this paper, we use the elementary methods and properties of classical Gauss sums to study the calculation problems of some mean values of character sums of special polynomials, and obtained several interesting calculation formulae for them. As an application, we give a criterion for determining that 2 is the cubic residue for any odd prime p.

Some Identities of the Degenerate Higher Order Derangement Polynomials and Numbers

Recently, Kim-Kim (J. Math. Anal. Appl. (2021), Vol. 493(1), 124521) introduced the λ-Sheffer sequence and the degenerate Sheffer sequence. In addition, Kim et al. (arXiv:2011.08535v1 17 November 2020) studied the degenerate derangement polynomials and numbers, and investigated some properties of those polynomials without using degenerate umbral calculus. In this paper, the y the degenerate derangement polynomials of order s (s∈N) and give a combinatorial meaning about higher order derangement numbers. In addition, the author gives some interesting identities related to the degenerate derangement polynomials of order s and special polynomials and numbers by using degenerate Sheffer sequences, and at the same time derive the inversion formulas of these identities.

A Note on q -Fubini-Appell Polynomials and Related Properties

The present article is aimed at introducing and investigating a new class of q -hybrid special polynomials, namely, q -Fubini-Appell polynomials. The generating functions, series representations, and certain other significant relations and identities of this class are established. Some members of q -Fubini-Appell polynomial family are investigated, and some properties of these members are obtained. Further, the class of 3-variable q -Fubini-Appell polynomials is also introduced, and some formulae related to this class are obtained. In addition, the determinant representations for these classes are established.

Fully degenerate Bell polynomials associated with degenerate Poisson random variables

Many mathematicians have studied degenerate versions of quite a few special polynomials and numbers since Carlitz’s work (Utilitas Math. 15 (1979), 51–88). Recently, Kim et al. studied the degenerate gamma random variables, discrete degenerate random variables and two-variable degenerate Bell polynomials associated with Poisson degenerate central moments, etc. This paper is divided into two parts. In the first part, we introduce a new type of degenerate Bell polynomials associated with degenerate Poisson random variables with parameter α > 0 \alpha \hspace{-0.15em}\gt \hspace{-0.15em}0 , called the fully degenerate Bell polynomials. We derive some combinatorial identities for the fully degenerate Bell polynomials related to the n n th moment of the degenerate Poisson random variable, special numbers and polynomials. In the second part, we consider the fully degenerate Bell polynomials associated with degenerate Poisson random variables with two parameters α > 0 \alpha \gt 0 and β > 0 \beta \hspace{-0.15em}\gt \hspace{-0.15em}0 , called the two-variable fully degenerate Bell polynomials. We show their connection with the degenerate Poisson central moments, special numbers and polynomials.