Extended Forms of Certain Hybrid Special Polynomials Related to Appell Sequences

In this article, the Legendre-Gould-Hopper polynomials are combined with Appell sequences to introduce certain mixed type special polynomials by using operational method. The generating functions, determinant definitions and certain other properties of Legendre-Gould-Hopper based Appell polynomials are derived. Operational rules providing connections between these formulae and known special polynomials are established. The 2-variable Hermite Kamp? de F?riet based Bernoulli polynomials are considered as an member of Legendre-Gould-Hopper based Appell family and certain results for this member are also obtained.

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Finding mixed families of special polynomials associated with Appell sequences

Journal of Mathematical Analysis and Applications ◽

10.1016/j.jmaa.2016.10.009 ◽

2017 ◽

Vol 447 (1) ◽

pp. 398-418 ◽

Cited By ~ 7

Author(s):

Subuhi Khan ◽

Nusrat Raza ◽

Mahvish Ali

Keyword(s):

Special Polynomials ◽

Appell Sequences ◽

Mixed Families

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Fractional calculus and generalized forms of special polynomials associated with Appell sequences

Georgian Mathematical Journal ◽

10.1515/gmj-2019-2028 ◽

2019 ◽

Vol 0 (0) ◽

Cited By ~ 1

Author(s):

Subuhi Khan ◽

Shahid Ahmad Wani

Keyword(s):

Fractional Calculus ◽

Generating Function ◽

Recurrence Relations ◽

Summation Formula ◽

Integral Transforms ◽

Operational Definition ◽

Euler Polynomials ◽

Appell Polynomials ◽

Special Polynomials ◽

Appell Sequences

Abstract In this article, an operational definition, generating function, explicit summation formula, determinant definition and recurrence relations of the generalized families of Hermite–Appell polynomials are derived by using integral transforms and some known operational rules. An analogous study of these results is also carried out for the generalized forms of the Hermite–Bernoulli and Hermite–Euler polynomials.

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Identities related to special polynomials and combinatorial numbers

Filomat ◽

10.2298/fil1715833y ◽

2017 ◽

Vol 31 (15) ◽

pp. 4833-4844 ◽

Cited By ~ 2

Author(s):

Eda Yuluklu ◽

Yilmaz Simsek ◽

Takao Komatsu

Keyword(s):

Generating Functions ◽

Functional Equations ◽

Stirling Numbers ◽

Bernoulli Numbers ◽

Euler Numbers ◽

Special Polynomials ◽

Central Factorial Numbers

The aim of this paper is to give some new identities and relations related to the some families of special numbers such as the Bernoulli numbers, the Euler numbers, the Stirling numbers of the first and second kinds, the central factorial numbers and also the numbers y1(n,k,?) and y2(n,k,?) which are given Simsek [31]. Our method is related to the functional equations of the generating functions and the fermionic and bosonic p-adic Volkenborn integral on Zp. Finally, we give remarks and comments on our results.

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The Mean Values of Character Sums and Their Applications

Mathematics ◽

10.3390/math9040318 ◽

2021 ◽

Vol 9 (4) ◽

pp. 318

Author(s):

Jiafan Zhang ◽

Yuanyuan Meng

Keyword(s):

Character Sums ◽

Gauss Sums ◽

Mean Values ◽

Special Polynomials ◽

Elementary Methods ◽

The Mean

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.

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Fully degenerate Bell polynomials associated with degenerate Poisson random variables

Open Mathematics ◽

10.1515/math-2021-0022 ◽

2021 ◽

Vol 19 (1) ◽

pp. 284-296

Author(s):

Hye Kyung Kim

Keyword(s):

Random Variables ◽

Random Variable ◽

Combinatorial Identities ◽

Bell Polynomials ◽

Poisson Random Variable ◽

Central Moments ◽

Special Polynomials ◽

Special Numbers And Polynomials ◽

New Type ◽

Two Parameters

Abstract 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.

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Degenerate poly-Bell polynomials and numbers

Advances in Difference Equations ◽

10.1186/s13662-021-03522-6 ◽

2021 ◽

Vol 2021 (1) ◽

Author(s):

Taekyun Kim ◽

Hye Kyung Kim

Keyword(s):

Combinatorial Identities ◽

Bell Polynomials ◽

Genocchi Polynomials ◽

Special Polynomials ◽

Explicit Representations

AbstractNumerous 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.

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RATIONAL SOLUTIONS OF THE BOUSSINESQ EQUATION

Analysis and Applications ◽

10.1142/s0219530508001250 ◽

2008 ◽

Vol 06 (04) ◽

pp. 349-369 ◽

Cited By ~ 14

Author(s):

PETER A. CLARKSON

Keyword(s):

Infinite Number ◽

Boussinesq Equation ◽

Nonlinear Schrödinger Equations ◽

Schrödinger Equations ◽

Rational Solutions ◽

Painleve Equations ◽

Special Polynomials ◽

Symmetry Reductions ◽

Nonlinear Schrodinger Equations ◽

De Vries

Rational solutions of the Boussinesq equation are expressed in terms of special polynomials associated with rational solutions of the second and fourth Painlevé equations, which arise as symmetry reductions of the Boussinesq equation. Further generalized rational solutions of the Boussinesq equation, which involve an infinite number of arbitrary constants, are derived. The generalized rational solutions are analogs of such solutions for the Korteweg–de Vries and nonlinear Schrödinger equations.

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