On (p, q)-Sine and (p, q)-Cosine Fubini Polynomials

Waseem Ahmad Khan,Ghulam Muhiuddin,Deena Al-Kadi,Ugur Duran

doi:10.3390/sym14030527

Waseem Ahmad Khan, Ghulam Muhiuddin + Show 2 more

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https://doi.org/10.3390/sym14030527

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Abstract

In recent years, (p,q)-special polynomials, such as p,q-Euler, p,q-Genocchi, p,q-Bernoulli, and p,q-Frobenius-Euler, have been studied and investigated by many mathematicians, as well physicists. It is important that any polynomial have explicit formulas, symmetric identities, summation formulas, and relations with other polynomials. In this work, the (p,q)-sine and (p,q)-cosine Fubini polynomials are introduced and multifarious abovementioned properties for these polynomials are derived by utilizing some series manipulation methods. p,q-derivative operator rules and p,q-integral representations for the (p,q)-sine and (p,q)-cosine Fubini polynomials are also given. Moreover, several correlations related to both the (p,q)-Bernoulli, Euler, and Genocchi polynomials and the (p,q)-Stirling numbers of the second kind are developed.

Highlights

In recent years, ( p, q)-calculus has been studied and examined widely by many physicists and mathematicians [1–12]. ( p, q)-special polynomials, such as ( p, q)-Euler, ( p, q)-Genocchi, ( p, q)-Bernoulli, ( p, q)-Frobenius-Euler, were firstly considered and developed by Duran et al [2,3]; many authors worked on other ( p, q)-special polynomials
The problems arising in mathematics, engineering and mathematical physics are framed in terms of differential equations
Most of these equations can only be treated by utilizing diverse families of special polynomials that give novel viewpoints of mathematical analysis

Summary

Introduction

In recent years, ( p, q)-calculus has been studied and examined widely by many physicists and mathematicians [1–12] (see the references cited therein). ( p, q)-special polynomials, such as ( p, q)-Euler, ( p, q)-Genocchi, ( p, q)-Bernoulli, ( p, q)-Frobenius-Euler, were firstly considered and developed by Duran et al [2,3]; many authors worked on other ( p, q)-special polynomials (see [6,8,10,12]). The problems arising in mathematics, engineering and mathematical physics are framed in terms of differential equations Most of these equations can only be treated by utilizing diverse families of special polynomials that give novel viewpoints of mathematical analysis. They are widely used in computational models of engineering and scientific problems. Fubini-type polynomials appear in combinatorial mathematics and play an important role in the theory and applications of mathematics; many number theory and combinatorics experts have extensively studied their properties and obtained a series of interesting results (see [6,13,14]).

These functions hold the following properties

Conclusions