Q-Generalized Tangent Based Hybrid Polynomials

Ghazala Yasmin,Junesang Choi,Hibah Islahi

doi:10.3390/sym13050791

Ghazala Yasmin, Junesang Choi + Show 1 more

Open Access

https://doi.org/10.3390/sym13050791

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Journal: Symmetry	Publication Date: May 3, 2021
Citations: 3	License type: CC BY 4.0

Affiliation: Dongguk University, Aligarh Muslim University

Abstract

In this paper, we incorporate two known polynomials to introduce so-called 2-variable q-generalized tangent based Apostol type Frobenius–Euler polynomials. Next we present a number of properties and formulas for these polynomials such as explicit expressions, series representations, summation formulas, addition formula, q-derivative and q-integral formulas, together with numerous particular cases of the new polynomials and their associated formulas demonstrated in two tables. Further, by using computer-aided programs (for example, Mathematica or Matlab), we draw graphs of some particular cases of the new polynomials, mainly, in order to observe in several angles how zeros of these polynomials are distributed and located. Lastly we provide numerous observations and questions which naturally arise amid the present investigation.

Highlights

A remarkably large number of a variety of polynomials, numbers and functions, and their generalizations and variants have been introduced and investigated, due mainly to their potential usefulness and direct applications in a wide range of research subjects
We provide a number of properties and formulas for these polynomials such as explicit representations, series representations, summation formulas, addition formula, q-derivative, q-integral formulas, numerous particular cases of the new polynomials and their related formulas illustrated with Tables 1 and 2
From Definition 6, we find that the 2-variable q-generalized tangent based Apostol type Frobenius–Euler polynomials CHn(,αq,m)(u, v; ρ; λ) of order α in the variables u and v are given by

Summary

Introduction

A remarkably large number of a variety of polynomials, numbers and functions, and their generalizations and variants have been introduced and investigated, due mainly to their potential usefulness and direct applications in a wide range of research subjects (see, e.g., [1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19] and the references therein). New (p, q)-Stirling polynomials of the second kind fitting for the (p, q)-analogue of Bernstein polynomials are introduced and studied in [17], which includes an extensive list of references about q-and (p, q)-extenstions of some known polynomials and numbers, in particular, q-and (p, q)-Stirling polynomials and q-and (p, q)-Bernstein polynomials. We couple the polynomials in Definitions 1 and 5 to introduce new polynomials which are called the 2-variable q-generalized tangent based Apostol type Frobenius– Euler polynomials of order α in the variables u and v, denoted by CHn(,αq,m)(u, v; ρ; λ), in Definition 6. We use computeraided programs (for example, Mathematica or Matlab) to draw graphs of some particular cases of the new polynomials, mainly, in order to observe in several angles how zeros of these polynomials are distributed and located. We give a number of observations and questions which naturally occurs amid this investigation

Preliminaries

Summation Formulae

Explicit Representations

Graphical Representations and Locations of Zeros