Truncated-Exponential-Based Appell-Type Changhee Polynomials

Tabinda Nahid,Junesang Choi,Parvez Alam

doi:10.3390/sym12101588

Tabinda Nahid, Junesang Choi + Show 1 more

Open Access

https://doi.org/10.3390/sym12101588

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Abstract

The truncated exponential polynomials em(x) (1), their extensions, and certain newly-introduced polynomials which combine the truncated exponential polynomials with other known polynomials have been investigated and applied in various ways. In this paper, by incorporating the Appell-type Changhee polynomials Chn*(x) (10) and the truncated exponential polynomials in a natural way, we aim to introduce so-called truncated-exponential-based Appell-type Changhee polynomials eCn*(x) in Definition 1. Then, we investigate certain properties and identities for these new polynomials such as explicit representation, addition formulas, recurrence relations, differential and integral formulas, and some related inequalities. We also present some integral inequalities involving these polynomials eCn*(x). Further we discuss zero distributions of these polynomials by observing their graphs drawn by Mathematica. Lastly some open questions are suggested.

Highlights

Introduction and PreliminariesA number of special polynomials have found many vital applications in a variety of fields such as mathematics, applied mathematics, mathematical physics and engineering
The Changhee polynomials and numbers have been generalized to yield more complicated polynomials and numbers which are found to have a number of identities including, especially, certain differential equations and proved to be connected with various problems in the areas of engineering and physics
By incorporating the Appell-type Changhee polynomials C h∗n ( x ) (10) and the truncated exponential polynomials em ( x ) (1) in a natural way, we aim to introduce so-called truncated-exponential-based Appell-type Changhee polynomials e Cn∗ ( x ) in Definition 1

Summary

Introduction and Preliminaries

A number of special polynomials have found many vital applications in a variety of fields such as mathematics, applied mathematics, mathematical physics and engineering. The generating relation (5) can be derived by taking the Cauchy product of two Maclaurin series ex t and 1/(1 − t) Differentiating both sides of the identity (5) with regard to the variable t and x, respectively, yields the following differential-recursive relations (see [2], Equation (5)). The Changhee polynomials and numbers have been generalized to yield more complicated polynomials and numbers which are found to have a number of identities including, especially, certain differential equations and proved to be connected with various problems in the areas of engineering and physics (see, e.g., [5,9,10,14,18,40]).

Truncated-Exponential-Based Appell-Type Changhee Polynomials

Differential Formulas

Integral Formulas

Inequalities Involving Integrals