Derivation of computational formulas for Changhee polynomials and their functional and differential equations

Ji Suk So,Yilmaz Simsek

doi:10.1186/s13660-020-02415-8

Ji Suk So, Yilmaz Simsek

Open Access

https://doi.org/10.1186/s13660-020-02415-8

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Abstract

The goal of this paper is to demonstrate many explicit computational formulas and relations involving the Changhee polynomials and numbers and their differential equations with the help of functional equations and partial derivative equations for generating functions of these polynomials and numbers. These formulas also include the Euler polynomials, the Stirling numbers, the Bernoulli numbers and polynomials of the second kind, the Changhee polynomials of higher order, and the Daehee polynomials of higher order, which are among the well known polynomial families. By using PDEs of these generating functions, not only some recurrence relations for derivative formulas of the Changhee polynomials of higher order, but also two open problems for partial derivative equations for generating functions are given. Moreover, by using functional equations of the generating functions, two inequalities including combinatorial sums, the Changhee numbers of negative order, and the Stirling numbers of the second kind are provided. Finally, further remarks and observations for the results of this paper are given.

Highlights

IntroductionMany kinds of partial differential equations (PDEs), ordinary differential equations (ODEs), and stochastic differential equation (SDEs), including boundary-value problems, initial-value problem, and discrete boundary-value problems have been studied and investigated
1 Introduction Using generating functions, many kinds of partial differential equations (PDEs), ordinary differential equations (ODEs), and stochastic differential equation (SDEs), including boundary-value problems, initial-value problem, and discrete boundary-value problems have been studied and investigated. By using these equations many properties of the generating functions have been investigated. Generating functions, their functional equations and their PDEs including special numbers and polynomials have been studied in many different areas
(2020) 2020:149 with their PDEs and functional functions, we investigate and study many new formulas and relations involving the Bernoulli numbers and polynomials of the second kind, the Euler numbers and polynomials, the Stirling numbers, the Peters polynomials, the Boole polynomials and numbers, the Daehee numbers, and the Changhee polynomials

Summary

Introduction

Many kinds of partial differential equations (PDEs), ordinary differential equations (ODEs), and stochastic differential equation (SDEs), including boundary-value problems, initial-value problem, and discrete boundary-value problems have been studied and investigated. By using these equations many properties of the generating functions have been investigated. Simsek Journal of Inequalities and Applications (2020) 2020:149 with their PDEs and functional functions, we investigate and study many new formulas and relations involving the Bernoulli numbers and polynomials of the second kind, the Euler numbers and polynomials, the Stirling numbers, the Peters polynomials, the Boole polynomials and numbers, the Daehee numbers, and the Changhee polynomials. The following notations and definitions are used in this paper: Let N = {1, 2, 3, . The falling and rising factorials functions, often used in the theory of the hypergeometric functions and partition theory, are defined as follows:

Objectives

Conclusion