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

[摘要] 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.

[学科分类] 电力

[关键词] Generating function;Bernoulli numbers and polynomials of the second kind;Euler numbers and polynomials;Stirling numbers;Peters polynomials and numbers;Boole polynomials and numbers;Changhee polynomials and numbers;Daehee numbers and numbers