Some New Families of Special Polynomials and Numbers Associated with Finite Operators

Yilmaz Simsek

doi:10.3390/sym12020237

Abstract

The aim of this study was to define a new operator. This operator unify and modify many known operators, some of which were introduced by the author. Many properties of this operator are given. Using this operator, two new classes of special polynomials and numbers are defined. Many identities and relationships are derived, including these new numbers and polynomials, combinatorial sums, the Bernoulli numbers, the Euler numbers, the Stirling numbers, the Daehee numbers, and the Changhee numbers. By applying the derivative operator to these new polynomials, derivative formulas are found. Integral representations, including the Volkenborn integral, the fermionic p-adic integral, and the Riemann integral, are given for these new polynomials.

Highlights

Special numbers, special functions, and operators are widely used in mathematics, physics, and engineering
Substituting f ( x ) = ∑ dl x l (n ∈ N0 and dl ∈ R) into (12), we define a new class of special l =1 polynomials as follows: n
J =0 we obtain a derivative formula for the polynomials Pn ( x, k; a, b; λ, β) by the following theorem: Theorem 1

Summary

Introduction

Special numbers, special functions, and operators are widely used in mathematics, physics, and engineering. By applying a derivative operator and p-adic integrals to these new special polynomials, many interesting identities, relations, and formulas were found. The remainder of this paper is structured as follows: Section 2 outlines a new finite operator is defined. Some properties of this operator are given. Using these integral representations, many new identities and formulas are derived including the Bernoulli numbers, the Euler numbers, the Stirling numbers, the Daehee numbers, and the Changhee numbers.

A New Operator

New Families of Special Polynomials and Numbers

Conclusions