Some Relations of Two Type 2 Polynomials and Discrete Harmonic Numbers and Polynomials

Taekyun Kim,Dae San Kim

doi:10.3390/sym12060905

Taekyun Kim, Dae San Kim

Open Access

https://doi.org/10.3390/sym12060905

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Abstract

Harmonic numbers appear, for example, in many expressions involving Riemann zeta functions. Here, among other things, we introduce and study discrete versions of those numbers, namely the discrete harmonic numbers. The aim of this paper is twofold. The first is to find several relations between the Type 2 higher-order degenerate Euler polynomials and the Type 2 high-order Changhee polynomials in connection with the degenerate Stirling numbers of both kinds and Jindalrae–Stirling numbers of both kinds. The second is to define the discrete harmonic numbers and some related polynomials and numbers, and to derive their explicit expressions and an identity.

Highlights

The degenerate Bernoulli and degenerate Euler polynomials were studied by Carlitz [1], as degenerate versions of the usual Bernoulli and Euler polynomials with their arithmetic and combinatorial interest
Degenerate Stirling numbers of the first and second kinds, degenerate central factorial numbers of the second kind, degenerate Cauchy numbers, and degenerate Bernstein polynomials. They have been explored by means of different methods such as generating functions, umbral calculus, combinatorial methods, differential equations, probability theory, p-adic integrals, p-adic q-integrals and special functions
The novelty of the present paper is the introduction of degenerate versions of harmonic numbers, called the degenerate harmonic numbers given by

Summary

Introduction

The degenerate Bernoulli and degenerate Euler polynomials were studied by Carlitz [1], as degenerate versions of the usual Bernoulli and Euler polynomials with their arithmetic and combinatorial interest. We note here that we use the orthogonality relations of the degenerate Stirling numbers in order to derive several corollaries from the obtained theorems. These include the discrete exponential functions, the discrete Stirling numbers of both kinds, the discrete logarithm function, the harmonic numbers and their generating function, the degenerate Bell polynomials, Jindalrae–Stirling numbers of both kinds, the Type 2 higher-order degenerate Euler polynomials, and the Type 2 higher-order Changhee polynomials. We introduce the generalized degenerate harmonic numbers and find an identity relating these numbers, the Type 2 higher-order Changhee polynomials, and the degenerate Stirling numbers of the first kind. From Equations (4) and (5), we can derive the generating functions of the degenerate Stirling numbers of both kinds which are given by (see [8]).

Discrete Harmonic Numbers and Related Polynomials and Numbers

Further Remark

Conclusions