Identities and recurrence relations of special numbers and polynomials of higher order by analysis of their generating functions

Yilmaz Simsek,Daeyeoul Kim

doi:10.1186/s13660-018-1815-7

Yilmaz Simsek, Daeyeoul Kim

Open Access

https://doi.org/10.1186/s13660-018-1815-7

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Abstract

The aim of this is to give generating functions for new families of special numbers and polynomials of higher order. By using these generating functions and their functional equations, we derive identities and relations for these numbers and polynomials. Relations between these new families of special numbers and polynomials and Bernoulli numbers and polynomials are given. Finally, recurrence relations and derivative formulas, which are related to these numbers and polynomials, are given.

Highlights

Many special numbers and polynomials have been found by researchers in recent years
We define a new family of special numbers and polynomials of higher order with their generating functions
We summarize the content of this paper as follows: In Sect. 2, relations between a new family of specials numbers and polynomials, Apostol–Bernoulli numbers and polynomials, and Bernoulli numbers and polynomials are given

Summary

Introduction

Many special numbers and polynomials have been found by researchers in recent years. These numbers and polynomials have various applications in mathematics and related areas. We define a new family of special numbers and polynomials of higher order with their generating functions. Motivation of this paper is to derive some identities and recurrence relations for these new families of special numbers and polynomials of higher order by using generating functions, functional equations, and partial derivative formulas. In order to derive the results of this paper, we need generating functions for new families of special numbers and polynomials.

Results

Conclusion