Identities associated with Milne\u2013Thomson type polynomials and special numbers

Yilmaz Simsek,Nenad Cakic

doi:10.1186/s13660-018-1679-x

Yilmaz Simsek, Nenad Cakic

Open Access

https://doi.org/10.1186/s13660-018-1679-x

Copy DOI

Abstract

The purpose of this paper is to give identities and relations including the Milne–Thomson polynomials, the Hermite polynomials, the Bernoulli numbers, the Euler numbers, the Stirling numbers, the central factorial numbers, and the Cauchy numbers. By using fermionic and bosonic p-adic integrals, we derive some new relations and formulas related to these numbers and polynomials, and also the combinatorial sums.

Highlights

Many authors have studied special numbers and polynomials with their generating functions
For combinatorial interpretations of these special numbers and polynomials with their generating functions see for details [1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31], and the references therein
We summarize the results of this paper as follows: In Sect. 2, by using generating functions and their functional equations, we give some identities including the three-variable polynomials y6(n; x, y, z; a, b, v), the Hermite polynomials, the array polynomials and the Stirling numbers of the second kind

Summary

Introduction

Many authors have studied special numbers and polynomials with their generating functions Because these special numbers and polynomials including the Bernoulli numbers and polynomials, the Euler numbers and polynomials, the Stirling numbers, the Milne–Thomson numbers and polynomials, the Hermite numbers and polynomials, Central factorial numbers, Cauchy numbers, and the others have many applications in mathematics, and in other related areas. It is well-known that there are many combinatorial interpretations of these special numbers especially, the Stirling numbers and the central factorial numbers in partition theory, in set theory, in probability theory and in other sciences.

Objectives

Conclusion