Powers under umbral composition and degeneration for Sheffer sequences

Dae San Kim,Toufik Mansour,Hyuck In Kwon,Taekyun Kim

doi:10.1186/s13662-016-0795-5

Abstract

In this paper we study the powers under umbral composition and degeneration for Sheffer sequences, where we presented several applications related to Bernoulli polynomials, Frobenius-Euler polynomials, falling factorial polynomials and Bell polynomials and their degeneration cases.

Highlights

The aim of this paper is to use umbral calculus and to study powers under umbral composition and degeneration for Sheffer sequences
Umbral calculus is considered for some special Sheffer polynomials such as Bell polynomials, Bernoulli polynomials, Frobenius-Euler polynomials, Korobov polynomials, degenerate Bernoulli polynomials, and falling factorial polynomials
The main goal of this paper is to study the powers under umbral composition and degeneration for Sheffer sequences

Summary

Introduction

The aim of this paper is to use umbral calculus and to study powers under umbral composition and degeneration for Sheffer sequences. For each nonnegative integer m, the mth power of an invertible series g(t) will be indicated by (g(t))m, while the compositional powers of a delta series f (t) will be denoted by f m(t) = f ◦ f ◦ · · · ◦ f (t). The main goal of this paper is to study the powers under umbral composition and degeneration for Sheffer sequences. Observe here that the notation rn(m)(x) for the mth power of rn(x) under umbral composition agrees with that for mth order polynomial of rn(x). Let us take the Frobenius-Euler polynomials Hn(α)(x|λ) of order α, = λ ∈ C (see [ – ]).

Powers under umbral composition applied to associated sequences

Degenerate Bell polynomials

Degenerate falling factorial polynomials

Generalized Bell polynomials