Identities between polynomials related to Stirling and harmonic numbers

Abstract

We consider two types of polynomials $F_n (x) = \sum_{ν=1}^n ν! S_2(n,ν) x^ν$ and $\hat{F}_n (x) = \sum_{ν=1}^n ν! S_2(n,ν) H_νx^ν$, where $S_2(n,ν)$ are the Stirling numbers of the second kind and $H_ν$ are the harmonic numbers. We show some properties and relations between these polynomials. Especially, the identity $\hat{F}_n (-\tfrac{1}{2}) = - (n-1)/2 \cdot F_{n-1} (-\tfrac{1}{2})$ is established for even $n$, where the values are connected with Genocchi numbers. For odd $n$ the value of $\hat{F}_n (-\tfrac{1}{2})$ is given by a convolution of these numbers. Subsequently, we discuss some of these convolutions, which are connected with Miki type convolutions of Bernoulli and Genocchi numbers, and derive some 2-adic valuations of them.

Figures (2)

• Figure A.1: Functions $\mathbf{F}_n$
• Figure A.2: Functions $\widehat{\mathbf{F}}_n$

Theorems & Definitions (59)

• Lemma 1.1
• proof
• Theorem 1.2
• Theorem 1.3
• Theorem 1.5
• Remark
• Proposition 2.1: Worpitzky Worpitzky:1883
• proof
• Proposition 2.2
• proof