Identities between polynomials related to Stirling and harmonic numbers

TL;DR

This paper explores properties and identities of polynomials involving Stirling and harmonic numbers, establishing new relations and connections with Genocchi numbers, and analyzing their convolutions and 2-adic valuations.

Contribution

It introduces new identities between polynomials related to Stirling and harmonic numbers, including a key relation involving Genocchi numbers and convolution formulas.

Findings

Established an identity linking $ ilde{F}_n(-1/2)$ and $F_{n-1}(-1/2)$ for even $n$

Connected polynomial values with Genocchi numbers and convolutions

Derived 2-adic valuations of convolutions involving Bernoulli and Genocchi numbers

Abstract

We consider two types of polynomials $F_n (x) = \sum_{\nu=1}^n \nu! S_2(n,\nu) x^\nu$ and $\hat{F}_n (x) = \sum_{\nu=1}^n \nu! S_2(n,\nu) H_\nu x^\nu$, where $S_2(n,\nu)$ are the Stirling numbers of the second kind and $H_\nu$ 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.