Identities between polynomials related to Stirling and harmonic numbers

Bernd C. Kellner

Published 2012-09-05, updated 2014-10-29Version 2

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.