On Harmonic Geometric Polynomials and Numbers
33èmes Journées Arithmétiques, Luxembourg, Luxembourg, 30 June - 04 July 2025, pp.46, (Summary Text)
- Publication Type: Conference Paper / Summary Text
- City: Luxembourg
- Country: Luxembourg
- Page Numbers: pp.46
- Akdeniz University Affiliated: Yes
Abstract
We obtain new recurrence relations, explicit formulas, and convolution identities for harmonic geometric polynomials $H w_n(x)$ defined by
\[
H w_n(x)=\sum_{k=1}^{n}
\left\{\begin{matrix}
n\\
k
\end{matrix}\right\}
k!H_k x^k,
\]
where
\[
\left\{\begin{matrix}
n\\
k
\end{matrix}\right\}
\]
denotes the Stirling number of the second kind and $H_n$ is the $n$th harmonic number. Moreover, we relate these polynomials to Bernoulli and Euler numbers, and through these relations, we obtain new explicit representations and recurrence relations for Bernoulli and Euler numbers.