C-polynomials and LC-functions: towards a generalization of the Hurwitz zeta function

Lahcen Lamgouni

Published 2022-12-08Version 1

Let $f(t)=\sum_{n=0}^{+\infty}\frac{C_{f,n}}{n!}t^n$ be an analytic function at $0$, let $C_{f,n}(x)=\sum_{k=0}^{n} \binom{n}{k}C_{f,k} x^{n-k}$ be the sequence of Appell polynomials, which we call $\textit{C-polynomials associated to}$ $f$, constructed from the sequence of the coefficients $C_{f,n}$ of $f$, and let $P_{f,n}(x)$ be the sequence of C-polynomials associated to the function $p_{f}(t)=\frac{e^t-1}{t}f(t)$ which we call \textit{P-polynomials associated to $f$}. This work addresses three main topics. The first concerns the study of these two types of polynomials and the connection between them. In the second, inspired by the definition of the P-polynomials and under an additional condition on $f$, we introduce and study a function $P_{f}(s,z)$ of complex variables which generalizes the function $s^z$ and which we denote by $s^{(z,f)}$. In the third part we generalize the Hurwitz zeta function as well as its fundamental properties, the most remarkable being the Hurwitz's formula, by constructing a new class of functions $L(z,f)=\sum_{n=n_{f }}^{+\infty}n^{(-z,f)}$ related to the C-polynomials and which we call $\textit{LC-functions associated to}$ $f$ ($n_{f}$ being a positive integer depending on the choice of $f$).