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

Abstract

Let $f(t)=\sum_{n=0}^{+\infty}\frac{C_{f,n}}{n!}t^n$ be an analytic function at $0$, and let $C_{f, n}(x)=\sum_{k=0}^{n}\binom{n}{k}C_{f,k} x^{n-k}$ be the sequence of Appell polynomials, referred to as $\textit{C-polynomials associated to f}$, constructed from the sequence of coefficients $C_{f,n}$. We also define $P_{f,n}(x)$ as the sequence of C-polynomials associated to the function $p_{f}(t)=f(t)(e^t-1)/t$, called $\textit{P-polynomials associated to f}$. This work investigates three main topics. Firstly, we examine the properties of C-polynomials and P-polynomials and the underlying features that connect them. Secondly, drawing inspiration from the definition of P-polynomials and subject to an additional condition on $f$, we introduce and study the complex-variable function $P_{f}(s,z)=\sum_{k=0}^{+\infty}\binom{z}{k}P_{f,k}s^{z-k}$, which generalizes the $s^z$ function and is denoted by $s^{(z,f)}$. Thirdly, the paper's significant contribution is the generalization of the Hurwitz zeta function and its fundamental properties, most notably Hurwitz's formula, by constructing a novel class of functions defined by $L(z,f)=\sum_{n=n_{f}}^{+\infty}n^{(-z,f)}$, which are intrinsically linked to C-polynomials and referred to as $\textit{LC-functions associated to f}$ (the constant $n_{f}$ is a positive integer dependent on the choice of $f$). This research offers a detailed analysis of C-polynomials, P-polynomials, and LC-functions associated to a given analytic function $f$, thoroughly examining their interrelations and introducing unexplored research directions for a novel and expansive class of LC-functions possessing a functional equation equivalent to that of the Riemann zeta function, thereby highlighting the potential applications and implications of the findings.