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

Let $f(t)={\sum }_{n=0}^{+\mathrm{\infty }}\frac{C_{f,n}}{n!}{t}^n$ be an analytic function at $0$, and let $C_{f,n}(x)={\sum }_{k=0}^n\left(\genfrac{}{}{0px}{}{n}{k}\right)C_{f,k}{x}^{n-k}$ be the sequence of Appell polynomials, referred to as $\mathit{\text{C-polynomials}}\mathit{\text{associated}}\mathit{\text{to}}\mathit{\text{f}}$, 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 $p_f(t)=f(t)(e^t-1)\mathrm{/}t$, called $\mathit{\text{P-polynomials}}\mathit{\text{associated}}\mathit{\text{to}}\mathit{\text{f}}$. 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 $P_f(s,z)={\sum }_{k=0}^{+\mathrm{\infty }}\left(\genfrac{}{}{0px}{}{z}{k}\right)P_{f,k}{s}^{z-k}$, 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 $L(z,f)={\sum }_{n=n_f}^{+\mathrm{\infty }}{n}^{(-z,f)}$, which are intrinsically linked to C-polynomials and referred to as $\mathit{\text{LC-functions}}\mathit{\text{associated}}\mathit{\text{to}}\mathit{\text{f}}$ (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.