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 \(C_{f, n}(x)=\sum _{k=0}^{n}\left( {\begin{array}{c}n\\ k\end{array}}\right) C_{f,k} x^{n-k}\) be the sequence of Appell polynomials, referred to as 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 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 bivariate complex function \(P_{f}(s,z)=\sum _{k=0}^{+\infty }\left( {\begin{array}{c}z\\ k\end{array}}\right) P_{f,k}s^{z-k}\), which generalizes the \(s^z\) function and is denoted by \(s^{(z,f)}\). Thirdly, the paper’s main 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 LC-functions associated to f (the constant \(n_{f}\) is a positive integer dependent on the choice of f).