C-polynomials and LC-functions: towards a generalization of the Hurwitz zeta function (Q6624950)

The \(C\)-polynomials associated to \(f(t)=\sum_{n=0}^{\infty}\frac{C_{f,n}}{n!}t^{n}\) are defined by \N\[\NC_{f,n}(x)=\sum_{k=0}^{n}\binom{n}{k}C_{f,k}x^{n-k}.\N\]\NLet \(P_{f,n}(x)\) denotes the \(P\)-polynomials associated to \(f\) defined as the sequence of \(C\)-polynomials associated to the function \(P_{f}(t)=f(t)(e^{t}-1)/t\).\N\NIn this paper under review, the author first studies the properties of \(C\)-polynomials and \(P\)-polynomials. Second, he introduces and studies the bivariate complex 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 \(s^{(z, f )}\). Furthermore, he gives a generalization of the Hurwitz zeta function and its fundamental properties.