On a \(p\)-adic interpolation function for the \(q\)-extension of the generalized Bernoulli polynomials and its derivative (Q1024475)

Let \(\chi\) be a Dirichlet character. In this paper, for \(q\in\mathbb{C}\) with \(|q|<1\) and \(h\in\mathbb{Z}\), the author introduces a \(q\)-extension \(B^{(h)}_{n,q,\chi}(x)\) of the generalized Bernoulli polynomial \(B_{n,\chi}(x)\) via a generating function and also a \(q\)-extension \(L^{(h)}_{q}(s,x|\chi)\) of the Dirichlet \(L\)-function \(L(s,x|\chi)\) of Hurwitz's type satisfying the condition \[ L^{(h)}_{q}(1-n,x|\chi)=-\frac{B^{(h)}_{n,q,\chi}(x)}{n}. \] Moreover, for a prime number \(p\), the author constructs a \(p\)-adic analogue \(L^{(h)}_{p,q}(s,x|\chi)\) of \(L^{(h)}_{q}(s,x|\chi)\) (in this case, \(q\) is suitably taken from \(\mathbb{C}_p\)) which interpolates \(B^{(h)}_{n,q,\chi}(x)\) at negative integers and, by introducing a \(q\)-extension of the Diamond gamma function via a \(p\)-adic \(q\)-integral, evaluates the derivative \(\frac{\partial}{\partial s}L^{(h)}_{p,q}(0,x|\chi)\) explicitly.