On generalized Arakawa-Kaneko zeta functions with parameters \(a\), \(b\), \(c\) (Q2205129)

Summary: For \(k\in\mathbb{Z}\), the generalized Arakawa-Kaneko zeta functions with \(a\), \(b\), \(c\) parameters are given by the Laplace-Mellin integral \[\xi_k(s,x;a,b,c)= (1/\Gamma (s)) \int_0^\infty (\mathrm{Li}_k (1-(ab)^{-t}) /b^t -a^{-t}) c^{-xt} t^{s-1} \,dt, \] where \(\Re(s)>0\) and \(x>0\) if \(k\geq 1\), and \(\Re(s)>0\) and \(x>|k|+1\) if \(k\leq 0\). In this paper, an interpolation formula between these generalized zeta functions and the poly-Bernoulli polynomials with \(a\), \(b\), \(c\) parameters is obtained. Moreover, explicit, difference, and Raabe's formulas for \(\xi_k(s,x;a,b,c)\) are derived.