On Arakawa–Kaneko zeta-functions associated with $GL_2(mathbb{C})$ and their functional relations

We construct a certain class of Arakawa Kaneko zeta functions associated with GL(2)(C), which includes the ordinary Arakawa Kaneko zeta-function. We also define poly-Bernoulli polynomials associated with GL(2)(C) which appear in their special values of these zeta-functions. We prove some functional relations for these zeta-functions, which are regarded as interpolation formulas of various relations among poly-Bernoulli numbers. Considering their special values, we prove difference relations and duality relations for poly-Bernoulli polynomials associated with GL(2)(C).