  For these two decades, the Arakawa-Kaneko zeta function has been studied
actively. Recently Kaneko and Tsumura constructed its variants from the
viewpoint of poly-Bernoulli numbers. In this paper, we generalize their zeta
functions of Arakawa-Kaneko type to those with indices in which positive and
negative integers are mixed. We show that values of these functions at positive
integers can be expressed in terms of the multiple Hurwitz zeta star values.
