  Integral representation is one of the powerful tools for studying analytic
continuation of the zeta functions. It is known that Hurwitz zeta function
generalizes the famous Riemann zeta function which plays an important role in
analytic number theory. They both have several multiple versions in the
literature. In this paper, we introduce three novel multiple zeta functions for
Barnes type and study their integral representations through hyperbolic
probability distributions given by Pitman and Yor (2003, Canad. J. Math., 55,
292-330). The analytically continued properties of the three multiple zeta
functions are also investigated. Surprisingly, two of them, unlike the previous
results, can extend analytically to entire functions in the whole complex
plane.

Number Theory;Probability