Integral expressions for Hilbert-type infinite multilinear form and related multiple Hurwitz-Lerch Zeta functions

The article deals with different kinds integral expressions concerning multiple Hurwitz– Lerch Zeta function (introduced originally by Barnes [4]), Hilbert–type infinite multilinear form and its power series extension. Here Laplace integral forms and multiple Mellin– Barnes type integral representation are derived for these special functions. As a special cases of our investigations we deduce the integral expressions for the Matsumoto’s multiple Mordell–Tornheim Zeta function, that is, for Tornheim’s double sum i.e. Mordell–Witten Zeta, for the multiple Hurwitz Zeta and for the multiple Hurwitz–Euler Eta function, recently studied by Choi and Srivastava [7].