Multiple Stieltjes constants and Laurent type expansion of the multiple zeta functions at integer points

Abstract: In this article, we study the local behaviour of the multiple zeta functions at integer points and write down a Laurent type expansion of the multiple zeta functions around these points. Such an expansion involves a convergent power series whose coefficients are obtained by a regularisation process, similar to the one used in defining the classical Stieltjes constants for the Riemann zeta function. We therefore call these coefficients {\it multiple Stieltjes constants}. The remaining part of the above mentioned Laurent type expansion is then expressed in terms of the multiple Stieltjes constants arising in smaller depths.