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

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