Multiple zeta functions at regular integer points

We show recurrence relations for Euler–Zagier multiple zeta functions which describe the $r$-fold function with one variable specialized to a non-positive integer as a rational linear combination of $(r-1)$-fold functions, extending the previous results of Akiyama–Egami–Tanigawa and Matsumoto. As an application, we obtain an explicit method to calculate the special values of the multiple zeta function at any integer points (the arguments could be neither all-positive nor all-non-positive) as a rational linear combination of multiple zeta values.