Laurent series expansions of multiple zeta-functions of Euler-Zagier type at integer points

Values at non-positive integers of generalized Euler–Zagier multiple zeta-functions

“…Moreover, it is known that for n ≥ 2, almost all n−tuples of non-positive integers lie on the singular locus above and are points of discontinuity (see [1], Th.1). The evaluation of (limit) values of multiple zeta-functions at those points was first considered by S. Akiyama, S. Egami and Y. Tanigawa [1], and then studied further by [2], [19], [20], [14], [18], and [17].…”

“…Deduction of Corollary 1 from Proposition 1: The corollary follows from point 2 of Proposition 1 by applying Cauchy's formula which expresses the coefficients of Laurent's expansion of a given one variable meromorphic function in terms of its integrals on small disks around its singular point. The identity (17) follows by using in addition the equality (16).…”

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

“…Next let r ≥ 1. It is enough to prove that the power series (7) converges in a neighbourhood of (1, . .…”

“…, ρ r ) such that ρ 1 + • • •+ ρ r < 1. We deduce from formula (12) and Lemma 2 (for K = D, A = 0 and k 0 = 1) that, for N ≥ 2, the function and of the formal power series (s 1 − 1)v, where v is the formal power series (7). This implies that the formal power series (s 1 − 1)v converges on D. Hence v converges on D and if ζ Reg (1,...,1) (s 1 , .…”

“…Remark 10. A statement formally equivalent to Corollary 1, has also been stated in [7,Theorem 1.3]. But it does not imply Theorem 7.…”