Abstract. We study the function ζ(k1, . . . , kn−1; s) =and show that the poly-Bernoulli numbers introduced in our previous paper are expressed as special values at negative arguments of certain combinations of these functions. As a consequence of our study, we obtain a series of relations among multiple zeta values.

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O. Introduction 0.!The so-called generalized eta-functions were studied by Lewittes [8] and Berndt [3,4] from a view point of transformation formulae. In the present paper we construct certain ray class invariants of real quadratic fields utilizing the generalized eta-functions and their transformation formulae. For the construction we employ the method of Ramachandra [9] for imaginary quadratic fields.On the other hand Stark 1-16-18] introduced certain ray class invariants for totally real fields in terms of the value at s = 0 of the derivative of some L-series of the fields and presented a striking conjecture on the arithmetic of the invariants. In his papers [14,15], Shintani represented the invariants as some products of certain natural special values of multiple gamma functions based upon his remarkable investigations [10][11][12]. He also proved the Stark conjecture for real quadratic fields in a special (but non-trivial) case in [14].One of our aim is to show that the Stark-Shintani ray class invariants for real quadratic fields can be represented as some products of our ray class invariants. 0.2We summarize our results. For an irrational real algebraic number e and a pair where we put e[w] = exp(2rciw) (wE IF). It is proved that the infinite series t/(~, s, p, q) is absolutely convergent if Re(s)<0, thanks to the Thue-Siegel-Roth theorem in the diophantine approximation theory. For a real number x, denote by (resp.

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