Abstract. Extended double shuffle relations for multiple zeta values are obtained by using the fact that any product of regularized multiple zeta values has two different representations. In this paper, we give two formulas for the generating function of the triple zeta values of any fixed weight by the use of the extended double shuffle relations obtained as two-fold products of double and single zeta values and also as three-fold products of single zeta values. As applications of the formulas, we also obtain parameterized, weighted, and restricted sum formulas for triple zeta values.

In this paper we introduce an elliptic analogue of the generalized Dedekind-Rademacher sums which satisfy reciprocity laws. In these sums, Kronecker's double series play a role of elliptic Bernoulli functions. This paper gives an answer to the problem of S. Fukuhara and N. Yui concerning the elliptic ApostolDedekind sums. We also mention a relation between the generating function of Kronecker's double series and that of the (Debye) elliptic polylogarithms studied by A. Levin.

We give some restricted sum formulas for double zeta values whose arguments satisfy certain congruence conditions modulo 2 or 6, and also give an application to identities showed by Ramanujan for sums of products of Bernoulli numbers with a gap of 6.Comment: ver.

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Closed expressions are obtained for sums of products of Kronecker's double series of the form, where the summation ranges over all nonnegative integers j 1 , . . . , j N with j 1 + · · · + j N = n. Corresponding results are derived for functions which are an elliptic analogue of the periodic Euler polynomials. As corollaries, we reproduce the formulas for sums of products of Bernoulli numbers, Bernoulli polynomials, Euler numbers, and Euler polynomials, which were given by K. Dilcher.