97 research outputs found
An Elliptic Analogue Of Generalized Cotangent Dirichlet Series And Its Transformation Formulae At Some Integer Arguments
B.C. Berndt evaluated special values of the cotangent Dirichlet series. T.
Arakawa studied a generalization of the series, or generalized cotangent
Dirichlet series, and gave its transformation formulae. In this paper, we
establish an elliptic analogue of the generalized cotangent Dirichlet series
and give its transformation formulae at some integer arguments. As a corollary,
we obtain the transformation formulae of the generalized cotangent Dirichlet
series at some integer arguments which are the part of Arakawa's transformation
formulae. Those transformation formulae give the special values of the
cotangent Dirichlet series evaluated by B.C. Berndt.Comment: 34page
Functional Equations and the Harmonic Relations for Multiple Zeta Values
Let denote Jacobi's theta function.
We show that the function
satisfies functional equations,
which is a generalization of the harmonic relations
for multiple zeta values
Elliptic Bernoulli Functions And Their Identities
We introduce an elliptic analogue of the Bernoulli functions, which we call elliptic Bernoulli functions. They are defined by using the modified generating function of the elliptic polylogarithms. By degeneration of the elliptic Bernoulli functions, we obtain standard properties and new identities for the Bernoulli functions
Sums of Products of Kronecker's Double Series
Closed expressions are obtained for sums of products of Kronecker's double series.
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
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