Generalized log sine integrals and the Mordell-Tornheim zeta values
نویسندگان
چکیده
منابع مشابه
Generalized Log Sine Integrals and the Mordell-tornheim Zeta Values
We introduce certain integrals of a product of the Bernoulli polynomials and logarithms of Milnor’s multiple sine functions. It is shown that all the integrals are expressed by the Mordell-Tornheim zeta values at positive integers and that the converse is also true. Moreover, we apply the theory of the integral to obtain various new results for the Mordell-Tornheim zeta values.
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We prove that the Mordell-Tornheim zeta value of depth r can be expressed as a rational linear combination of products of the Mordell-Tornheim zeta values of lower depth than r when r and its weight are of different parity.
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In this paper the authors present several algorithmic formulas which are potentially useful in computing the following Mordell-Tornheim zeta values: ζMT,r(s1, · · · , sr ; s) := ∞ ∑ m1, ··· ,mr=1 1 m1 1 · · ·m sr r (m1 + · · ·+mr)s for the special cases ζMT,r(1, · · · , 1; s) and ζMT,r(0, · · · , 0; s). Some interesting (known or new) consequences and illustrative examples are also considered.
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RÉSUMÉ. Nous prouvons que toute somme de Mordell-Tornheim avec des arguments entiers positifs peut s’écrire comme une combinaison linéaire rationnelle de valeurs prises par des fonctions multi-zêta ayant le même poids et la même profondeur. Selon un résultat de Tsumura, il s’ensuit que toute somme de Mordell-Tornheim ayant un poids et une profondeur de parité différente peut s’exprimer comme un...
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ژورنال
عنوان ژورنال: Transactions of the American Mathematical Society
سال: 2011
ISSN: 0002-9947
DOI: 10.1090/s0002-9947-2010-05176-1