Linear differential equations and multiple zeta values. I. Zeta(2)
نویسندگان
چکیده
منابع مشابه
Formal differential operators, vertex operator algebras and zeta–values, I
We study relationships between spinor representations of certain Lie algebras and Lie superalgebras of differential operators on the circle and values of ζ–functions at the negative integers. By using formal calculus techniques we discuss the appearance of values of ζ–functions at the negative integers underlying the construction. In addition we provide a conceptual explanation of this phenomen...
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Multiple zeta values (MZVs, also called Euler sums or multiple harmonic series) are nested generalizations of the classical Riemann zeta function evaluated at integer values. The fact that an integral representation of MZVs obeys a shuue product rule allows the possibility of a combi-natorial approach to them. Using this approach we prove a longstanding conjecture of Don Zagier about MZVs with ...
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for any collection of positive integers s1, s2, . . . , sl. By definition, Lis(1) = ζ(s), s ∈ Z, s1 ≥ 2, s2 ≥ 1, . . . , sl ≥ 1. (4.2) Taking, as before for multiple zeta values, Lixs(z) := Lis(z), Li1(z) := 1, (4.3) let us extend action of the map Li : w 7→ Liw(z) by linearity on the graded algebra H (not H, since multi-indices are coded by words in H). Lemma 4.1. Let w ∈ H be an arbitrary non...
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It is now a good time to go back to the MZV story. where F (a, b; c; z) denotes the hypergeometric function and i = √ −1. Proof. Routine verification (with a help of Lemma 4.1 for the left-hand side) shows that the both sides of the required equality are annihilated by action of the differential operator (1 − z) d dz 2 z d dz 2 − t 4 ; in addition, the first terms in z-expansions of the both si...
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ژورنال
عنوان ژورنال: Fundamenta Mathematicae
سال: 2010
ISSN: 0016-2736,1730-6329
DOI: 10.4064/fm210-3-1