Multiple zeta values, Padé approximation and Vasilyev's conjecture
نویسندگان
چکیده
منابع مشابه
Multiple Zeta Values, Padé Approximation and Vasilyev’s Conjecture
Sorokin gave in 1996 a new proof that π is transcendental. It is based on a simultaneous Padé approximation problem involving certain multiple polylogarithms, which evaluated at the point 1 are multiple zeta values equal to powers of π. In this paper we construct a Padé approximation problem of the same flavour, and prove that it has a unique solution up to proportionality. At the point 1, this...
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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: ANNALI SCUOLA NORMALE SUPERIORE - CLASSE DI SCIENZE
سال: 2016
ISSN: 2036-2145,0391-173X
DOI: 10.2422/2036-2145.201309_009