Integrals of logarithmic functions and alternating multiple zeta values
نویسندگان
چکیده
منابع مشابه
Euler-type Multiple Integrals as Linear Forms in Zeta Values
0. In 1978, Apéry showed the irrationality of ζ(3) = ∑∞ n=1 1 n3 by giving the approximants `n = unζ(3) − vn ∈ Qζ(3) + Q, un, dnvn ∈ Z, dn = l.c.m.(1, 2, . . . , n), with the property |`n| → ( √ 2 − 1) < 1/e as n → ∞. A similar approach was put forward to show the irrationality of ζ(2) (which is π/6, hence transcendental thanks to Lindemann) but I will concentrate on the case of ζ(3). A few mon...
متن کاملQuasi-symmetric functions, multiple zeta values, and rooted trees
The algebra Sym of symmetric functions is a proper subalgebra of QSym: for example, M11 and M12 +M21 are symmetric, but M12 is not. As an algebra, QSym is generated by those monomial symmetric functions corresponding to Lyndon words in the positive integers [11, 6]. The subalgebra of QSym ⊂ QSym generated by all Lyndon words other than M1 has the vector space basis consisting of all monomial sy...
متن کاملAspectsof Multiple Zeta Values
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 ...
متن کاملMultiple Zeta Values
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...
متن کاملMultiple Zeta Values 17
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...
متن کاملذخیره در منابع من
با ذخیره ی این منبع در منابع من، دسترسی به آن را برای استفاده های بعدی آسان تر کنید
ژورنال
عنوان ژورنال: Mathematica Slovaca
سال: 2019
ISSN: 0139-9918,1337-2211
DOI: 10.1515/ms-2017-0227