1,823 research outputs found
Multiple harmonic sums and Wolstenholme's theorem
We give a family of congruences for the binomial coefficients in terms of multiple harmonic sums, a generalization of the harmonic
numbers. Each congruence in this family (which depends on an additional
parameter ) involves a linear combination of multiple harmonic sums, and
holds . The coefficients in these congruences are integers
depending on and , but independent of . More generally, we construct
a family of congruences for , whose members
contain a variable number of terms, and show that in this family there is a
unique "optimized" congruence involving the fewest terms. The special case
and recovers Wolstenholme's theorem , valid for all primes . We also characterize those triples
for which the optimized congruence holds modulo an extra power of
: they are precisely those with either dividing the numerator of the
Bernoulli number , or .Comment: 22 page
Families of weighted sum formulas for multiple zeta values
Euler's sum formula and its multi-variable and weighted generalizations form
a large class of the identities of multiple zeta values. In this paper we prove
a family of identities involving Bernoulli numbers and apply them to obtain
infinitely many weighted sum formulas for double zeta values and triple zeta
values where the weight coefficients are given by symmetric polynomials. We
give a general conjecture in arbitrary depth at the end of the paper.Comment: The conjecture at the end is reformulate
On Multiple Zeta Values of Even Arguments
For k <= n, let E(2n,k) be the sum of all multiple zeta values with even
arguments whose weight is 2n and whose depth is k. Of course E(2n,1) is the
value of the Riemann zeta function at 2n, and it is well known that E(2n,2) =
(3/4)E(2n,1). Recently Z. Shen and T. Cai gave formulas for E(2n,3) and
E(2n,4). We give two formulas form E(2n,k), both valid for arbitrary k <=n, one
of which generalizes the Shen-Cai results; by comparing the two we obtain a
Bernoulli-number identity. We also give explicit generating functions for the
numbers E(2n,k) and for the analogous numbers E*(2n,k) defined using multiple
zeta-star values of even arguments.Comment: DESY number added; misprints fixed; reference added. Second revision
(2016): New result on multiple zeta-star values adde
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