5,689 research outputs found
Evaluation of binomial double sums involving absolute values
We show that double sums of the form can always be
expressed in terms of a linear combination of just four functions, namely
, , , and , with
coefficients that are rational in . We provide two different proofs: one is
algorithmic and uses the second author's computer algebra package Sigma; the
second is based on complex contour integrals. In many instances, these results
are extended to double sums of the above form where is
replaced by with independent parameter .Comment: AmS-LaTeX, 42 pages; substantial revision: several additional and
more general results, see Proposition 11 and Theorems 15-1
Some binomial sums involving absolute values
We consider several families of binomial sum identities whose definition
involves the absolute value function. In particular, we consider centered
double sums of the form obtaining new results in the cases . We show that there is a close connection between these double sums in the
case and the single centered binomial sums considered by Tuenter.Comment: 15 pages, 19 reference
On Differences of Zeta Values
Finite differences of values of the Riemann zeta function at the integers are
explored. Such quantities, which occur as coefficients in Newton series
representations, have surfaced in works of Maslanka, Coffey, Baez-Duarte, Voros
and others. We apply the theory of Norlund-Rice integrals in conjunction with
the saddle point method and derive precise asymptotic estimates. The method
extends to Dirichlet L-functions and our estimates appear to be partly related
to earlier investigations surrounding Li's criterion for the Riemann
hypothesis.Comment: 18 page
Lattice Green functions in all dimensions
We give a systematic treatment of lattice Green functions (LGF) on the
-dimensional diamond, simple cubic, body-centred cubic and face-centred
cubic lattices for arbitrary dimensionality for the first three
lattices, and for for the hyper-fcc lattice. We show that there
is a close connection between the LGF of the -dimensional hypercubic lattice
and that of the -dimensional diamond lattice. We give constant-term
formulations of LGFs for all lattices and dimensions. Through a still
under-developed connection with Mahler measures, we point out an unexpected
connection between the coefficients of the s.c., b.c.c. and diamond LGFs and
some Ramanujan-type formulae for Comment: 30 page
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