16 research outputs found
The Evaluation of the Sums of More General Series by Bernstein Polynomials
Let n,k be the positive integers, and let S_{k}(n) be the sums of the k-th
power of positive integers up to n. By means of that we consider the evaluation
of the sum of more general series by Bernstein polynomials. Additionally we
show the reality of our idea with some examples.Comment: 6 pages, submitte
Multivariate p-dic L-function
We construct multivariate p-adic L-function in the p-adic number fild by
using Washington method.Comment: 9 page
Faulhaber's Theorem on Power Sums
We observe that the classical Faulhaber's theorem on sums of odd powers also
holds for an arbitrary arithmetic progression, namely, the odd power sums of
any arithmetic progression is a polynomial in
. While this assertion can be deduced from the original
Fauhalber's theorem, we give an alternative formula in terms of the Bernoulli
polynomials. Moreover, by utilizing the central factorial numbers as in the
approach of Knuth, we derive formulas for -fold sums of powers without
resorting to the notion of -reflexive functions. We also provide formulas
for the -fold alternating sums of powers in terms of Euler polynomials.Comment: 12 pages, revised version, to appear in Discrete Mathematic
Combinatorial Interpretations of the q-Faulhaber and q-Salie Coefficients
Recently, Guo and Zeng discovered two families of polynomials featuring in a
q-analogue of Faulhaber's formula for the sums of powers and a q-analogue of
Gessel-Viennot's formula involving Salie's coefficients for the alternating
sums of powers. In this paper, we show that these are polynomials with
symmetric, nonnegative integral coefficients by refining Gessel-Viennot's
combinatorial interpretations.Comment: 15 page