648 research outputs found
A simple combinatorial proof of Shapiro's Catalan convolution
Shapiro proved an elegant convolution formula involving Catalan numbers of
even index. This paper gives a simple combinatorial proof of his formula. In
addition, we show that it is equivalent with the alternating convolution
formula of central binomial coefficients
Central Binomial Sums, Multiple Clausen Values and Zeta Values
We find and prove relationships between Riemann zeta values and central
binomial sums. We also investigate alternating binomial sums (also called
Ap\'ery sums). The study of non-alternating sums leads to an investigation of
different types of sums which we call multiple Clausen values. The study of
alternating sums leads to a tower of experimental results involving
polylogarithms in the golden ratio. In the non-alternating case, there is a
strong connection to polylogarithms of the sixth root of unity, encountered in
the 3-loop Feynman diagrams of {\tt hep-th/9803091} and subsequently in
hep-ph/9910223, hep-ph/9910224, cond-mat/9911452 and hep-th/0004010.Comment: 17 pages, LaTeX, with use of amsmath and amssymb packages, to appear
in Journal of Experimental Mathematic
Remarks on the Vietoris sequence and corresponding convolution formulas
In this paper we consider the so-called Vietoris sequence, a sequence of rational numbers of the form ck=12k(k⌊k2⌋), k= 0, 1, ⋯. This sequence plays an important role in many applications and has received a lot of attention over the years. In this work we present the main properties of the Vietoris sequence, having in mind its role in the context of hypercomplex function theory. Properties and patterns of the convolution triangles associated with (ck)k are also presented.Research at CMAT was partially financed by Portuguese funds through FCT - Funda¸c˜ao para a Ciˆencia e a
Tecnologia, within the Projects UIDB/00013/2020 and UIDP/00013/2020. Research at CIDMA has been
financed by FCT, within the Projects UIDB/04106/2020 and UIDP/04106/2020
The Euler and Springer numbers as moment sequences
I study the sequences of Euler and Springer numbers from the point of view of
the classical moment problem.Comment: LaTeX2e, 30 pages. Version 2 contains some small clarifications
suggested by a referee. Version 3 contains new footnotes 9 and 10. To appear
in Expositiones Mathematica
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