1,717 research outputs found
A CMV--based eigensolver for companion matrices
In this paper we present a novel matrix method for polynomial rootfinding. By
exploiting the properties of the QR eigenvalue algorithm applied to a suitable
CMV-like form of a companion matrix we design a fast and computationally simple
structured QR iteration.Comment: 14 pages, 4 figure
The modular properties and the integral representations of the multiple elliptic gamma functions
We show the modular properties of the multiple 'elliptic' gamma functions,
which are an extension of those of the theta function and the elliptic gamma
function. The modular property of the theta function is known as Jacobi's
transformation, and that of the elliptic gamma function was provided by Felder
and Varchenko. In this paper we deal with the multiple sine functions, since
the modular properties of the multiple elliptic gamma functions result from the
equivalence between two ways to represent the multiple sine functions as
infinite product.
We also derive integral representations of the multiple sine functions and
the multiple elliptic gamma functions. We introduce correspondences between the
multiple elliptic gamma functions and the multiple sine functions.Comment: 20 pages, LaTeX, To appear in "Adv. in Math.
On equal values of power sums of arithmetic progressions
In this paper we consider the Diophantine equation \begin{align*}b^k
+\left(a+b\right)^k &+ \cdots + \left(a\left(x-1\right) + b\right)^k=\\ &=d^l +
\left(c+d\right)^l + \cdots + \left(c\left(y-1\right) + d\right)^l,
\end{align*} where are given integers. We prove that, under some
reasonable assumptions, this equation has only finitely many integer solutions.Comment: This version differs slightly from the published version in its
expositio
Denominators of Bernoulli polynomials
For a positive integer let where
runs over all primes and is the sum of the base digits of .
For all we prove that is divisible by all "small" primes
with at most one exception. We also show that is large, has
many prime factors exceeding , with the largest one exceeding
. We establish Kellner's conjecture, which says that the number of
prime factors exceeding grows asymptotically as for some constant with . Further, we
compare the sizes of and , leading to the
somewhat surprising conclusion that although tends to infinity
with , the inequality is more frequent
than its reverse.Comment: 25 page
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