Recent zbMATH articles in MSC 26Chttps://www.zbmath.org/atom/cc/26C2021-04-16T16:22:00+00:00WerkzeugOn the absolute continuity of random nodal volumes.https://www.zbmath.org/1456.601362021-04-16T16:22:00+00:00"Angst, Jürgen"https://www.zbmath.org/authors/?q=ai:angst.jurgen"Poly, Guillaume"https://www.zbmath.org/authors/?q=ai:poly.guillaumeSummary: We study the absolute continuity with respect to the Lebesgue measure of the distribution of the nodal volume associated with a smooth, nondegenerate and stationary Gaussian field \((f(x), x \in \mathbb{R}^d)\). Under mild conditions, we prove that in dimension \(d \geq 3\), the distribution of the nodal volume has an absolutely continuous component plus a possible singular part. This singular part is actually unavoidable bearing in mind that some Gaussian processes have a positive probability to keep a constant sign on some compact domain. Our strategy mainly consists in proving closed Kac-Rice type formulas allowing one to express the volume of the set \(\{f=0\}\) as integrals of explicit functionals of \((f, \nabla f, \text{Hess}(f))\) and next to deduce that the random nodal volume belongs to the domain of a suitable Malliavin gradient. The celebrated Bouleau-Hirsch criterion then gives conditions ensuring the absolute continuity.The irreducibility of some Wronskian Hermite polynomials.https://www.zbmath.org/1456.112062021-04-16T16:22:00+00:00"Grosu, Codruţ"https://www.zbmath.org/authors/?q=ai:grosu.codrut"Grosu, Corina"https://www.zbmath.org/authors/?q=ai:grosu.corinaSummary: We study the irreducibility in \(\mathbb{Z}[x]\) of Wronskian Hermite polynomials labelled by partitions. It is known that these polynomials factor as a power of \(x\) times a remainder polynomial. We show that the remainder polynomial is irreducible for the partitions \((n,m)\) with \(m\leq 2\), and \((n,n)\) when \(n+1\) is a square.
Our main tools are two theorems that we prove for all partitions. The first result gives a sharp upper bound for the slope of the edges of the Newton polygon for the remainder polynomial. The second result is a Schur-type congruence for Wronskian Hermite polynomials.
We also explain how irreducibility determines the number of real zeros of Wronskian Hermite polynomials, and prove Veselov's conjecture for partitions of the form \((n,k,k-1,\ldots,1)\).Identities and relations for Hermite-based Milne-Thomson polynomials associated with Fibonacci and Chebyshev polynomials.https://www.zbmath.org/1456.050072021-04-16T16:22:00+00:00"Kilar, Neslihan"https://www.zbmath.org/authors/?q=ai:kilar.neslihan"Simsek, Yilmaz"https://www.zbmath.org/authors/?q=ai:simsek.yilmazSummary: The aim of this paper is to give many new and interesting identities, relations, and combinatorial sums including the Hermite-based Milne-Thomson type polynomials, the Chebyshev polynomials, the Fibonacci-type polynomials, trigonometric type polynomials, the Fibonacci numbers, and the Lucas numbers. By using Wolfram Mathematica version 12.0, we give surfaces graphics and parametric plots for these polynomials and generating functions. Moreover, by applying partial derivative operators to these generating functions, some derivative formulas for these polynomials are obtained. Finally, suitable connections of these identities, formulas, and relations of this paper with those in earlier and future studies are designated in detail remarks and observations.