Hankel Determinants of Random Moment Sequences

For \(t \in [0,1]\) let \(\underline{H}_{2\lfloor nt \rfloor } = (m_{i+j})_{i,j=0}^{\lfloor nt \rfloor }\) denote the Hankel matrix of order \(2\lfloor nt \rfloor \) of a random vector \((m_1,\ldots ,m_{2n})\) on the moment space \(\mathcal {M}_{2n}(I)\) of all moments (up to the order 2n) of probability measures on the interval \(I \subset \mathbb {R}\). In this paper we study the asymptotic properties of the stochastic process \(\{ \log \det \underline{H}_{2\lfloor nt \rfloor } \}_{t\in [0,1]}\) as \(n \rightarrow \infty \). In particular weak convergence and corresponding large deviation principles are derived after appropriate standardization.     

Third Hankel Determinants for Subclasses of Univalent Functions

The main aim of this paper is to discuss the third Hankel determinants for three classes: \(S^*\) of starlike functions, \(\mathcal {K}\) of convex functions and \(\mathcal {R}\) of functions whose derivative has a positive real part. Moreover, the sharp results for twofold and threefold symmetric functions from these classes are obtained.     

On New Arithmetic Properties of Determinants of Hankel Matrices

We construct an infinite-dimensional Hankel matrix H_∞ with elements of the form m/2^l, where m, l ∈ ?, with the following property: the divisors of the numerators of its principal minors contain all prime numbers.     

Compact Hankel Operators on Generalized Bergman Spaces of the Polydisc

Let ${\vartheta}$ be a measure on the polydisc ${\mathbb{D}^n}$ which is the product of n regular Borel probability measures so that ${\vartheta([r,1)^n\times\mathbb{T}^n) >0 }$ for all 0 < r < 1. The Bergman space ${A^2_{\vartheta}}$ consists of all holomorphic functions that are square integrable with respect to ${\vartheta}$ . In one dimension, it is well known that if f is continuous on the closed disc ${\overline{\mathbb{D}}}$ , then the Hankel operator H _f is compact on ${A^2_\vartheta}$ . In this paper we show that for n ≥ 2 and f a continuous function on ${{\overline{\mathbb{D}}}^n}$ , H _f is compact on ${A^2_\vartheta}$ if and only if there is a decomposition f = h + g, where h belongs to ${A^2_\vartheta}$ and ${\lim_{z\to\partial\mathbb{D}^n}g(z)=0}$ .     

Hankel Determinants of a Linear Combination of Three Successive Catalan Numbers

We evaluate the determinants of Hankel matrices, whose elements are a linear combination of three successive shifted Catalan numbers. This is done by finding a Jacobi linear functional, such that their moments are, up to a multiplicative constant, the Catalan numbers. The values of such determinants are then expressed in terms of Jacobi polynomials.     

Prime Divisors of Shifted Factorials

For any positive integer n we let P(n) be the largest primefactor of n. We improve and generalize several results of P.Erds and C. Stewart on P(n!+1). In particular, we show thatlim sup_ninfin; P(n!+1)/n2.5, which improves their lower boundof lim sup_ninfin; P(n!+1)/n2. 2000 Mathematics Subject Classification11A05, 11A07, 11J86.     

A Congruence for Factorials

The methods of p-adic analysis are used to prove a congruencefor (pn)!(pⁿn!) modulo a power of a prime p.