Thorsten Neuschel

Plancherel-Rotach formulae for average characteristic polynomials of products of Ginibre random matrices and the Fuss-Catalan distribution

Formulae of Plancherel-Rotach type are established for the average characteristic polynomials of certain Hermitian products of rectangular Ginibre random matrices on the region of zeros. These polynomials form a general class of multiple orthogonal hypergeometric polynomials generalizing the classical Laguerre polynomials. The proofs are based on a multivariate version of the complex method of saddle points. After suitable rescaling the asymptotic zero distributions for the polynomials are studied and shown to coincide with the Fuss-Catalan distributions. Moreover, introducing appropriate coordinates, elementary and explicit characterizations are derived for the densities as well as for the distribution functions of the Fuss-Catalan distributions of general order.

Asymptotics for characteristic polynomials of Wishart type products of complex Gaussian and truncated unitary random matrices

Based on the multivariate saddle point method we study the asymptotic behavior of the characteristic polynomials associated to Wishart type random matrices that are formed as products consisting of independent standard complex Gaussian and a truncated Haar distributed unitary random matrix. These polynomials form a general class of hypergeometric functions of type 2 F r . We describe the oscillatory behavior on the asymptotic interval of zeros by means of formulae of Plancherel-Rotach type and subsequently use it to obtain the limiting distribution of the suitably rescaled zeros. Moreover, we show that the asymptotic zero distribution lies in the class of Raney distributions and by introducing appropriate coordinates elementary and explicit characterizations are derived for the densities as well as for the distribution functions.

On the asymptotic normality of the Legendre-Stirling numbers of the second kind

For the Legendre-Stirling numbers of the second kind asymptotic formulae are derived in terms of a local central limit theorem. Thereby, supplements of the recently published asymptotic analysis of the Chebyshev-Stirling numbers are established. Moreover, we provide results on the asymptotic normality and unimodality for modified Legendre-Stirling numbers.

Complete monotonicity and zeros of sums of squared Baskakov functions

We prove complete monotonicity of sums of squares of generalized Baskakov basis functions by deriving the corresponding results for hypergeometric functions. Moreover, in the central Baskakov case we study the distribution of the complex zeros for large values of a parameter. We finally discuss the extension of some results for sums of higher powers.

Euler–Frobenius numbers

These numbers are defined as the coefficients of the Euler–Frobenius polynomials which usually are introduced via the rational function expansion n being a nonnegative integer and λ∈[0, 1). The special case An, l (0) is known from combinatorics (Eulerian numbers) and the general one An, l (λ) occurs, for example, in approximation theory, summability, and rounding error analysis. By supplementing and extending known results on Eulerian numbers, various theorems for the Euler–Frobenius numbers An, l(λ) and related quantities are established including unimodality, monotonicity properties, and asymptotic expansions given by a local central limit theorem.

Asymptotic zero distribution of Jacobi-Piñeiro and multiple Laguerre polynomials

We give the asymptotic distribution of the zeros of Jacobi-Pineiro polynomials and multiple Laguerre polynomials of the first kind. We use the nearest neighbor recurrence relations for these polynomials and a recent result on the ratio asymptotics of multiple orthogonal polynomials. We show how these asymptotic zero distributions are related to the Fuss-Catalan distribution.

On a conjecture on sparse binomial-type polynomials by Brown, Dilcher and Manna

We prove a conjecture by Brown, Dilcher and Manna on the asymptotic behavior of sparse binomial-type polynomials arising naturally in a graph-theoretical context in connection with the expected number of independent sets of a graph.