Wolfgang Gawronski

The Legendre-Stirling numbers

The Legendre-Stirling numbers are the coefficients in the integral Lagrangian symmetric powers of the classical Legendre second-order differential expression. In many ways, these numbers mimic the classical Stirling numbers of the second kind which play a similar role in the integral powers of the classical second-order Laguerre differential expression. In a recent paper, Andrews and Littlejohn gave a combinatorial interpretation of the Legendre-Stirling numbers. In this paper, we establish several properties of the Legendre-Stirling numbers; as with the Stirling numbers of the second kind, they have interesting generating functions and recurrence relations. Moreover, there are some surprising and intriguing results relating these numbers to some classical results in algebraic number theory.

The Jacobi-Stirling numbers

The Jacobi-Stirling numbers were discovered as a result of a problem involving the spectral theory of powers of the classical second-order Jacobi differential expression. Specifically, these numbers are the coefficients of integral composite powers of the Jacobi expression in Lagrangian symmetric form. Quite remarkably, they share many properties with the classical Stirling numbers of the second kind which are the coefficients of integral powers of the Laguerre differential expression. In this paper, we establish several properties of the Jacobi-Stirling numbers and its companions including combinatorial interpretations, thereby extending and supplementing known recent contributions to the literature.

Smoothing histograms by means of lattice-and continuous distributions

SummaryUsing lattice distributions or an auxiliary density function each satisfying certain moment conditions a general type of estimator for a one dimensional density functionf is developed. This estimator can be looked at as a smoothed histogram. As a measure of quality the exact order of magnitude for the mean squared error is established (pointwise and uniformly) in terms of the size of an iid sample drawn fromf and depending on a design parameter. The methods in deriving the asymptotic behaviour of the mean squared error are based on Edgeworth expansions for the auxiliary distributions.

STRONG ASYMPTOTICS AND THE ASYMPTOTIC ZERO DISTRIBUTIONS OF LAGUERRE POLYNOMIALS Ln(an+α) AND HERMITE POLYNOMIALS Ηn(an+α)

QueAtioni oi con&tnained on. weighted polynomial approximation an.e clo&ely related to orthogonal polynomials with degree dependent weight functions, and various of it& asymptotic. properties. Supplementing earlier and recent re&ulti strong asymptotia ion. the. generalized Laguerre and Hermite polynomials L^^(z), H^\z) axe derived, thai iL· asymptotic ionmi i f , ζ is in the interval oi zeros, [rn> sn] say, and ii ζ belongs to the cut plane

Strong asymptotics for Laguerre polynomials with varying weights

[rn> sn]. Thereby in particular generali.zatU.oni o{ the classical Planchenel-Rotach iormulae aAe established and the limit distribution oi the zeros is computed as the degn.ee η tend!) to infinity. AMS 1980 subject classification: 30 C 15, 33 A 65.

Strong laws for density estimators of bernstein type

Abstract Supplementing and extending classical and recent results strong asymptotics for the Laguerre polynomials L n ( α n ) are established, as n → ∞, when the parameter α n depends on the degree n suitably. A case of particular interest is the one for which α n grows faster than n . Rescaling the argument z appropriately the resulting asymptotic forms are described by elementary functions, thereby extending the classical formulae of Plancherel-Rotach type for Laguerre polynomials L n ( α ) ( z ), α being fixed.

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

Starting from the classical theorem of Weierstrass (and its modifications) on approximation of continuous functions by means of Bernstein polynomials a smoothed histogram type estimator is developed for estimating probability densities and its derivatives. Consistency results are obtained in form of various strong laws. In particular, one gets estimates for the rates for pointwise and uniform strong convergence of estimators for the derivatives. Moreover, for approximating the density itself the exact order of consistency is established. This is done by a law of iterated logarithm for pointwise approximation and by a law of logarithm in case of uniform approximation.

Complete monotonicity and zeros of sums of squared Baskakov functions

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.

Euler–Frobenius numbers

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.

On the zeros of Lerch's transcendental function of real parameters

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.