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Christian Berg

Publications and source records attributed to Christian Berg.

47 records · Page 3Linked to original sources
arXiv2008

Iteration of the rational function z-1/z and a Hausdorff moment sequence

In a previous paper we considered a positive function f, uniquely determined for s>0 by the requirements f(1)=1, log(1/f) is convex and the functional equation f(s)=psi(f(s+1)) with psi(s)=s-1/s. We prove that the meromorphic extension of f to the whole complex plane is given by the formula f(z)=lim_{n\to\infty}psi^{\circ n}(lambda_n(lambda_{n+1}/lambda_n)^z), where the numbers lambda_n are defined by lambda_0=0 and the recursion lambda_{n+1}=(1/2)(lambda_n+sqrt{lambda_n^2+4}). The numbers m_n=1/lambda_{n+1} form a Hausdorff moment sequence of a probability measure μsuch that \int t^{z-1}dμ(t)=1/f(z)

math.CV↗
arXiv2007

Bounds on Tur{á}n determinants

Let μdenote a symmetric probability measure on [-1,1] and let (p_n) be the corresponding orthogonal polynomials normalized such that p_n(1)=1. We prove that the normalized Tur{á}n determinant Δ_n(x)/(1-x^2), where Δ_n=p_n^2-p_{n-1}p_{n+1}, is a Tur{á}n determinant of order n-1 for orthogonal polynomials with respect to (1-x^2)dμ(x). We use this to prove lower and upper bounds for the normalized Tur{á}n determinant in the interval -1<x<1.

math.CA↗
arXiv2007

Quantum Hilbert matrices and orthogonal polynomials

Using the notion of quantum integers associated with a complex number $q\neq 0$, we define the quantum Hilbert matrix and various extensions. They are Hankel matrices corresponding to certain little $q$-Jacobi polynomials when $|q|<1$, and for the special value $q=(1-\sqrt{5})/(1+\sqrt{5})$ they are closely related to Hankel matrices of reciprocal Fibonacci numbers called Filbert matrices. We find a formula for the entries of the inverse quantum Hilbert matrix.

math.CA↗
arXiv2007

The fixed point for a transformation of Hausdorff moment sequences and iteration of a rational function

We study the fixed point for a non-linear transformation in the set of Hausdorff moment sequences, defined by the formula: $T((a_n))_n=1/(a_0+... +a_n)$. We determine the corresponding measure $μ$, which has an increasing and convex density on $]0,1[$, and we study some analytic functions related to it. The Mellin transform $F$ of $μ$ extends to a meromorphic function in the whole complex plane. It can be characterized in analogy with the Gamma function as the unique log-convex function on $]-1,\infty[$ satisfying $F(0)=1$ and the functional equation $1/F(s)=1/F(s+1)-F(s+1), s>-1$.

math.CA↗
arXiv2005

Linearization coefficients of Bessel polynomials

We prove positivity results about linearization and connection coefficients for Bessel polynomials. The proof is based on a recursion formula and explicit formulas for the coefficients in special cases. The result implies that the distribution of a convex combination of independent Student-t random variables with arbitrary odd degrees of freedom has a density which is a convex combination of certain Student-t densities with odd degrees of freedom.

math.PR↗
arXiv2004

On powers of Stieltjes moment sequences, II

We consider the set of Stieltjes moment sequences, for which every positive power is again a Stieltjes moment sequence, we and prove an integral representation of the logarithm of the moment sequence in analogy to the Lévy-Khinchin representation. We use the result to construct product convolution semigroups with moments of all orders and to calculate their Mellin transforms. As an application we construct a positive generating function for the orthonormal Hermite polynomials.

math.CA↗
arXiv2004

The Chen-Rubin conjecture in a continuous setting

We study the median in the Gamma distribution as a function of the parameter. Using methods from complex analysis we show that the derivative of the median is less than 1 and we thereby solve a conjecture of Chen and Rubin from 1986. We also study the asymptotic behaviour of the median at infinity and at zero.

math.CA↗
arXiv1999

Small eigenvalues of large Hankel matrices:The indeterminate case

In this paper we characterise the indeterminate case by the eigenvalues of the Hankel matrices being bounded below by a strictly positive constant. An explicit lower bound is given in terms of the orthonormal polynomials and we find expresions for this lower bound in a number of indeterminate moment problems.

math.CA↗