Inequalities for ratios of Gamma functions
We prove that the function $$ \log \frac{Γ(x+a+b)Γ(x)}{Γ(x+a)Γ(x+b)}-b\log\left(1+\frac{a}{x}\right) $$ is a generalized Stieltjes function of order $3$ for $a>0$ and $0 0$ and $b>1$.
math.GM↗
arXiv subjects
Henrik Laurberg Pedersen
Publications and source records attributed to Henrik Laurberg Pedersen.
On the entropy of the log-normal distribution
In a paper by Novi Inverardi and Tagliani from 2024 it is claimed that the log-normal density has maximal entropy in the class of densities having the same moments as the log-normal density. We disprove the claim by finding a family of densities with larger entropy.
math.PR↗
Special Functions from a Complex Viewpoint
This document contains the lecture notes for a mini-course on special functions from a complex viewpoint given at the OPSFA Summer School OPSFA-S10 2024, in the period July 29th -- August 2nd, 2024. The summer school was hosted by Section of Mathematics -- Luciano Modica at the Uninettuno University.
math.CA↗
Approximations of generalized Bernstein functions
We establish sharp inequalities involving the incomplete Beta and Gamma functions. These inequalities arise in the approximation of generalized Bernstein functions by higher order Thorin-Bernstein functions. Furthermore, new properties of a related function, namely $x^λΓ(x)/Γ(x+λ)$ are derived.
math.CA↗
Higher order Thorin-Bernstein Functions
We investigate subclasses of generalized Bernstein functions related to complete Bernstein and Thorin-Bernstein functions. Representations in terms of incomplete beta and gamma as well as hypergeometric functions are presented. Several special cases and examples are discussed.
math.CA↗
On the asymptotic behaviour of the quantiles in the gamma distribution
The asymptotic behaviour of the quantiles in the gamma distribution are investigated as the shape parameter tends to zero. Some remarks about the behaviour at infinity are given.
math.PR↗
Filled Julia sets of Chebyshev polynomials
We study the possible Hausdorff limits of the Julia sets and filled Julia sets of subsequences of the sequence of dual Chebyshev polynomials of a non-polar compact set K in C and compare such limits to K. Moreover, we prove that the measures of maximal entropy for the sequence of dual Chebyshev polynomials of K converges weak* to the equilibrium measure on K.
math.DS↗
Julia sets of Orthogonal polynomials
For a probability measure with compact and non-polar support in the complex plane we relate dynamical properties of the associated sequence of orthogonal polynomials $\{P_n\}$ to properties of the support. More precisely we relate the Julia set of $P_n$ to the outer boundary of the support, the filled Julia set to the polynomial convex hull $K$ of the support, and the Green's function associated with $P_n$ to the Green's function for the complement of $K$.
math.CV↗