Produit Beta-Gamma et régularité du signe

Thomas Simon (1)

1 LPP - Laboratoire Paul Painlevé - UMR 8524

We study the total positivity of the multiplicative convolution kernel T associated with the independent product of two random variables ${\bf B}(a,b)$ and ${\bf \Gamma}(c).$ This kernel is totally positive of infinite order if $b$ or $d = a+b -c$ are integers. Otherwise the sign-regularity of T has always a finite order, which is here computed. More precisely, for every $n\ge 1$ it is shown that T is totally positive of order $n + 1$ if and only if $(d,b)$ lies above a certain stairway ${\mathcal E}_n$ plotted in the upper half-plane. This stairway also characterizes the sign-invariance of several determinants associated with the confluent hypergeometric function of the second kind.

Mots clés: Déterminant de Hankel; Fonction hypergéométrique confluente; Inégalité de Turán; Positivité complète; Produit Beta-Gamma; Régularité du signe; Transformation de Kummer; Wronskien

Domaines: Probabilités [math.PR]; Analyse classique [math.CA]

Thomas Simon. Produit Beta-Gamma et régularité du signe. 2012. ⟨hal-00721209⟩