Kasraoui, Stanton and Zeng, and Kim, Stanton and Zeng introduced certain q-analogues of Laguerre and Charlier polynomials. The moments of these orthogonal polynomials have combinatorial models in terms of crossings in permutations and set partitions. The aim of this article is to prove simple formulas for the moments of the q-Laguerre and the q-Charlier polynomials, in the style of the Touchard...

We consider discrete orthogonal polynomial ensembles which are discrete analogues of the orthogonal polynomial ensembles in random matrix theory. These ensembles occur in certain problems in combinatorial probability and can be thought of as probability measures on partitions. The Meixner ensemble is related to a two-dimensional directed growth model, and the Charlier ensemble is related to the...

We show that the function h(x) = ∏ i<j(xj − xi) is harmonic for any random walk in Rk with exchangeable increments, provided the required moments exist. For the subclass of random walks which can only exit the Weyl chamber W = {x : x1 < x2 < · · · < xk} onto a point where h vanishes, we define the corresponding Doob h-transform. For certain special cases, we show that the marginal distribution ...

A model of a q-harmonic oscillator based on q-Charlier poly-nomials of Al-Salam and Carlitz is discussed. Simple explicit realization of q-creation and q-annihilation operators, q-coherent states and an ana-log of the Fourier transformation are found. A connection of the kernel of this transform with biorthogonal rational functions is observed. Models of q-harmonic oscillators are being develop...

We use the method of positive quadratic forms and discrete analogues of the Laguerre inequality recently obtained by the author, to give bounds on the zeros of the Charlier polynomials, which are uniform in all parameters involved.

Recensé : Bernard Charlier, Christine Grard, Frédéric Laugrand, Pierre-Joseph Laurent, Saskia Simon (dir.), Écritures anthropologiques, Louvain-la-Neuve, Academia/L’Harmat- tan, 2020 (coll. « Investigations d’anthropologie prospective », n°20), 340 p.

Large integration of solar Photo Voltaic (PV) in distribution network has resulted in over-voltage problems. Several control techniques are developed to address over-voltage problem using Deterministic Load Flow (DLF). However, intermittent characteristics of PV generation require Probabilistic Load Flow (PLF) to introduce variability in analysis that is ignored in DLF. The traditional PLF tech...