Related papers: Multivariate Hahn polynomials and difference equat…

Univariate delta Gon\v{c}arov polynomials arise when the classical Gon\v{c}arov interpolation problem in numerical analysis is modified by replacing derivatives with delta operators. When the delta operator under consideration is the…

We introduce orthogonal polynomials $M_j^{\mu,\ell}(x)$ as eigenfunctions of a certain self-adjoint fourth order differential operator depending on two parameters $\mu\in\mathbb{C}$ and $\ell\in\mathbb{N}_0$. These polynomials arise as…

The one variable Krawtchouk polynomials, a special case of the $_2F_1$ function did appear in the spectral representation of the transition kernel for a Markov chain studied a long time ago by M. Hoare and M. Rahman. A multivariable…

A supersymmetric extension of the Hahn algebra is introduced. This quadratic superalgebra, which we call the Hahn superalgebra, is constructed using the realization provided by the Dunkl oscillator model in the plane, whose Hamiltonian…

The aim of this paper is to bring into the picture a new phenomenon in the theory of orthogonal matrix polynomials satisfying second order differential equations. The last few years have witnessed some examples of a (fixed) family of…

We call Krawtchouk-Griffiths systems, KG-systems, systems of multivariate polynomials orthogonal with respect to corresponding multinomial distributions. The original Krawtchouk polynomials are orthogonal with respect to a binomial…

The type III Hermite $X_m$ exceptional orthogonal polynomial family is generalized to a double-indexed one $X_{m_1,m_2}$ (with $m_1$ even and $m_2$ odd such that $m_2 > m_1$) and the corresponding rational extensions of the harmonic…

Orthogonal polynomials for the multinomial distribution m(x, p) of N balls dropped into d boxes (box i has probability p(i)) are called multivariate Krawtchouk polynomials. This paper gives an introduction to their properties, collections…

We investigate polynomials, called m-polynomials, whose generator polynomial has coefficients that can be arranged as a matrix, where q is a positive integer greater than one. Orthogonality relations are established and coefficients are…

This paper studies new Lancaster characterizations of bivariate multivariate Poisson, negative binomial and normal distributions which have diagonal expansions in multivariate orthogonal polynomials. The characterizations extend classical…

We present an algebraic theory of orthogonal polynomials in several variables that includes classical orthogonal polynomials as a special case. Our bottom line is a straightforward connection between apolarity of binary forms and the inner…

For a long time it has been a challenging goal to identify all orthogonal polynomial systems that occur as eigenfunctions of a linear differential equation. One of the widest classes of such eigenfunctions known so far, is given by…

Kerov Hamiltonians are defined as a set of commuting operators which have Kerov functions as common eigenfunctions. In the particular case of Macdonald polynomials, well known are the exponential Ruijsenaars Hamiltonians, but the…

We construct new families of (q-) difference and (contour) integral operators having nice actions on Koornwinder's multivariate orthogonal polynomials. We further show that the Koornwinder polynomials can be constructed by suitable…

The goal of this work is to characterize all second order difference operators of several variables that have discrete orthogonal polynomials as eigenfunctions. Under some mild assumptions, we give a complete solution of the problem.

A family of bi-differential operators from $C^\infty\big(\Mat(m,\mathbb R)\times\Mat(m,\mathbb R)\big)$ into $C^\infty\big(\Mat(m,\mathbb R)\big)$ which are covariant for the projective action of the group $SL(2m,\mathbb R)$ on…

This paper explores a factorization using bidiagonal matrices of the recurrence matrix of Hahn multiple orthogonal polynomials. The factorization is expressed in terms of ratios involving the generalized hypergeometric function ${}_3F_2$…

In [Castillo \& Mbouna, Indag. Math. {\bf 31} (2020) 223-234], the concept of $\pi_N$-coherent pairs of order $(m,k)$ with index $M$ is introduced. This definition, implicitly related with the standard derivative operator, automatically…

We generalize some previous results on random polynomials in several complex variables. A standard setting is to consider random polynomials $H_n(z):=\sum_{j=1}^{m_n} a_jp_j(z)$ that are linear combinations of basis polynomials $\{p_j\}$…

The Motzkin numbers can be derived as coefficients of hybrid polynomials. Such an identification allows the derivation of new identities for this family of numbers and offers a tool to investigate previously unnoticed links with the theory…