Related papers: Differential properties of matrix orthogonal polyn…
The concept of permutograph is introduced and properties of integral functions on permutographs are established. The central result characterizes the class of integral functions that are representable as lattice polynomials. This result is…
A linear functional $\bf u$ is classical if there exist polynomials, $\phi$ and $\psi$, with $\deg \phi\le 2$, $\deg \psi=1$, such that ${\mathscr D}\left(\phi(x) {\bf u}\right)=\psi(x){\bf u}$, where ${\mathscr D}$ is a certain…
Sigmoid functions play an important role in many areas of applied mathematics, including machine learning, population dynamics and probability. We place the study of sigmoid functions in the context of the derivative sub-group of the group…
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…
Let D_v the difference operator and q-difference operators defined by D_\omega p(x) = \frac{p(x+\omega)-p(x)}{\omega} and D_q p(x) = \frac{p(qx)-p(x)}{(q-1)x}, respectively. Let U and V be two moment regular linear functionals and let…
We give a survey of the analytic theory of matrix orthogonal polynomials.
In a previous paper we have introduced matrix-valued analogues of the Chebyshev polynomials by studying matrix-valued spherical functions on SU(2)\times SU(2). In particular the matrix-size of the polynomials is arbitrarily large. The…
By using Fourier transforms of two symmetric sequences of finite orthogonal polynomials, we introduce two new classes of finite orthogonal functions and obtain their orthogonality relations via Parseval's identity.
Skew orthogonal polynomials arise in the calculation of the $n$-point distribution function for the eigenvalues of ensembles of random matrices with orthogonal or symplectic symmetry. In particular, the distribution functions are completely…
We study the discrete semiclassical orthogonal polynomials of class s=1. By considering all possible solutions of the Pearson equation, we obtain five canonical families. We also consider limit relations between these and other families of…
This semi-expository paper surveys results concerning three classes of orthogonal polynomials: in one non-hermitian variable, in several isometric non-commuting variables, and in several hermitian non-commuting variables. The emphasis is on…
The theory of orthogonal polynomials on the unit circle is developed for a general class of weights leading to systems of recurrence relations and derivatives of the polynomials and their associated functions, and to functional-difference…
Conjectured links between the distribution of values taken by the characteristic polynomials of random orthogonal matrices and that for certain families of L-functions at the centre of the critical strip are used to motivate a series of…
Planar functions are special functions from a finite field to itself that give rise to finite projective planes and other combinatorial objects. We consider polynomials over a finite field $K$ that induce planar functions on infinitely many…
The $m$-symmetric Macdonald polynomials form a basis of the space of polynomials that are symmetric in the variables $x_{m+1},x_{m+2},\dots$ (while having no special symmetry in the variables $x_1,\dots,x_m$).We establish in this article…
We present a new approach to solving polynomial ordinary differential equations by transforming them to linear functional equations and then solving the linear functional equations. We will focus most of our attention upon the first-order…
In this contribution we consider sequences of monic polynomials orthogonal with respect to a Sobolev-type inner product \[ \langle f,g \rangle _{S}:= \langle {\bf u}, f g\rangle +N (\mathscr D_q f)(\alpha) (\mathscr D _{q}g)(\alpha),\qquad…
We consider Koornwinder's method for constructing orthogonal polynomials in two variables from orthogonal polynomials in one variable. If semiclassical orthogonal polynomials in one variable are used, then Koornwinder's construction…
The tridiagonal representation approach is an algebraic method for solving second order differential wave equations. Using this approach in the solution of quantum mechanical problems, we encounter two new classes of orthogonal polynomials…
We study functional graphs generated by several quadratic polynomials, acting simultaneously on a finite field of odd characteristic. We obtain several results about the number of leaves in such graphs. In particular, in the case of graphs…