相关论文: Multi-indexed Orthogonal Polynomials of a Discrete…
We introduce two kinds of multiple little q-Jacobi polynomials by imposing orthogonality conditions with respect to r discrete little q-Jacobi measures on the exponential lattice q^k (k=0,1,2,3,...), where 0 < q < 1. We show that these…
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…
In this Letter we identify special systems of (an arbitrary number) N of first-order Ordinary Differential Equations with homogeneous polynomials of arbitrary degree M on their right-hand sides, which feature very simple explicit solutions;…
Some particular examples of classical and quantum systems on the lattice are solved with the help of orthogonal polynomials and its connection to continuous models are explored.
The multi-indexed Jacobi polynomials are the main part of the eigenfunctions of exactly solvable quantum mechanical systems obtained by certain deformations of the P\"oschl-Teller potential (Odake-Sasaki). By fine-tuning the parameter(s) of…
A new interpretation and applications of the ``Diophantine'' and factorisation properties of {\em finite} orthogonal polynomials in the Askey scheme are explored. The corresponding twelve polynomials are the ($q$-)Racah, (dual, $q$-)Hahn,…
Polynomials known as Multiple Orthogonal Polynomials in a single variable are polynomials that satisfy orthogonality conditions concerning multiple measures and play a significant role in several applications such as Hermite-Pad\'e…
In this contribution we consider sequences of monic polynomials orthogonal with respect to Sobolev-type inner product \[ \left\langle f,g\right\rangle= \langle {\bf u}^{\tt M},fg\rangle+\lambda \mathscr T^j f (\alpha)\mathscr…
By using a generalization of Sturm-Liouville problems in discrete spaces, a basic class of symmetric orthogonal polynomials of a discrete variable with four free parameters, which generalizes all classical discrete symmetric orthogonal…
A new set of multiple orthogonal polynomials of both type I and type II with respect to two weight functions involving Gauss' hypergeometric function on the interval $(0,1)$ is studied. This type of polynomials have direct applications in…
We introduce the quadratic harness condition and show that integrable quadratic harnesses have orthogonal martingale polynomials with a three step recurrence that satisfies a q-commutation relation. This implies that quadratic harnesses are…
We study a class of stationary Markov processes with marginal distributions identifiable by moments such that every conditional moment of degree say $m$ is a polynomial of degree at most $m\;\text{.}\;$ We show that then under some…
In this work, we introduce and construct specific $q$-polynomials that are desired from the well-established families of $q$-orthogonal polynomials, namely little $q$-Jacobi polynomials and $q$-Laguerre polynomials, respectively. We examine…
We consider orthogonal polynomials with respect to a linear differential operator $$\mathcal{L}^{(M)}=\sum_{k=0}^{M}\rho_{k}(z)\frac{d^k}{dz^k}, $$ where $\{\rho_k\}_{k=0}^{M}$ are complex polynomials such that $deg[\rho_k]\leq k, 0\leq k…
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…
In a recent paper in the Journal of Theoretical Probability Gong, Mao and Zhang, using the theory of Dirichlet forms, extended Karlin and McGregor's classical results on first-hitting times of a birth-death process on the nonnegative…
By generalising the one to one correspondence between exactly solvable hermitian matrices $\mathcal{H}=\mathcal{H}^\dagger$ and exactly solvable spinless fermion systems $\mathcal{H}_f=\sum_{x,y}c_x^\dagger\mathcal{H}(x,y)c_y$, four types…
Recursive algebraic construction of two infinite families of polynomials in $n$ variables is proposed as a uniform method applicable to every semisimple Lie group of rank $n$. Its result recognizes Chebyshev polynomials of the first and…
We study the monic orthogonal polynomials with respect to a singularly perturbed Airy weight. By using Chen and Ismail's ladder operator approach, we derive a discrete system satisfied by the recurrence coefficients for the orthogonal…
Two sets of infinitely many exceptional orthogonal polynomials related to the Wilson and Askey-Wilson polynomials are presented. They are derived as the eigenfunctions of shape invariant and thus exactly solvable quantum mechanical…