Related papers: The bilateral birth-death chain generated by the a…
Orthogonal polynomials on the real line always satisfy a three-term recurrence relation. The recurrence coefficients determine a tridiagonal semi-infinite matrix (Jacobi matrix) which uniquely characterizes the orthogonal polynomials. We…
Many examples of exactly solvable birth and death processes, a typical stationary Markov chain, are presented together with the explicit expressions of the transition probabilities. They are derived by similarity transforming exactly…
Duality relations between continuous-state and discrete-state stochastic processes with continuous-time have already been studied and used in various research fields. We propose extended duality relations, which enable us to derive…
We consider positive Jacobi matrices $J$ with compact inverses and consequently with purely discrete spectra. A number of properties of the corresponding sequence of orthogonal polynomials is studied including the convergence of their…
A selfadjoined block tridiagonal matrix with positive definite blocks on the off-diagonals is by definition a Jacobi matrix with matrix entries. Transfer matrix techniques are extended in order to develop a rotation number calculation for…
We establish connections between the absorption probabilities of a class of birth-death processes with killing, and the stationary tail of a related class of birth-death processes with catastrophes. The major ingredients of the proofs are a…
We consider a bilateral birth-death process characterized by a constant transition rate $\lambda$ from even states and a possibly different transition rate $\mu$ from odd states. We determine the probability generating functions of the even…
For a fixed $p \in \mathbb{N}$, sequences of polynomials $\{P_n\}$, $n \in \mathbb{N}$, defined by a $(p+2)$-term recurrence relation are related to several topics in Approximation Theory. A $(p+2)$-banded matrix $J$ determines the…
We present results on co-recursive associated Laguerre and Jacobi polynomials which are of interest for the solution of the Chapman-Kolmogorov equations of some birth and death processes with or without absorption. Explicit forms,…
For any distribution $\pi$ with support equal to $[n] = \{1, 2,..., n \}$, we study the set $\mathcal{A}_{\pi}$ of tridiagonal stochastic matrices $K$ satisfying $\pi(i) K[i,j] = \pi(j) K[j,i]$ for all $i, j \in [n]$. These matrices…
This paper investigates stochastic finite matrices and the corresponding finite Markov chains constructed using recurrence matrices for general families of orthogonal polynomials and multiple orthogonal polynomials. The paper explores the…
Multivariate Krawtchouk polynomials are constructed explicitly as Birth and Death polynomials, which have the nearest neighbour interactions. They form the complete set of eigenpolynomials of a birth and death process with the birth and…
In this paper I shall give the complete solution of the equations governing the bilateral birth and death process on path set $\mathbb{R}_q=\{q^n,\quad n\in\mathbb{Z}\}$ in which the birth and death rates $\lambda_n=q^{2\nu-2n}$ and…
The random walk to be considered takes place in the d- spherical dual of the group U(n + 1), for a fixed finite dimensional irreducible representation d of U(n). The transition matrix comes from the three term recursion relation satisfied…
We consider the unital associative algebra $\mathcal{A}$ with two generators $\mathcal{X}$, $\mathcal{Z}$ obeying the defining relation $[\mathcal{Z},\mathcal{X}]=\mathcal{Z}^2+\Delta$. We construct irreducible tridiagonal representations…
The aim of this paper is to derive (by using two operators, representable by a Jacobi matrix) a family of q-orthogonal polynomials, which turn to be dual to alternative q-Charlier polynomials. A discrete orthogonality relation and a…
We study LU-type factorizations of the infinitesimal generator of a birth--death process on $\mathbb{N}_0$. Our goal is to characterize those factorizations whose Darboux transformations (that is, inverting the order of the factors) yield…
We consider a continuous time Markov process on $\mathbb{N}_0$ which can be interpreted as generalized alternating birth-death process in a non-autonomous random environment. Depending on the status of the environment the process either…
The orthogonal polynomials with recurrence relation \[(\la\_n+\mu\_n-z) F\_n(z)=\mu\_{n+1} F\_{n+1}(z)+\la\_{n-1} F\_{n-1}(z)\] with two kinds of cubic transition rates $\la\_n$ and $\mu\_n,$ corresponding to indeterminate Stieltjes moment…
The list of physically motivated urn models that can be solved in terms of classical orthogonal polynomials is very small. It includes a model proposed by D. Bernoulli and further analyzed by S. Laplace and a model proposed by P. and T.…