Related papers: Exactly solvable discrete time Birth and Death pro…

We study a family of quasi-birth-and-death (QBD) processes associated with the so-called first family of Jacobi-Koornwinder bivariate polynomials. These polynomials are orthogonal on a bounded region typically known as the swallow tail. We…

We consider a stochastic spatial point process with births and deaths on $\mathbb{R}^d$, with the hard-core property that at any time the balls of radius half of any two points do not overlap. We give explicit construction of the process.…

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,…

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…

We consider consistent dynamics for non-intersecting birth and death chains, originating from dualities of stochastic coalescing flows and one dimensional orthogonal polynomials. As corollaries, we obtain unified and simple probabilistic…

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…

We study a general class of birth-and-death processes with state space $\mathbb{N}$ that describes the size of a population going to extinction with probability one. This class contains the logistic case. The scale of the population is…

We give a probabilistic interpretation of the associated Jacobi polynomials, which can be constructed from the three-term recurrence relation for the classical Jacobi polynomials by shifting the integer index $n$ by a real number $t$. Under…

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 explicit criteria for several types of ergodicity of one-dimensional diffusions or birth-death processes have been found out recently in a surprisingly short period. One of the criteria is for exponential ergodicity of birth-death…

We introduce a family of discrete determinantal point processes related to orthogonal polynomials on the real line, with correlation kernels defined via spectral projections for the associated Jacobi matrices. For classical weights, we show…

The paper is devoted to the further study of the remarkable classes of orthogonal polynomials recently discovered by Bender and Dunne. We show that these polynomials can be generated by solutions of arbitrary quasi - exactly solvable…

We present new exactly solvable systems of the discrete quantum mechanics with pure imaginary shifts, whose physical range of the coordinate is the whole real line. These systems are shape invariant and their eigenfunctions are described by…

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…

A new type of exact solvability is reported. We study the general central polynomial potentials (with 2q anharmonic terms) which satisfy the Magyari's partial exact solvability conditions (this means that they possess a…

Birth-death processes take place ubiquitously throughout the universe. In general, birth and death rates depend on the system size (corresponding to the number of products or customers undergoing the birth-death process) and thus vary every…

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…

We explore probabilistic consequences of correspondences between $q$-Whittaker measures and periodic and free boundary Schur measures established by the authors in the recent paper [arXiv:2106.11922]. The result is a comprehensive theory of…

We study almost automorphic solutions of the discrete delayed neutral dynamic system% \[ x(t+1)=A(t)x(t)+\Delta Q(t,x(t-g(t)))+G(t,x(t),x(t-g(t))) \] by means of a fixed point theorem due to Krasnoselskii. Using discrete variant of…

We introduce a pair of novel difference equations, whose solutions are expressed in terms of Racah or Wilson polynomials depending on the nature of the finite-difference step. A number of special cases and limit relations are also examined,…