QBD processes associated with Jacobi-Koornwinder bivariate polynomials and urn models. (English) Zbl 1526.60049
Summary: 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 will explicitly compute the coefficients of the three-term recurrence relations generated by these QBD polynomials and study the conditions under we can produce families of discrete-time QBD processes. Finally, we show an urn model associated with one special case of these QBD processes.
MSC:
| 60J80 | Branching processes (Galton-Watson, birth-and-death, etc.) |
| 33C45 | Orthogonal polynomials and functions of hypergeometric type (Jacobi, Laguerre, Hermite, Askey scheme, etc.) |
| 42C05 | Orthogonal functions and polynomials, general theory of nontrigonometric harmonic analysis |
Keywords:
quasi-birth-and-death (QBD) processes; multivariate orthogonal polynomials; Jacobi-Koornwinder polynomials; urn modelsReferences:
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