Matrix Valued Orthogonal Polynomials Arising from Group Representation Theory and a Family of Quasi-Birth-and-Death Processes

Abstract

Abstract. We consider a family of matrix valued orthogonal polynomials obtained by Pacharoni and Tirao in connection with spherical functions for the pair (SU(N + 1), U(N )); see [I. Pacharoni and J. A. Tirao, Constr. Approx., 25 (2007), pp. 177-192]. After an appropriate conjugation, we obtain a new family of matrix valued orthogonal polynomials where the corresponding block Jacobi matrix is stochastic and has special probabilistic properties. This gives a highly nontrivial example of a nonhomogeneous quasi-birth-and-death process for which we can explicitly compute its "nstep transition probability matrix" and its invariant distribution. The richness of the mathematical structures involved here allows us to give these explicit results for a several parameter family of quasi-birth-and-death processes with an arbitrary (finite) number of phases. Some of these results are plotted to show the effect that choices of the parameter values have on the invariant distribution.

Stochastic Darboux transformations for quasi-birth-and-death processes and urn models

“…We would like to remark on a special property of the matrix-valued orthogonal polynomials P n generated by the Darboux transformation of P for the case s 12 = 1 above. It is well known that the original matrix-valued orthogonal polynomials P n satisfy a second-order differential equation of the form P ′′ n (x)F 2 (x) + P ′ n (x)F 1 (x) + P n (x)F 0 = Λ n P n (x), (4.17) where F 2 (x) = x(1 − x)I and F 1 , F 0 certain matrix polynomials of degree 1 and 0, respectively (see for instance [24] or [9]). In this situation (and only in this situation) the matrix-valued polynomials P n obtained by performing the Darboux transformation also satisfy a second-order differential equation of the form (4.17) with coefficients F 2 , F 1 , F 0 given by…”

“…Finally we can also study recurrence associated with the process P . According to Theorem 8.1 of [9], the QBD process that results from P is never positive recurrent. If −1 < β ≤ 0, then the process is null recurrent.…”

Matrix differential equations and scalar polynomials satisfying higher order recursions

“…For a situation where the matrix valued orthogonal polynomials happen to satisfy matrix valued differential equations which along with the orthogonality measure are an outgrowth of the work started in [16], see [15].…”

Matrix polynomials orthogonal with respect to a non-symmetric matrix of measures

“…MOPs -which are known to satisfy a three-term recurrence relation of type (1.3) -are investigated for example in the theory of random walks on graphs or in the theory of multiple birth-and-death processes (cf. [5,8]), where such thre term formulae appear. Many properties of random walk or birth-and-death processes can be obtained from the matrix of measures of orthogonality.…”