Paper Info
Title
Matrix Valued Orthogonal Polynomials Arising from Group Representation Theory and a Family of Quasi-Birth-and-Death Processes
Abstract
We consider a family of matrix valued orthogonal polynomials obtained by Pacharoni and Tirao in connection with spherical functions for the pair ($\mathrm{SU}(N+1)$, $\mathrm{U}(N)$); 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 “n-step 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.
Year
DOI
Venue
2008
10.1137/070697604
SIAM J. Matrix Analysis Applications
Keywords
Field
DocType
parameter value,quasi-birth-and-death process,group representation theory,orthogonal polynomial,parameter family,new family,corresponding block jacobi matrix,quasi-birth-and-death processes,n-step transition probability matrix,j. a,invariant distribution,nonhomogeneous quasi-birth-and-death process,jacobi matrix,markov chains,markov chain,spherical function
Combinatorics,Orthogonal matrix,Polynomial matrix,Stochastic matrix,Orthogonal polynomials,Matrix (mathematics),Mathematical analysis,Pure mathematics,Invariant (mathematics),Orthogonal group,Mathematics,Block matrix
Journal
Volume
Issue
ISSN
30
2
0895-4798
Citations 
PageRank 
References 
3
0.84
1
Authors
2
Name
Order
Citations
PageRank
F. Alberto Grünbaum1199.14
Manuel D. de la Iglesia2134.96