Matrix valued orthogonal polynomials arising from group representation theory and a family of quasi-birth-and-death processes

Copyright © by SIAM. Unauthorized reproduction of this article is prohibited. SIAM J. MATRIX ANAL. APPL.c 2008 Society for Industrial and Applied Mathematics Vol. 30, No. 2, pp. 741–761 MATRIX VALUED ORTHOGONAL POLYNOMIALS ARISING FROM GROUP REPRESENTATION THEORY AND A FAMILY OF QUASI-BIRTH-AND-DEATH PROCESSES∗ F. ALBERTO GR¨ UNBAUM†AND MANUEL D. DE LA IGLESIA‡ 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. Key words. matrix valued orthogonal polynomials, Markov chains, block tridiagonal transition matrix, quasi-birth-and-death processes AMS subject classifications. 60J10, 42C05 DOI. 10.1137/070697604 1. Purpose and contents of the paper. The aim of this paper is to tie together two subjects that have received quite a bit of attention recently. We will not give a detailed explanation of either one of them, since this would require too much space and it has been done properly in the literature already. Besides, since these two subjects require rather different backgrounds, an ab-initio exposition would be a formidable task. To compensate for this we give a brief historical view of how these topics developed and then combine them at the appropriate point. The contents of this paper can be divided into three parts. A first part gives a brief account of the subjects that are going to play a role in this paper. The introduction contains some historical developments tying the moment problem with spectral theory and a quick look at birth-and-death processes, including the appearance of the appropriate orthogonal polynomials. Section 3 reviews very briefly Kre˘ın’s theory of matrix valued orthogonal polynomials and discusses the first example relevant to our considerations. Section 4 gives a minimal description of the class of Markov chains known as quasi-birth-and-death processes and talks about the very natural connection between this and the previous section. The second part introduces the family of examples arising from group representation theory that we are going to use in this paper. Section 5 gives a guide to the literature on matrix valued spherical functions aimed at showing how the examples discussed in section 6 arose. Section 6 gives the bare-bones details of the extensive ∗Received by the editors July 18, 2007; accepted for publication (in revised form) by H. J. Woerdeman March 14, 2008; published electronically July 2, 2008. http://www.siam.org/journals/simax/30-2/69760.html †Department of Mathematics, University of California, Berkeley, Berkeley, CA 94720 (grunbaum@ math.berkeley.edu). This author’s work was partially supported by NSF grant DMS-0603901. ‡Departamento de An´alisis Matem´atico, Universidad de Sevilla, Apdo (P.O. BOX) 1160, 41080 Sevilla, Spain ([email protected]). This author’s work was partially supported by D.G.E.S., ref. BFM2003-06335-C03-01, FQM-262 P06-FQM-01735, FQM-481 (Junta de Andaluc´ıa). 741 Downloaded 06/16/16 to 150.214.182.169. Redistribution subject to SIAM license or copyright; see http://www.siam.org/journals/ojsa.php Copyright © by SIAM. Unauthorized reproduction of this article is prohibited. 742 F. ALBERTO GR¨ UNBAUM AND MANUEL D. DE LA IGLESIA work carried out in [30] in the case of the complex projective space. Section 7 shows how to conjugate the weight matrix from [30] to obtain a family of matrix valued orthogonal polynomials with extra probabilistic properties. The third part concentrates on the probabilistic aspects of our family of examples. Section 8 deals with a number of issues of probabilistic nature and displays the network associated with our examples. In particular we find explicit expressions for the invariant distribution. Section 9 gives graphical displays of some results obtained from the exact formulas in the previous sections. The goal here is to show that by varying the parameters afforded by the group theoretical situation one can obtain quite a range of different probabilistic behaviors. Finally, section 10 gives a summary of the results of the paper and the challenges that lie ahead. 2. Introduction. The classical Hausdorff moment problem, that of determining a measure dψ(x) in the interval [−1,1] from its moments σn=1 −1 xndψ(x), originated in very concrete problems at the end of the 19th century and was discussed by people such as Chebyshev, Markov, and Stieltjes. In the hands of Weyl and a few others, this showed the far reaching power of the modern theory of functional analysis in the early part of the 20th century. The main ingredient here is to connect this problem with the spectral theory of a second order difference operator (built from the moments σn) acting on functions defined on the nonnegative integers. The moments in question determine (up to scalars) a family of polynomials {Qn(x)}n≥0, and these polynomials are the eigenfunctions of the second order difference operator alluded to above. In the appropriate Hilbert space this operator is symmetric, and the problem of finding dψ(x) is the problem of finding self-adjoint extensions of this symmetric operator. Under certain conditions there is a unique such extension and thus a unique solution to the moment problem we started from, but at any rate any extension gives a measure dψ(x) that makes the polynomials orthogonal with respect to each other. To get closer to our subject we need a few more ingredients. One of them is given in the rest of this section, and the other two in sections 3 and 4. The presence of a second order difference operator acting on the space of functions defined on the nonnegative integers, i.e., a semi-infinite tridiagonal matrix, makes it natural to think of a very special kind of Markov chain on the space of nonnegative integers. These are the so-called birth-and-death processes where at each discrete unit of time a transition is allowed from state ito state jwith probability Pij and we put Pij =0if|i−j|>1.The one-step transition probability matrix is given by (2.1) P=⎛ ⎜ ⎜ ⎜ ⎜ ⎝ r0p0 q1r1p1 q2r2p2 ......... ⎞ ⎟ ⎟ ⎟ ⎟ ⎠. We will assume that pj>0, qj+1 >0, and rj≥0 for j≥0. We also assume pj+rj+qj= 1 for j≥1 and by putting p0+r0≤1 we allow for the state j=0to be an absorbing state (with probability 1 −p0−r0). Some of these conditions can be relaxed. Downloaded 06/16/16 to 150.214.182.169. Redistribution subject to SIAM license or copyright; see http://www.siam.org/journals/ojsa.php Copyright © by SIAM. Unauthorized reproduction of this article is prohibited. MATRIX VALUED ORTHOGONAL POLYNOMIALS AND QBDs 743 The problem here is to obtain an expression for the so-called “n-step transition probability matrix,” giving the probability of going between any two states in nsteps. By making use of the ideas mentioned above, i.e., by bringing in an appropriate Hilbert space and applying then the spectral theorem, Karlin and McGregor [20] obtained a neat representation formula for the quantity of interest, as recalled below. If one introduces the polynomials {Qn(x)}n≥0by the conditions Q−1(x)=0, Q0(x) = 1 and by using the notation φ=⎛ ⎜ ⎜ ⎝ Q0(x) Q1(x) . . . ⎞ ⎟ ⎟ ⎠, one insists on the recursion relation Pφ=xφ, it is possible to prove the existence of a unique measure dψ(x) supported in [−1,1] such that 1 −1 Qi(x)Qj(x)dψ(x)1 −1 Qj(x)2dψ(x)=δij and one gets the Karlin–McGregor representation formula (2.2) Pn ij =1 −1 xnQi(x)Qj(x)dψ(x)1 −1 Qj(x)2dψ(x). If time is taken to be continuous, as it is done in other papers by Karlin and McGregor, then this formula and the matrix Psuffer only cosmetic changes. It is interesting to notice that this seminal paper of Karlin and McGregor refers both to the standard text on the moment problem at the time [33], as well as to the fact that Feller and McKean had already recognized the relevance of the Hilbert space setup in the study of diffusion processes; see [7, 26]. One can mention other papers, such as [8, 17, 19, 25], where similar ideas were at play. The last section of [20] deals with the case of a finite state space and the case when the nonnegative integers are replaced by the set of all integers. Since one is using a very powerful tool such as the spectral theorem it is clear that an adaptation of the ideas from birth-and-death processes will work here too. In the case of the integers, one is dealing with a state space with two singular points (one at each end of the line), and in this case Weyl and others had already found the correct tool: one replaces the spectral measure dψ(x)bya2×2 nonnegative matrix. The paper of Karlin and McGregor concludes with the explicit computation of this matrix in the case of the doubly infinite random walk. The general formula is given in expression (12) of [20] for the case of discrete time and also in (6.8) of [18] for continuous time. The representation formula given above is of intrinsic interest: the computation of the left-hand side of (2.2) for fixed i, j and arbitrary values of ninvolves all of the entries of (2.1). However, if dψ(x) is known, then the right-hand side of (2.2) gives a way of computing this quantity using only a fixed number of entries of (2.1). The applicability of (2.2) depends to a large extent on our ability to obtain useful expressions for the orthogonal polynomials and the orthogonality measure associated Downloaded 06/16/16 to 150.214.182.169. Redistribution subject to SIAM license or copyright; see http://www.siam.org/journals/ojsa.php Copyright © by SIAM. Unauthorized reproduction of this article is prohibited. 744 F. ALBERTO GR¨ UNBAUM AND MANUEL D. DE LA IGLESIA with P. If one looks around in the literature one discovers that the number of cases where this is possible is rather small. We close this section by showing how one can compute in the case of a stochastic matrix Pits invariant (stationary) distribution, i.e., the (unique up to scalars) row vector π=(π0,π 1,π 2,...) such that πP=π. Recall that a matrix Pwith nonnegative entries is called stochastic if the sum of the elements in any row equals unity. We first obtain, using r0+p0= 1, that π1=π0p0/q1. Then one proves by induction that for i≥1wehave πi=π0(p0p1...p i−1)/(q1q2...q i). This has the consequence that πi+1/πi=pi/qi+1. Now for i≥0wehave xQi(x)=piQi+1(x)+riQi(x)+qiQi−1(x) with q0= 0. Integrating this after multiplication by Qi+1 or Qi−1gives 1 −1 xQi(x)Qi+1(x)dψ(x)=pi1 −1 Q2 i+1(x)dψ(x)=qi+1 1 −1 Q2 i(x)dψ(x). Combining these two results we get that the ratio of the two integrals above is given by the common value qi+1/pi=πi/πi+1. The moral of this is that the solution to πP=πcan be computed (up to a common multiplicative scalar) either from the matrix Pitself or from the knowledge of the integrals 1 −1 Q2 i(x)dψ(x). In particular if we have an homogeneous birth-and-death process where pi=pand qi=qindependently of the value of i, then we have that the components of πare given by πi=π0(p/q)i,i≥0. 3. Matrix valued orthogonal polynomials. We need two more characters to be able to start our tale. The first one is the theory of matrix valued orthogonal polynomials, whose bare-bones development is given in two papers by Kre˘ın [21, 22]. There is no written account of the motivation that led Kre˘ın to this theory, but one can easily see the connection with the spectral theory of difference operators on the integers. This is very nicely discussed in the book by Berezans’ki˘ı [2]. In fact the study Downloaded 06/16/16 to 150.214.182.169. Redistribution subject to SIAM license or copyright; see http://www.siam.org/journals/ojsa.php Copyright © by SIAM. Unauthorized reproduction of this article is prohibited. MATRIX VALUED ORTHOGONAL POLYNOMIALS AND QBDs 745 of the classical second order difference operator on the integers is done in detail in [2] and may constitute the first example of the theory of Kre˘ın, where the polynomials and their orthogonality weight matrix W(x) are both explicitly given. This is, of course, a special case of the matrix that appears in the last section of [20] for general values of pand q(p+q= 1). We give now a brief account of Kre˘ın’s theory. Given a positive definite matrix valued measurable weight function W=W(x) with finite moments we can consider the skew symmetric bilinear form defined for any pair of matrix valued polynomial functions P(x) and Q(x) by the numerical matrix (P,Q)=(P,Q)W=R P(x)W(x)Q∗(x)dx, where Q∗(x) denotes the conjugate transpose of Q(x). We define the matrix valued norm of Pby (3.1) P2=(P,P)W. One can also deal with a more general weight matrix W(x); see [4]. This leads, using the Gram–Schmidt process, to the existence of a sequence of matrix valued orthogonal polynomials with nonsingular leading coefficients. Given an orthogonal sequence {Qn(x)}n≥0one gets a three term recursion relation (3.2) xQn(x)=AnQn+1(x)+BnQn(x)+CnQn−1(x), where Anis nonsingular. We will denote by Lthe corresponding Jacobi matrix, defined by the following block tridiagonal semi-infinite matrix: L=⎛ ⎜ ⎜ ⎜ ⎜ ⎝ B0A0 C1B1A1 C2B2A2 ......... ⎞ ⎟ ⎟ ⎟ ⎟ ⎠. Using the notation Φ=⎛ ⎜ ⎜ ⎝ Q0(x) Q1(x) . . . ⎞ ⎟ ⎟ ⎠ the relation (3.2) becomes (3.3) LΦ=xΦ. We will reserve the symbol Pfor the case where Lbecomes a one-step transition probability matrix, thought of as a scalar matrix. The corresponding Markov chain (to appear in section 4) will have a state space that is more complicated than the set {0,1,2,...}corresponding to a birth-and-death process featured in section 2. In the scalar case, concrete examples of orthogonal polynomials, including explicit formulas for them as well as their orthogonality measure preceded the development of any general theory. Prominent examples are the Hermite, Laguerre, and Jacobi Downloaded 06/16/16 to 150.214.182.169. Redistribution subject to SIAM license or copyright; see http://www.siam.org/journals/ojsa.php Copyright © by SIAM. Unauthorized reproduction of this article is prohibited. 746 F. ALBERTO GR¨ UNBAUM AND MANUEL D. DE LA IGLESIA polynomials. These examples arose from concrete problems in the eighteenth and nineteenth centuries and played a fundamental role, in the hands of Schr¨odinger, in the development of quantum mechanics around 1925. The situation in the matrix valued case is entirely different: the general theory just described above came first. Until a few years ago, it may be that the only nontrivial example was the one included in Berezans’ki˘ı’s book [2], alluded to above and recalled below for the benefit of the reader. Consider the block tridiagonal matrix L=⎛ ⎜ ⎜ ⎜ ⎜ ⎝ B0I C1B1I C2B2I ......... ⎞ ⎟ ⎟ ⎟ ⎟ ⎠ with 2 ×2 blocks given as follows: B0=1 201 10 ,B n=0 ifn≥1, Cn=1 4Iif n≥1, where Istands for the identity matrix. In this case one can compute explicitly the matrix valued polynomials {Qn(x)}n≥0given by xQn(x)=Qn+1(x)+BnQn(x)+CnQn−1(x),Q −1(x)=0,Q 0(x)=I. One gets Qn(x)= 1 2nUn(x)−Un−1(x) −Un−1(x)Un(x), where Un(x) are the Chebyshev polynomials of the second kind. The orthogonality measure is read off from the identity 4i π1 −1 Qi(x)1 √1−x21x x1Q∗ j(x)dx =δijI. Proceeding as in [3, 10, 20] one obtains a Karlin–McGregor representation. We get, for n=0,1,2,..., Ln ij =4i π1 −1 xnQi(x)1 √1−x21x x1Q∗ j(x)dx, where Ln ij stands for the (i, j) block of the matrix Ln. As is usual for birth-and-death processes, the indices i, j run from 0 on. In this way, as noticed in [10], one can compute the entries of the powers Lnwith Downloaded 06/16/16 to 150.214.182.169. Redistribution subject to SIAM license or copyright; see http://www.siam.org/journals/ojsa.php Copyright © by SIAM. Unauthorized reproduction of this article is prohibited. MATRIX VALUED ORTHOGONAL POLYNOMIALS AND QBDs 747 Lthought of as a pentadiagonal scalar matrix, namely L= ⎛ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎝ 01 210 1 20010 1 40001 ... 1 4000 ... ............ ⎞ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎠ . Lis not a stochastic matrix since its rows do not add up to unity. Nevertheless, defining Δ to be the 2 ×2 block diagonal matrix with Δii =2 iIfor every block, we get from (3.3) that ΔLΔ−1ΔΦ = xΔΦ and thus if P=ΔLΔ−1and  Φ = ΔΦ, we have P Φ=x Φ. The scalar version of Pis now the stochastic matrix P= ⎛ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎝ 01 2 1 20 1 200 1 20 1 200 0 1 2 ... 1 2000 ... ............ ⎞ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎠ . Observe that the norm of  Qn, defined in (3.1), satisfies  Qn2=π. This is nothing but the example considered at the end of [20] in the special case of p=q=1/2. In the last few years a number of new families of matrix valued orthogonal polynomials have been computed explicitly along with their orthogonality measure. Typically they are joint eigenfunctions of some fixed differential operator with matrix coefficients. This search was initiated in [5], but nontrivial examples were not discovered until [6] and [13, 16]. The family of examples we will consider later is related to one of these examples and is obtained by modifying the situation discussed in [30]. 4. Quasi-birth-and-death processes. Now we come to the last character of our story. For our purposes, we consider a two dimensional Markov chain with discrete time. The state space consists of the pair of integers (i, j), i∈{0,1,2,...},j∈ {1,...,d}. The first component is usually called the level and the second one the phase. The one-step transition probability matrix, which we will denote (as before) by Phas a block tridiagonal structure (see (4.2)). This indicates that in one unit of time a transition can change the phase without changing the level, or can change the level (and possibly the phase) to either of the adjacent levels. The probability of going in one step from state (i, j) to state (i,j) is given by the (j, j) element of the block Pi,i. Clearly in the case when the number of phases dis one we are back to the case of an ordinary birth-and-death process. In general, these processes are known as (discrete time) quasi-birth-and-death processes. For a much more detailed presentation of this field, as well as its connections with queueing problems in network theory as well as the general field of communication systems, the reader should consult [24, 27] and some of the references in [3]. Once one has the notions introduced in the previous sections it is very natural to connect them and to analyze these interesting Markov chains in terms of the Downloaded 06/16/16 to 150.214.182.169. Redistribution subject to SIAM license or copyright; see http://www.siam.org/journals/ojsa.php Copyright © by SIAM. Unauthorized reproduction of this article is prohibited. 748 F. ALBERTO GR¨ UNBAUM AND MANUEL D. DE LA IGLESIA corresponding spectral properties of the resulting family of matrix valued orthogonal polynomials. As pointed out before one can see the seed for this in the work of Kre˘ın, as well as in the original paper [20]. We have identified two references where the corresponding Karlin–McGregor representation formula has been explicitly given, but there may be others since this is such a natural extension of the scalar theory; see [3] and [10]. In the case of quasibirth-and-death processes one replaces (2.2) by (4.1) Pn ij =xnQi(x)W(x)Q∗ j(x)dxQj(x)W(x)Q∗ j(x)dx−1 . A different but related path to this circle of ideas in connection with network models can be seen in [1]. In [3] one finds some interesting examples where this representation formula is computed explicitly, including a new derivation of the result dealing with the case of random walk on the integers. In [10] one finds an example, taken from [9], where the observation is made that one has a stochastic matrix. The family of examples to be considered in the following sections is an extension of this example. Given the block transition probability matrix P, the problem of computing an invariant distribution row vector, i.e., a vector with nonnegative entries πi j, π=(π0;π1;...)≡(π0 1,π0 2,...,π0 d;π1 1,π1 2,...,π1 d;...) such that πP=π leads to a complicated system of equations. If (4.2) P=⎛ ⎜ ⎜ ⎜ ⎜ ⎝ B0A0 C1B1A1 C2B2A2 ......... ⎞ ⎟ ⎟ ⎟ ⎟ ⎠, we have π0B0+π1C1=π0 and then, for n≥1, πn−1An−1+πnBn+πn+1Cn+1 =πn. This gives, as is easy to check, the rather unpleasant expressions π1=π0(I−B0)C−1 1, π2=π0[(I−B0)C−1 1(I−B1)−A0]C−1 2, π3=π0[(I−B0)C−1 1(I−B1)C−1 2(I−B2)−A0C−1 2(I−B2)−(I−B0)C−1 1A1]C−1 3. These formulas require that the matrices Cnbe invertible. Under these conditions, one can derive nicer looking expressions for the invariant distribution (see [23]). There are Downloaded 06/16/16 to 150.214.182.169. Redistribution subject to SIAM license or copyright; see http://www.siam.org/journals/ojsa.php Copyright © by SIAM. Unauthorized reproduction of this article is prohibited. MATRIX VALUED ORTHOGONAL POLYNOMIALS AND QBDs 749 many issues here that we leave untouched in this analysis. For instance, the possibility of choosing π0with nonnegative entries so that all the subsequent πnwill have this property requires extra conditions. The general theory of quasi-birth-and-death processes is not restricted to the case when the matrices An,Bn, and Cnare all square matrices of the same size and An and Cnare nonsingular. It remains an interesting challenge to find a mathematical setup that can accommodate such a situation. Now that we have seen how to extend the Karlin–McGregor representation formula to the block tridiagonal case it is important to get examples where the polynomials and the orthogonality matrix can be explicitly written down. This is the purpose of the next three sections. 5. Matrix valued spherical functions and matrix valued orthogonal polynomials. The theory of matrix valued spherical functions, initially discussed in [34], is one of the routes leading to explicit families of matrix valued orthogonal polynomials and their orthogonality measure. The first family of examples appeared in [13, 16] in connection with G= SU(3). The size of the matrices here is already arbitrary, and the orthogonality matrix has a scalar factor of the form xα(1 −x). The extension to the case where this scalar factor can be taken to be xα(1 −x)βfor arbitrary α, β > −1 was undertaken in [9] in the 2 ×2 case without any reference to group representation theory. Further examples of this kind are given in [14]. The role of group representation theory in getting away from the 2×2 case can be seen in [15]. Finally, [30] displays for the case of G=SU(N+1) families of orthogonal polynomials depending on three parameters, α, β, and k. In the special case of k=β+1 2, one recovers the results of [9]. These orthogonal polynomials are given by properly “packaged and conjugated” sets of matrix valued spherical functions. These spherical functions correspond to irreducible representations of U(N) and therefore are parameterized by partitions μ=(m1,m 2,...,m N)∈ZNsuch that m1≥m2≥···≥mN. In this paper, following [30], we use only “one step” representations given by a partition μ=(m+,...,m+   k ,m,...,m   N−k ),1≤k≤N−1. In terms of the parameters αand β, one has α=mand β=N−1. The remaining free parameter will determine the size of the corresponding matrix valued orthogonal polynomials and is independent of N. In the next section it will be related to the parameter dappearing in [3]. The examples that have been worked out so far indicate that the matrix valued orthogonal polynomials that result from matrix valued spherical functions lead to a block tridiagonal matrix that can be made into a stochastic one. This will be seen, for our family of examples, in section 8. Although it is possible to obtain examples of stochastic matrices arising in a different fashion, see, for instance [2], we are not aware of any other general scheme that would produce these desirable kinds of matrices in a systematic fashion. 6. A family of examples arising from the complex projective space. In what follows we shall use Eij to denote the matrix with entry (i, j), which is equal to 1 and 0 elsewhere, where the indices i, j run from 0 on. Downloaded 06/16/16 to 150.214.182.169. Redistribution subject to SIAM license or copyright; see http://www.siam.org/journals/ojsa.php Copyright © by SIAM. Unauthorized reproduction of this article is prohibited. 756 F. ALBERTO GR¨ UNBAUM AND MANUEL D. DE LA IGLESIA The state space and the corresponding one-step transitions appear as follows: . . . . . . . . . . . . 1d+1 2d+1 3d+1 2d+2 2d+2 3d+2 3d+3 2d+3 3d+3 d2d3d4d 9. The shape of the invariant distribution. In this section we will study in more detail the behavior of the invariant distribution when the number of phases d is equal to two, a luxury we can afford since we have an analytic expression. In this case, the associated network takes the form 1357 2468 Downloaded 06/16/16 to 150.214.182.169. Redistribution subject to SIAM license or copyright; see http://www.siam.org/journals/ojsa.php Copyright © by SIAM. Unauthorized reproduction of this article is prohibited. MATRIX VALUED ORTHOGONAL POLYNOMIALS AND QBDs 757 0 1 2 3 4 5 6 7 0 2.5 5 7.5 10 12.5 15 Π 1 n Π 2 n Fig. 9.1.α=−0.9,β =0.1,k=0.8. 0 1 2 3 4 5 6 7 0 2.5 5 7.5 10 12.5 15 Π 1 n Π 2 n Fig. 9.2.α=2.5,β =0.1,k=0.55. The invariant distribution πsuch that πP=πis given by π=(π0;π1;...), where πn,n≥0, is the 2-dimensional vector given by πn=Γ(n+α+β+2)Γ(n+β+2) Γ(n+α+1)Γ(n+1)Γ(β+2)2(n+α+β−k+2) k(2n+α+β+2) n+k,(β−k+1)(2n+α+β+3)(n+α+β+2) (n+α+1)(n+k+1) . From this explicit expression we can easily obtain several quantities. Data of special interest may be the initial value and the asymptotic behavior. The initial value is given by π0=Γ(α+β+3) Γ(α+1)Γ(β+2)(α+β−k+2) 1,(β−k+1)(α+β+3) (α+1)(k+1) . The asymptotic behavior follows using asymptotic formulas for the Gamma function such as Γ(z+α) Γ(z)≈zαas |z|→∞. Hence, we have lim n→∞ πn=⎧ ⎪ ⎨ ⎪ ⎩ (∞,∞)ifβ>−1 2, 4 π(2k,1−2k)if β=−1 2, (0,0) if −1<β<−1 2. In what follows we shall include plots of the two components πn 1and πn 2, as functions of n, in a few representative cases. The general shape of both curves can change depending on the values of the parameters α, β, and k. A look at the role of the parameter βgives rise to four regions, namely −1<β<−1/2, β=−1/2, −1/2< β<0, and β≥0. The parameter αonly has influence on cosmetic changes like curvature and initial values depending on −1<α<0orα≥0, while kaffects the shape of the plots when its value is the middle point β+1 2of its possible range and the situation in the rest of values is quite symmetric. Figures 9.1 and 9.2 show the most interesting situations when β>0. Figures 9.3 and 9.4 show how the situation can change for small perturbations around β=−1/2. In Figure 9.3 both curves have a logarithmic growth and the second component has a minimum, while in Figure 9.4 both curves tend to 0. Downloaded 06/16/16 to 150.214.182.169. Redistribution subject to SIAM license or copyright; see http://www.siam.org/journals/ojsa.php Copyright © by SIAM. Unauthorized reproduction of this article is prohibited. 758 F. ALBERTO GR¨ UNBAUM AND MANUEL D. DE LA IGLESIA 0 1 2 3 4 5 6 7 0.5 0.6 0.7 0.8 0.9 1 1.1 Π1 n Π2 n Fig. 9.3.α=−0.8,β =−0.4,k=0.3. 0 1 2 3 4 5 6 7 0.35 0.4 0.45 0.5 0.55 0.6 0.65 0.7 Π 1 n Π 2 n Fig. 9.4.α=−0.9,β =−0.6,k=0.2. 0 1 2 3 4 5 6 7 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 Π1 n Π2 n Fig. 9.5.α=−0.98,β =−0.6,k=0.3. 0 1 2 3 4 5 6 7 0 0.2 0.4 0.6 Π1 n Π2 n Fig. 9.6.α=−0.9,β =−0.8,k=0.05. 0 1 2 3 4 5 6 7 0.3 0.4 0.5 0.6 0.7 0.8 0.9 Π1 n Π2 n Fig. 9.7.α=−0.92,β =−0.5,k=0.3. 0 1 2 3 4 5 6 7 0.55 0.6 0.65 0.7 0.75 Π1 n Π2 n Fig. 9.8.α=−0.6857,β =−0.5,k=0.3. Downloaded 06/16/16 to 150.214.182.169. Redistribution subject to SIAM license or copyright; see http://www.siam.org/journals/ojsa.php Copyright © by SIAM. Unauthorized reproduction of this article is prohibited. MATRIX VALUED ORTHOGONAL POLYNOMIALS AND QBDs 759 0 1 2 3 4 5 6 7 0.5 0.6 0.7 0.8 0.9 Π1 n Π2 n Fig. 9.9.α=−0.4857,β =−0.5,k=0.3. 0 1 2 3 4 5 6 7 0.45 0.5 0.55 0.6 0.65 0.7 0.75 0.8 Π 1 n Π 2 n Fig. 9.10.α=−0.9,β =−0.5,k=0.25. 0 1 2 3 4 5 6 7 0.4 0.5 0.6 0.7 0.8 Π 1 n Π 2 n Fig. 9.11.α=−0.8,β =−0.5,k=0.3421. 0 1 2 3 4 5 6 7 0.65 0.675 0.7 0.725 0.75 0.775 Π 1 n Π 2 n Fig. 9.12.α=−0.7,β =−0.5,k=0.25. We observe that Figure 9.5, with know approaching β+1, is similar to Figure 9.4. In Figure 9.6 we observe the consequences of kbeing very small. The remaining figures refer to the case β=−1/2, where both components converge for large n. In Figure 9.7 we observe that the initial value of the first component is always lower than that of the second component and that the initial value of the second component is always greater than that of the first component. A small change of the value of αhas the effect that the first component is always greater than the second component, as we can see in Figure 9.8. Small perturbations on αchange the curvatures of the components in Figure 9.9 with respect to Figure 9.8. In Figure 9.10 both components tend to the same value without ever touching, a consequence of choosing k=β+1 2. The last two figures show how the situation can change for small perturbations of αand k. In Figure 9.11 both curves start from the same value and then they converge to different limits, while in Figure 9.12 both components converge to the same limit. 10. Concluding remarks. A block tridiagonal matrix Lwith nonnegative entries and individual rows that add up to 1 gives rise to a quasi-birth-and-death process. The explicit evaluation of Ln, for arbitrary n=1,2,3,..., can be greatly simplified by using ideas that go back to Karlin and McGregor and have been explicitly set forth in [3, 10]. The only major difficulty here is that of computing the weight matrix W(x). In this paper we start from a rich group theoretical situation that yields W(x)aswell Downloaded 06/16/16 to 150.214.182.169. Redistribution subject to SIAM license or copyright; see http://www.siam.org/journals/ojsa.php Copyright © by SIAM. Unauthorized reproduction of this article is prohibited. 760 F. ALBERTO GR¨ UNBAUM AND MANUEL D. DE LA IGLESIA as a matrix Lof the type envisaged previously. There are enough free parameters here to give instances of transient as well as recurrent Markov chains. It remains as interesting challenge to find some real life applications to this large collection of examples. REFERENCES [1] S. Basu and N. K. Bose,Matrix Stieltjes series and network models, SIAM J. Matrix Anal. Appl., 14 (1983), pp. 209–222. [2] Ju. M. Berezans’ki˘ ı,Expansions in Eigenfunctions of Selfadjoint Operators, Transl. Math. Monogr. 17, American Mathematical Society, Providence, RI, 1968. [3] H. 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