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SIAM J. MATRIX ANAL. APPL.c
2008 Socie y o Indus ial and Applied Ma hema ics
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‡
Abs ac . We conside a amily o ma ix alued o hogonal polynomials ob ained by Pacha oni
and Ti ao in connec ion wi h sphe ical unc ions o he pai (SU(N+ 1), U(N)); see [I. Pacha oni
and J. A. Ti ao, Cons . App ox., 25 (2007), pp. 177–192]. A e an app op ia e conjuga ion, we
ob ain a new amily o ma ix alued o hogonal polynomials whe e he co esponding block Jacobi
ma ix is s ochas ic and has special p obabilis ic p ope ies. This gi es a highly non i ial example
o a nonhomogeneous quasi-bi h-and-dea h p ocess o which we can explici ly compu e i s “n-
s ep ansi ion p obabili y ma ix” and i s in a ian dis ibu ion. The ichness o he ma hema ical
s uc u es in ol ed he e allows us o gi e hese explici esul s o a se e al pa ame e amily o
quasi-bi h-and-dea h p ocesses wi h an a bi a y (fini e) numbe o phases. Some o hese esul s
a e plo ed o show he effec ha choices o he pa ame e alues ha e on he in a ian dis ibu ion.
Key wo ds. ma ix alued o hogonal polynomials, Ma ko chains, block idiagonal ansi ion
ma ix, quasi-bi h-and-dea h p ocesses
AMS subjec classifica ions. 60J10, 42C05
DOI. 10.1137/070697604
1. Pu pose and con en s o he pape . The aim o his pape is o ie o-
ge he wo subjec s ha ha e ecei ed qui e a bi o a en ion ecen ly. We will no
gi e a de ailed explana ion o ei he one o hem, since his would equi e oo much
space and i has been done p ope ly in he li e a u e al eady. Besides, since hese
wo subjec s equi e a he diffe en backg ounds, an ab-ini io exposi ion would be a
o midable ask. To compensa e o his we gi e a b ie his o ical iew o how hese
opics de eloped and hen combine hem a he app op ia e poin . The con en s o
his pape can be di ided in o h ee pa s.
A fi s pa gi es a b ie accoun o he subjec s ha a e going o play a ole in
his pape . The in oduc ion con ains some his o ical de elopmen s ying he momen
p oblem wi h spec al heo y and a quick look a bi h-and-dea h p ocesses, including
he appea ance o he app op ia e o hogonal polynomials. Sec ion 3 e iews e y
b iefly K e˘ın’s heo y o ma ix alued o hogonal polynomials and discusses he fi s
example ele an o ou conside a ions. Sec ion 4 gi es a minimal desc ip ion o he
class o Ma ko chains known as quasi-bi h-and-dea h p ocesses and alks abou he
e y na u al connec ion be ween his and he p e ious sec ion.
The second pa in oduces he amily o examples a ising om g oup ep esen-
a ion heo y ha we a e going o use in his pape . Sec ion 5 gi es a guide o he
li e a u e on ma ix alued sphe ical unc ions aimed a showing how he examples
discussed in sec ion 6 a ose. Sec ion 6 gi es he ba e-bones de ails o he ex ensi e
∗Recei ed by he edi o s July 18, 2007; accep ed o publica ion (in e ised o m) by H. J. Wo-
e deman Ma ch 14, 2008; published elec onically July 2, 2008.
h p://www.siam.o g/jou nals/simax/30-2/69760.h ml
†Depa men o Ma hema ics, Uni e si y o Cali o nia, Be keley, Be keley, CA 94720 (g unbaum@
ma h.be keley.edu). This au ho ’s wo k was pa ially suppo ed by NSF g an DMS-0603901.
‡Depa amen o de An´alisis Ma em´a ico, Uni e sidad de Se illa, Apdo (P.O. BOX) 1160, 41080
Se illa, Spain ([email p o ec ed]). This au ho ’s wo k was pa ially suppo ed by D.G.E.S., e .
BFM2003-06335-C03-01, FQM-262 P06-FQM-01735, FQM-481 (Jun a de Andaluc´ıa).
741
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742 F. ALBERTO GR¨
UNBAUM AND MANUEL D. DE LA IGLESIA
wo k ca ied ou in [30] in he case o he complex p ojec i e space. Sec ion 7 shows
how o conjuga e he weigh ma ix om [30] o ob ain a amily o ma ix alued
o hogonal polynomials wi h ex a p obabilis ic p ope ies.
The hi d pa concen a es on he p obabilis ic aspec s o ou amily o exam-
ples. Sec ion 8 deals wi h a numbe o issues o p obabilis ic na u e and displays he
ne wo k associa ed wi h ou examples. In pa icula we find explici exp essions o
he in a ian dis ibu ion. Sec ion 9 gi es g aphical displays o some esul s ob ained
om he exac o mulas in he p e ious sec ions. The goal he e is o show ha by
a ying he pa ame e s affo ded by he g oup heo e ical si ua ion one can ob ain
qui e a ange o diffe en p obabilis ic beha io s. Finally, sec ion 10 gi es a summa y
o he esul s o he pape and he challenges ha lie ahead.
2. In oduc ion. The classical Hausdo ff momen p oblem, ha o de e mining
a measu e dψ(x) in he in e al [−1,1] om i s momen s
σn=1
−1
xndψ(x),
o igina ed in e y conc e e p oblems a he end o he 19 h cen u y and was discussed
by people such as Chebyshe , Ma ko , and S iel jes. In he hands o Weyl and a ew
o he s, his showed he a eaching powe o he mode n heo y o unc ional analysis
in he ea ly pa o he 20 h cen u y. The main ing edien he e is o connec his
p oblem wi h he spec al heo y o a second o de diffe ence ope a o (buil om he
momen s σn) ac ing on unc ions defined on he nonnega i e in ege s. The momen s
in ques ion de e mine (up o scala s) a amily o polynomials {Qn(x)}n≥0, and hese
polynomials a e he eigen unc ions o he second o de diffe ence ope a o alluded o
abo e. In he app op ia e Hilbe space his ope a o is symme ic, and he p oblem
o finding dψ(x) is he p oblem o finding sel -adjoin ex ensions o his symme ic
ope a o . Unde ce ain condi ions he e is a unique such ex ension and hus a unique
solu ion o he momen p oblem we s a ed om, bu a any a e any ex ension gi es
a measu e dψ(x) ha makes he polynomials o hogonal wi h espec o each o he .
To ge close o ou subjec we need a ew mo e ing edien s. One o hem is gi en
in he es o his sec ion, and he o he wo in sec ions 3 and 4.
The p esence o a second o de diffe ence ope a o ac ing on he space o unc ions
defined on he nonnega i e in ege s, i.e., a semi-infini e idiagonal ma ix, makes i
na u al o hink o a e y special kind o Ma ko chain on he space o nonnega i e
in ege s. These a e he so-called bi h-and-dea h p ocesses whe e a each disc e e uni
o ime a ansi ion is allowed om s a e i o s a e jwi h p obabili y Pij and we pu
Pij =0i |i−j|>1.The one-s ep ansi ion p obabili y ma ix is gi en by
(2.1) P=⎛
⎜
⎜
⎜
⎜
⎝
0p0
q1 1p1
q2 2p2
.........
⎞
⎟
⎟
⎟
⎟
⎠.
We will assume ha pj>0, qj+1 >0, and j≥0 o j≥0. We also assume
pj+ j+qj= 1 o j≥1 and by pu ing p0+ 0≤1 we allow o he s a e j=0 o
be an abso bing s a e (wi h p obabili y 1 −p0− 0). Some o hese condi ions can be
elaxed.
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MATRIX VALUED ORTHOGONAL POLYNOMIALS AND QBDs 743
The p oblem he e is o ob ain an exp ession o he so-called “n-s ep ansi ion
p obabili y ma ix,” gi ing he p obabili y o going be ween any wo s a es in ns eps.
By making use o he ideas men ioned abo e, i.e., by b inging in an app op ia e Hilbe
space and applying hen he spec al heo em, Ka lin and McG ego [20] ob ained a
nea ep esen a ion o mula o he quan i y o in e es , as ecalled below.
I one in oduces he polynomials {Qn(x)}n≥0by he condi ions Q−1(x)=0,
Q0(x) = 1 and by using he no a ion
φ=⎛
⎜
⎜
⎝
Q0(x)
Q1(x)
.
.
.
⎞
⎟
⎟
⎠,
one insis s on he ecu sion ela ion
Pφ=xφ,
i is possible o p o e he exis ence o a unique measu e dψ(x) suppo ed in [−1,1]
such ha
1
−1
Qi(x)Qj(x)dψ(x)1
−1
Qj(x)2dψ(x)=δij
and one ge s he Ka lin–McG ego ep esen a ion o mula
(2.2) Pn
ij =1
−1
xnQi(x)Qj(x)dψ(x)1
−1
Qj(x)2dψ(x).
I ime is aken o be con inuous, as i is done in o he pape s by Ka lin and
McG ego , hen his o mula and he ma ix Psuffe only cosme ic changes.
I is in e es ing o no ice ha his seminal pape o Ka lin and McG ego e e s
bo h o he s anda d ex on he momen p oblem a he ime [33], as well as o he
ac ha Felle and McKean had al eady ecognized he ele ance o he Hilbe space
se up in he s udy o diffusion p ocesses; see [7, 26]. One can men ion o he pape s,
such as [8, 17, 19, 25], whe e simila ideas we e a play.
The las sec ion o [20] deals wi h he case o a fini e s a e space and he case
when he nonnega i e in ege s a e eplaced by he se o all in ege s. Since one is
using a e y powe ul ool such as he spec al heo em i is clea ha an adap a ion
o he ideas om bi h-and-dea h p ocesses will wo k he e oo. In he case o he
in ege s, one is dealing wi h a s a e space wi h wo singula poin s (one a each end
o he line), and in his case Weyl and o he s had al eady ound he co ec ool: one
eplaces he spec al measu e dψ(x)bya2×2 nonnega i e ma ix. The pape o
Ka lin and McG ego concludes wi h he explici compu a ion o his ma ix in he
case o he doubly infini e andom walk. The gene al o mula is gi en in exp ession
(12) o [20] o he case o disc e e ime and also in (6.8) o [18] o con inuous ime.
The ep esen a ion o mula gi en abo e is o in insic in e es : he compu a ion
o he le -hand side o (2.2) o fixed i, j and a bi a y alues o nin ol es all o he
en ies o (2.1). Howe e , i dψ(x) is known, hen he igh -hand side o (2.2) gi es a
way o compu ing his quan i y using only a fixed numbe o en ies o (2.1).
The applicabili y o (2.2) depends o a la ge ex en on ou abili y o ob ain use ul
exp essions o he o hogonal polynomials and he o hogonali y measu e associa ed
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744 F. ALBERTO GR¨
UNBAUM AND MANUEL D. DE LA IGLESIA
wi h P. I one looks a ound in he li e a u e one disco e s ha he numbe o cases
whe e his is possible is a he small.
We close his sec ion by showing how one can compu e in he case o a s ochas ic
ma ix Pi s in a ian (s a iona y) dis ibu ion, i.e., he (unique up o scala s) ow
ec o
π=(π0,π
1,π
2,...)
such ha
πP=π.
Recall ha a ma ix Pwi h nonnega i e en ies is called s ochas ic i he sum o he
elemen s in any ow equals uni y.
We fi s ob ain, using 0+p0= 1, ha π1=π0p0/q1. Then one p o es by
induc ion ha o i≥1weha e
πi=π0(p0p1...p
i−1)/(q1q2...q
i).
This has he consequence ha
πi+1/πi=pi/qi+1.
Now o i≥0weha e
xQi(x)=piQi+1(x)+ iQi(x)+qiQi−1(x)
wi h q0= 0. In eg a ing his a e mul iplica ion by Qi+1 o Qi−1gi es
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 hese wo esul s we ge ha he a io o he wo in eg als abo e is gi en
by he common alue
qi+1/pi=πi/πi+1.
The mo al o his is ha he solu ion o πP=πcan be compu ed (up o a common
mul iplica i e scala ) ei he om he ma ix Pi sel o om he knowledge o he
in eg als
1
−1
Q2
i(x)dψ(x).
In pa icula i we ha e an homogeneous bi h-and-dea h p ocess whe e pi=pand
qi=qindependen ly o he alue o i, hen we ha e ha he componen s o πa e
gi en by πi=π0(p/q)i,i≥0.
3. Ma ix alued o hogonal polynomials. We need wo mo e cha ac e s
o be able o s a ou ale. The fi s one is he heo y o ma ix alued o hogonal
polynomials, whose ba e-bones de elopmen is gi en in wo pape s by K e˘ın [21, 22].
The e is no w i en accoun o he mo i a ion ha led K e˘ın o his heo y, bu one
can easily see he connec ion wi h he spec al heo y o diffe ence ope a o s on he
in ege s. This is e y nicely discussed in he book by Be ezans’ki˘ı [2]. In ac he s udy
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MATRIX VALUED ORTHOGONAL POLYNOMIALS AND QBDs 745
o he classical second o de diffe ence ope a o on he in ege s is done in de ail in [2]
and may cons i u e he fi s example o he heo y o K e˘ın, whe e he polynomials
and hei o hogonali y weigh ma ix W(x) a e bo h explici ly gi en. This is, o
cou se, a special case o he ma ix ha appea s in he las sec ion o [20] o gene al
alues o pand q(p+q= 1).
We gi e now a b ie accoun o K e˘ın’s heo y.
Gi en a posi i e defini e ma ix alued measu able weigh unc ion W=W(x)
wi h fini e momen s we can conside he skew symme ic bilinea o m defined o any
pai o ma ix alued polynomial unc ions P(x) and Q(x) by he nume ical ma ix
(P,Q)=(P,Q)W=R
P(x)W(x)Q∗(x)dx,
whe e Q∗(x) deno es he conjuga e anspose o Q(x). We define he ma ix alued
no m o Pby
(3.1) P2=(P,P)W.
One can also deal wi h a mo e gene al weigh ma ix W(x); see [4].
This leads, using he G am–Schmid p ocess, o he exis ence o a sequence o
ma ix alued o hogonal polynomials wi h nonsingula leading coefficien s. Gi en an
o hogonal sequence {Qn(x)}n≥0one ge s a h ee e m ecu sion ela ion
(3.2) xQn(x)=AnQn+1(x)+BnQn(x)+CnQn−1(x),
whe e Anis nonsingula . We will deno e by L he co esponding Jacobi ma ix,
defined by he ollowing block idiagonal semi-infini e ma ix:
L=⎛
⎜
⎜
⎜
⎜
⎝
B0A0
C1B1A1
C2B2A2
.........
⎞
⎟
⎟
⎟
⎟
⎠.
Using he no a ion
Φ=⎛
⎜
⎜
⎝
Q0(x)
Q1(x)
.
.
.
⎞
⎟
⎟
⎠
he ela ion (3.2) becomes
(3.3) LΦ=xΦ.
We will ese e he symbol P o he case whe e Lbecomes a one-s ep ansi ion
p obabili y ma ix, hough o as a scala ma ix. The co esponding Ma ko chain
( o appea in sec ion 4) will ha e a s a e space ha is mo e complica ed han he se
{0,1,2,...}co esponding o a bi h-and-dea h p ocess ea u ed in sec ion 2.
In he scala case, conc e e examples o o hogonal polynomials, including explici
o mulas o hem as well as hei o hogonali y measu e p eceded he de elopmen
o any gene al heo y. P ominen examples a e he He mi e, Lague e, and Jacobi
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746 F. ALBERTO GR¨
UNBAUM AND MANUEL D. DE LA IGLESIA
polynomials. These examples a ose om conc e e p oblems in he eigh een h and
nine een h cen u ies and played a undamen al ole, in he hands o Sch ¨odinge , in
he de elopmen o quan um mechanics a ound 1925.
The si ua ion in he ma ix alued case is en i ely diffe en : he gene al heo y
jus desc ibed abo e came fi s . Un il a ew yea s ago, i may be ha he only
non i ial example was he one included in Be ezans’ki˘ı’s book [2], alluded o abo e
and ecalled below o he benefi o he eade .
Conside he block idiagonal ma ix
L=⎛
⎜
⎜
⎜
⎜
⎝
B0I
C1B1I
C2B2I
.........
⎞
⎟
⎟
⎟
⎟
⎠
wi h 2 ×2 blocks gi en as ollows:
B0=1
201
10
,B
n=0 i n≥1,
Cn=1
4Ii n≥1,
whe e Is ands o he iden i y ma ix. In his case one can compu e explici ly he
ma ix alued polynomials {Qn(x)}n≥0gi en by
xQn(x)=Qn+1(x)+BnQn(x)+CnQn−1(x),Q
−1(x)=0,Q
0(x)=I.
One ge s
Qn(x)= 1
2nUn(x)−Un−1(x)
−Un−1(x)Un(x),
whe e Un(x) a e he Chebyshe polynomials o he second kind.
The o hogonali y measu e is ead off om he iden i y
4i
π1
−1
Qi(x)1
√1−x21x
x1Q∗
j(x)dx =δijI.
P oceeding as in [3, 10, 20] one ob ains a Ka lin–McG ego ep esen a ion. We ge ,
o n=0,1,2,...,
Ln
ij =4i
π1
−1
xnQi(x)1
√1−x21x
x1Q∗
j(x)dx,
whe e Ln
ij s ands o he (i, j) block o he ma ix Ln. As is usual o bi h-and-dea h
p ocesses, he indices i, j un om 0 on.
In his way, as no iced in [10], one can compu e he en ies o he powe s Lnwi h
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MATRIX VALUED ORTHOGONAL POLYNOMIALS AND QBDs 747
L hough o as a pen adiagonal scala ma ix, namely
L=
⎛
⎜
⎜
⎜
⎜
⎜
⎜
⎜
⎜
⎜
⎝
01
210
1
20010
1
40001
...
1
4000
...
............
⎞
⎟
⎟
⎟
⎟
⎟
⎟
⎟
⎟
⎟
⎠
.
Lis no a s ochas ic ma ix since i s ows do no add up o uni y. Ne e heless,
defining Δ o be he 2 ×2 block diagonal ma ix wi h Δii =2
iI o e e y block, we
ge om (3.3) ha ΔLΔ−1ΔΦ = xΔΦ and hus i P=ΔLΔ−1and 
Φ = ΔΦ, we
ha e P
Φ=x
Φ. The scala e sion o Pis now he s ochas ic ma ix
P=
⎛
⎜
⎜
⎜
⎜
⎜
⎜
⎜
⎜
⎜
⎝
01
2
1
20
1
200 1
20
1
200 0 1
2
...
1
2000
...
............
⎞
⎟
⎟
⎟
⎟
⎟
⎟
⎟
⎟
⎟
⎠
.
Obse e ha he no m o 
Qn, defined in (3.1), sa isfies 
Qn2=π. This is no hing
bu he example conside ed a he end o [20] in he special case o p=q=1/2.
In he las ew yea s a numbe o new amilies o ma ix alued o hogonal poly-
nomials ha e been compu ed explici ly along wi h hei o hogonali y measu e. Typ-
ically hey a e join eigen unc ions o some fixed diffe en ial ope a o wi h ma ix
coefficien s. This sea ch was ini ia ed in [5], bu non i ial examples we e no disco -
e ed un il [6] and [13, 16]. The amily o examples we will conside la e is ela ed o
one o hese examples and is ob ained by modi ying he si ua ion discussed in [30].
4. Quasi-bi h-and-dea h p ocesses. Now we come o he las cha ac e o
ou s o y. Fo ou pu poses, we conside a wo dimensional Ma ko chain wi h disc e e
ime. The s a e space consis s o he pai o in ege s (i, j), i∈{0,1,2,...},j∈
{1,...,d}. The fi s componen is usually called he le el and he second one he
phase. The one-s ep ansi ion p obabili y ma ix, which we will deno e (as be o e)
by Phas a block idiagonal s uc u e (see (4.2)). This indica es ha in one uni
o ime a ansi ion can change he phase wi hou changing he le el, o can change
he le el (and possibly he phase) o ei he o he adjacen le els. The p obabili y o
going in one s ep om s a e (i, j) o s a e (i,j) is gi en by he (j, j) elemen o he
block Pi,i. Clea ly in he case when he numbe o phases dis one we a e back o he
case o an o dina y bi h-and-dea h p ocess. In gene al, hese p ocesses a e known as
(disc e e ime) quasi-bi h-and-dea h p ocesses.
Fo a much mo e de ailed p esen a ion o his field, as well as i s connec ions wi h
queueing p oblems in ne wo k heo y as well as he gene al field o communica ion
sys ems, he eade should consul [24, 27] and some o he e e ences in [3].
Once one has he no ions in oduced in he p e ious sec ions i is e y na u-
al o connec hem and o analyze hese in e es ing Ma ko chains in e ms o he
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748 F. ALBERTO GR¨
UNBAUM AND MANUEL D. DE LA IGLESIA
co esponding spec al p ope ies o he esul ing amily o ma ix alued o hogonal
polynomials. As poin ed ou be o e one can see he seed o his in he wo k o K e˘ın,
as well as in he o iginal pape [20].
We ha e iden ified wo e e ences whe e he co esponding Ka lin–McG ego ep-
esen a ion o mula has been explici ly gi en, bu he e may be o he s since his is
such a na u al ex ension o he scala heo y; see [3] and [10]. In he case o quasi-
bi h-and-dea h p ocesses one eplaces (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 diffe en bu ela ed pa h o his ci cle o ideas in connec ion wi h ne wo k
models can be seen in [1].
In [3] one finds some in e es ing examples whe e his ep esen a ion o mula is
compu ed explici ly, including a new de i a ion o he esul dealing wi h he case o
andom walk on he in ege s. In [10] one finds an example, aken om [9], whe e he
obse a ion is made ha one has a s ochas ic ma ix. The amily o examples o be
conside ed in he ollowing sec ions is an ex ension o his example.
Gi en he block ansi ion p obabili y ma ix P, he p oblem o compu ing an
in a ian dis ibu ion ow ec o , i.e., a ec o wi h nonnega i e en ies πi
j,
π=(π0;π1;...)≡(π0
1,π0
2,...,π0
d;π1
1,π1
2,...,π1
d;...)
such ha
πP=π
leads o a complica ed sys em o equa ions.
I
(4.2) P=⎛
⎜
⎜
⎜
⎜
⎝
B0A0
C1B1A1
C2B2A2
.........
⎞
⎟
⎟
⎟
⎟
⎠,
we ha e
π0B0+π1C1=π0
and hen, o n≥1,
πn−1An−1+πnBn+πn+1Cn+1 =πn.
This gi es, as is easy o check, he a he unpleasan exp essions
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 o mulas equi e ha he ma ices Cnbe in e ible. Unde hese condi ions, one
can de i e nice looking exp essions o he in a ian dis ibu ion (see [23]). The e a e
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MATRIX VALUED ORTHOGONAL POLYNOMIALS AND QBDs 749
many issues he e ha we lea e un ouched in his analysis. Fo ins ance, he possibili y
o choosing π0wi h nonnega i e en ies so ha all he subsequen πnwill ha e his
p ope y equi es ex a condi ions.
The gene al heo y o quasi-bi h-and-dea h p ocesses is no es ic ed o he case
when he ma ices An,Bn, and Cna e all squa e ma ices o he same size and An
and Cna e nonsingula . I emains an in e es ing challenge o find a ma hema ical
se up ha can accommoda e such a si ua ion.
Now ha we ha e seen how o ex end he Ka lin–McG ego ep esen a ion o -
mula o he block idiagonal case i is impo an o ge examples whe e he poly-
nomials and he o hogonali y ma ix can be explici ly w i en down. This is he
pu pose o he nex h ee sec ions.
5. Ma ix alued sphe ical unc ions and ma ix alued o hogonal
polynomials. The heo y o ma ix alued sphe ical unc ions, ini ially discussed
in [34], is one o he ou es leading o explici amilies o ma ix alued o hogonal
polynomials and hei o hogonali y measu e. The fi s amily o examples appea ed
in [13, 16] in connec ion wi h G= SU(3). The size o he ma ices he e is al eady
a bi a y, and he o hogonali y ma ix has a scala ac o o he o m xα(1 −x).
The ex ension o he case whe e his scala ac o can be aken o be xα(1 −x)β o
a bi a y α, β > −1 was unde aken in [9] in he 2 ×2 case wi hou any e e ence o
g oup ep esen a ion heo y. Fu he examples o his kind a e gi en in [14]. The ole
o g oup ep esen a ion heo y in ge ing away om he 2×2 case can be seen in [15].
Finally, [30] displays o he case o G=SU(N+1) amilies o o hogonal polynomials
depending on h ee pa ame e s, α, β, and k. In he special case o k=β+1
2, one
eco e s he esul s o [9].
These o hogonal polynomials a e gi en by p ope ly “packaged and conjuga ed”
se s o ma ix alued sphe ical unc ions. These sphe ical unc ions co espond o
i educible ep esen a ions o U(N) and he e o e a e pa ame e ized by pa i ions
μ=(m1,m
2,...,m
N)∈ZNsuch ha m1≥m2≥···≥mN.
In his pape , ollowing [30], we use only “one s ep” ep esen a ions gi en by a pa i-
ion
μ=(m+,...,m+
 
k
,m,...,m
 
N−k
),1≤k≤N−1.
In e ms o he pa ame e s αand β, one has α=mand β=N−1. The emaining
ee pa ame e will de e mine he size o he co esponding ma ix alued o hogonal
polynomials and is independen o N. In he nex sec ion i will be ela ed o he
pa ame e dappea ing in [3].
The examples ha ha e been wo ked ou so a indica e ha he ma ix alued
o hogonal polynomials ha esul om ma ix alued sphe ical unc ions lead o a
block idiagonal ma ix ha can be made in o a s ochas ic one. This will be seen,
o ou amily o examples, in sec ion 8.
Al hough i is possible o ob ain examples o s ochas ic ma ices a ising in a
diffe en ashion, see, o ins ance [2], we a e no awa e o any o he gene al scheme
ha would p oduce hese desi able kinds o ma ices in a sys ema ic ashion.
6. A amily o examples a ising om he complex p ojec i e space. In
wha ollows we shall use Eij o deno e he ma ix wi h en y (i, j), which is equal o
1 and 0 elsewhe e, whe e he indices i, j un om 0 on.
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756 F. ALBERTO GR¨
UNBAUM AND MANUEL D. DE LA IGLESIA
The s a e space and he co esponding one-s ep ansi ions appea as ollows:
.
.
.
.
.
.
.
.
.
.
.
.
1d+1 2d+1 3d+1
2d+2 2d+2 3d+2
3d+3 2d+3 3d+3
d2d3d4d
9. The shape o he in a ian dis ibu ion. In his sec ion we will s udy in
mo e de ail he beha io o he in a ian dis ibu ion when he numbe o phases d
is equal o wo, a luxu y we can affo d since we ha e an analy ic exp ession. In his
case, he associa ed ne wo k akes he o m
1357
2468
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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 in a ian dis ibu ion πsuch ha πP=πis gi en by
π=(π0;π1;...),
whe e πn,n≥0, is he 2-dimensional ec o gi en 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) .
F om his explici exp ession we can easily ob ain se e al quan i ies. Da a o special
in e es may be he ini ial alue and he asymp o ic beha io . The ini ial alue is
gi en 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 asymp o ic beha io ollows using asymp o ic o mulas o he Gamma unc-
ion such as Γ(z+α)
Γ(z)≈zαas |z|→∞. Hence, we ha e
lim
n→∞
πn=⎧
⎪
⎨
⎪
⎩
(∞,∞)i β>−1
2,
4
π(2k,1−2k)i β=−1
2,
(0,0) i −1<β<−1
2.
In wha ollows we shall include plo s o he wo componen s πn
1and πn
2, as unc ions
o n, in a ew ep esen a i e cases. The gene al shape o bo h cu es can change
depending on he alues o he pa ame e s α, β, and k. A look a he ole o he
pa ame e βgi es ise o ou egions, namely −1<β<−1/2, β=−1/2, −1/2<
β<0, and β≥0. The pa ame e αonly has influence on cosme ic changes like
cu a u e and ini ial alues depending on −1<α<0o α≥0, while kaffec s he
shape o he plo s when i s alue is he middle poin β+1
2o i s possible ange and he
si ua ion in he es o alues is qui e symme ic.
Figu es 9.1 and 9.2 show he mos in e es ing si ua ions when β>0.
Figu es 9.3 and 9.4 show how he si ua ion can change o small pe u ba ions
a ound β=−1/2. In Figu e 9.3 bo h cu es ha e a loga i hmic g ow h and he
second componen has a minimum, while in Figu e 9.4 bo h cu es end o 0.
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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.
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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 obse e ha Figu e 9.5, wi h know app oaching β+1, is simila o Figu e 9.4.
In Figu e 9.6 we obse e he consequences o kbeing e y small.
The emaining figu es e e o he case β=−1/2, whe e bo h componen s con-
e ge o la ge n. In Figu e 9.7 we obse e ha he ini ial alue o he fi s componen
is always lowe han ha o he second componen and ha he ini ial alue o he
second componen is always g ea e han ha o he fi s componen . A small change
o he alue o αhas he effec ha he fi s componen is always g ea e han he
second componen , as we can see in Figu e 9.8.
Small pe u ba ions on αchange he cu a u es o he componen s in Figu e 9.9
wi h espec o Figu e 9.8. In Figu e 9.10 bo h componen s end o he same alue
wi hou e e ouching, a consequence o choosing k=β+1
2.
The las wo figu es show how he si ua ion can change o small pe u ba ions o
αand k. In Figu e 9.11 bo h cu es s a om he same alue and hen hey con e ge
o diffe en limi s, while in Figu e 9.12 bo h componen s con e ge o he same limi .
10. Concluding ema ks. A block idiagonal ma ix Lwi h nonnega i e en-
ies and indi idual ows ha add up o 1 gi es ise o a quasi-bi h-and-dea h p ocess.
The explici e alua ion o Ln, o a bi a y n=1,2,3,..., can be g ea ly simplified by
using ideas ha go back o Ka lin and McG ego and ha e been explici ly se o h in
[3, 10]. The only majo difficul y he e is ha o compu ing he weigh ma ix W(x).
In his pape we s a om a ich g oup heo e ical si ua ion ha yields W(x)aswell
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760 F. ALBERTO GR¨
UNBAUM AND MANUEL D. DE LA IGLESIA
as a ma ix Lo he ype en isaged p e iously. The e a e enough ee pa ame e s
he e o gi e ins ances o ansien as well as ecu en Ma ko chains. I emains
as in e es ing challenge o find some eal li e applica ions o his la ge collec ion o
examples.
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