Multiple orthogonal polynomials as special functions

Multiple orthogonal polynomials as special 

functions 

Walter Van Assche 

April 7, 2011 

1 / 27

Multiple orthogonal polynomials 

Multiple orthogonal 

polynomials 

definition 

type II multiple 

orthogonal polynomials 

type I multiple 


Determinantal point 

processes 

Recurrence relations 

Various examples 

2 / 27

definition 

Polynomials of one variable but indexed by a multi-index 

n = (n1,n2,...,nr) ∈ N r (r ≥ 1) satisfying orthogonality 

conditions with respect tor positive measureµ1,...,µr on the 

real line. 

They appear naturally in Hermite-Padé approximation tor 

functions 

fj(z) = 

 

dµj(x) 

, 1 ≤ j ≤ r. 

z −x 

There are two types: type I and type II 



definition 






processes 



3 / 27

type II multiple orthogonal polynomials 

Pn is a monic polynomial of degree|n| = n1 +n2 +···+nr 

for which 

 

Pn(x)x k dµ1(x) = 0, k = 0,1,...,n1 −1 

 

. 

Pn(x)x k dµr(x) = 0, k = 0,1,...,nr −1 

|n| linear conditions for|n| unknowns. 

Solution exists and unique: n is a normal index for type II. 



definition 






processes 



4 / 27

type I multiple orthogonal polynomials 

(An,1,...,An,r) is a vector ofr polynomials, withAn,j of 

degreenj −1, for which 

 

 

x k 

x |n|−1 

r 

An,j dµj(x) = 0, k = 0,1,...,|n|−2 

j=1 

r 

An,j dµj(x) = 1. 

j=1 

|n| linear conditions for|n| unknowns. 

Solution exists and unique: n is a normal index for type I (⇔for 

type II). 

Notation: 

Qn(x) = 

r 

j=1 

An,j(x)wj(x), wj(x) = dµj(x) 

dµ(x) . 



definition 






processes 



5 / 27

Determinantal point processes 




processes 

what? 

definition 

biorthogonal ensembles 

random matrices 

random matrices with 

external source 

non-intersecting 

Brownian motions 



squared Bessel paths 



6 / 27

what? 

Surveys 

■ A. Borodin: Determinantal point processes, arXiv:0911.1153 [math.PR] 

■ T. Tao: http://terrytao.worldpress.com/2009/08/23/determinantal-processes 

■ A. Soshnikov: Determinantal random point fields, Russian Math. Surveys 55 

(2000) 

■ R. Lyons: Determinantal probability measures, Publ. Math. Inst. Hautes 

Etudes Sci. 98 (2003) 

■ K. Johansson: Random matrices and determinantal processes, 

arXiv:math-ph/0510038 

7 / 27

definition 

A point process onRis determinantal if there exists a kernel 

K : R×R → R such that 

Pr{∃ particle in each(xi,xi +dxi),1 ≤ i ≤ n} 

= det K(xi,xj) n 

i,j=1 dx1dx2...dxn. 

(provided these probabilities are positive, of course). 




processes 

what? 

definition 












8 / 27

iorthogonal ensembles 

If(φi)i=1,2,...,n and(ψi)i=1,2,...,n are functions onRand 

Zn = 

 

R n 

det φi(xj) n 

i,j=1 det ψi(xj) n 

i,j=1 dx1...dxn, 

then the point process with 

P(x1,...,xn) = Z −1 

n det φi(xj) n 

i,j=1 det ψi(xj) n 

i,j=1 

is determinantal with 

and 

K(x,y) = 

n 

i=1 

Gi,j = 

 

n 

(G −1 )i,jφi(x)ψj(y), 

j=1 

R 

φi(x)ψj(x)dx 




processes 

what? 

definition 












9 / 27

andom matrices 

For the Gaussian Unitary Ensemble (GUE) of Hermitiann×n 

matricesM with probability distribution 

Z −1 

n e −2TrV(M) dM 

the eigenvalues are a determinantal point process with 

K(x,y) = 

n−1 

i=0 

pi(x)pi(y)e −V(x)−V(y) 

wherep0(x),p1(x),p2(x),... are the orthonormal polynomials 

for the weighte −2V(x) : 

 

pi(x)pj(x)e −2V(x) dx = δm,n 




processes 

what? 

definition 












10 / 27

andom matrices with external source 

IfAis a given Hermitiann×n matrix and the probability 

distribution is 

Z −1 

n e −Tr(V(M)+AM) dM, 

then the eigenvalues are a determinantal process with 

K(x,y) = 

|n|−1 

 

i=0 

Pni (x)Qni+1 (x) 

wherePn andQn are multiple orthogonal polynomials for the 

measurese −V(x)−ajx (1 ≤ j ≤ r), witha1,...,ar the 

eigenvalues ofAandnj the multiplicity of the eigenvaluesaj. 

The multi-indices(n0,...,nn) are a path inN r fromn0 =0 to 

n |n| = n such thatni+1 −ni =ej for somej ∈ {1,...,r} 

ande1,...,er are the unit vectors inN r . 




processes 

what? 

definition 












11 / 27

non-intersecting Brownian motions 




processes 

what? 

definition 












12 / 26

non-intersecting Brownian motions 




processes 

what? 

definition 












13 / 26


Y(t) = X 2 1(t)+X 2 2(t)+···+X 2 d (t) 

where(X1(t),X2(t),...,Xd(t)) is ad-dimensional Brownian 

motion starting from(a1,...,ad) and ending at(0,0,...,0). 

This is a biorthogonal ensemble which uses modified Bessel 

functionsIα withd = 2(α+1). 




processes 

what? 

definition 












14 / 27





processes 


nearest neighbor 

recurrence relations 

compatibility relation 

Christoffel-Darboux 

formula 


15 / 27

nearest neighbor recurrence relations 

Orthogonal polynomials on the real line always satisfy a 

three-term recurrence relation. 

xpn(x) = an+1pn+1(x)+bnpn(x)+anpn−1(x). 

Multiple orthogonal polynomials (with all multi-indices normal) 

satisfy a system ofr recurrence relations 

xPn(x) = Pn+ek (x)+bn,kPn(x)+ 

r 

j=1 

xQn(x) = Qn−ek (x)+bn−ek,kQn(x)+ 

an,jPn−ej 

r 

j=1 

an,jQn+ej 

(x), 1 ≤ k ≤ r. 




processes 






formula 


(x), 1 ≤ k ≤ r. 

16 / 27


The recurrence coefficients satisfy some partial difference 

equations (Van Assche, 2011). 

Suppose1 ≤ i = j ≤ r, then 

bn+ei,j −bn,j = bn+ej,i −bn,i 

r r 

 

bn+ej,i bn,i 

an+ej,k − an+ei,k = det 

bn+ei,j bn,j 

j=1 k=1 

an,i 

an+ej,i 

= bn−ei,j −bn−ei,i 

bn,j −bn,i 

Reason: Pn+ei+ej (x) can be computed in two ways from the 

recurrence relations: 

■ first computePn+ei 

■ first computePn+ej 

(x) and from therePn+ei+ej (x) 

(x) and from therePn+ej+ei (x) 

 




processes 






formula 


17 / 27

Christoffel-Darboux formula 

For orthogonal polynomials there is the Christoffel-Darboux 

relation 

n−1 

i=0 

pi(x)pi(y) = an 

pn(x)pn−1(y)−pn−1(x)pn(y) 

x−y 

For multiple orthogonal polynomials there is a similar formula 

|n|−1 

 

i=0 

Pni (x)Qni+1 (y) = Pn(x)Qn(y)− r 

j=1 

an,jPn−ej (x)Qn+ej (y) 

x−y 

where(ni) i=0,1,...,|n| is a path inN r fromn0 =0 ton |n| = n 

such that for eachi ∈ {0,1,...,|n|−1} one has 

ni+1 −ni =ej for somej ∈ {1,2,...,r} (Kuijlaars & 

Daems). 




processes 






formula 


. 

18 / 27





processes 



multiple Hermite 


multiple Laguerre 

polynomials, first kind 


polynomials, second 

kind 

multiple Jacobi 




multiple orthogonal 

polynomials and 

modified Bessel 

functionsI 




functionsK 

references 

19 / 27

multiple Hermite polynomials 

∞ 

−∞ 

x k Hn(x)e −x2 +cjx dx = 0, k = 0,1,...,nj −1 

whereci ≤ cj wheneveri = j. 

random matrices with external source (Kuijlaars & Bleher), 

non-intersection Brownian motions (Kuijlaars, Daems, Delvaux, 

Bleher) 

Rodrigues formula 

e −x2 

Hn(x) = (−1) |n| 2 −|n| 

⎛ 

⎝ 

r 

e 

j=1 

Recurrence relations: for1 ≤ k ≤ r 

ck 1 

xHn(x) = Hn+ek (x)+ Hn(x)+ 

2 2 

−cjx dnj 

dx 

r 

j=1 

nj ecjx 

⎞ 

⎠e −x2 

. 

njHn−ej (x). 




processes 









kind 













references 

20 / 27

multiple Laguerre polynomials, first kind 

∞ 

0 

x k Ln(x)x αj e −x dx = 0, k = 0,1,...,nj −1 

whereα1,...,αr > −1 andαi −αj /∈ Z. 

(−1) |n| e −x Ln(x) = 

r 

j=1 

 

x 

−αj dnj 

dx 

xLn(x) = Ln+ek (x)+bn,kLn(x)+ 

an,j = nj(nj +αj) 

r 

i=1,i=j 

nj xnj+αj 

r 

j=1 

bn,j = |n|+nj +αj +1. 

nj +αj −αi 

 

e −x 

an,jLn−ej (x) 

nj −ni +αj −αi 




processes 









kind 













references 

21 / 27

multiple Laguerre polynomials, second kind 

∞ 

0 

x k Ln(x)x α e −cjx dx = 0, k = 0,1,...,nj −1 

withc1,...,cr > 0 andci = cj wheneveri = j. 

Wishart ensembles in random matrix theory. 

(−1) |n| 

⎛ 

⎝ 

r 

j=1 

c nj 

j 

⎞ 

⎠x α Ln(x) = 

r 

j=1 

 

e 

xLn(x) = Ln+ek (x)+bn,kLn(x)+ 

an,j = |n|+α)nj 

c 2 j 

cjx dnj 

dx 

r 

j=1 

, bn,j = |n|+α+1 

cj 

nj e−cjx 

 

x |n|+α 

an,jLn−ej (x) 

+ 

r 

i=1 

ni 

ci 




processes 









kind 













references 

22 / 27

multiple Jacobi polynomials 

1 

0 

x k Pn(x)x αj (1−x) β dx = 0, k = 0,1,...,nj −1 

withα1,...,αr,β > −1 andαi −αj /∈ Z. 

(−1) |n| 

r 

(|n|+αj +β +1)nj (1−x)β Pn(x) 

j=1 

= 

r 

j=1 

 

x 

−αj dnj 

dx 

nj xnj+αj 

 

(1−x) |n|+β 




processes 









kind 













references 

23 / 27

multiple Jacobi polynomials 

xPn(x) = Pn+ek (x)+bn,kPn(x)+ 

r 

j=1 

an,jPn−ej (x) 

Letqr(x) = r 

j=1 (x−αj) andQr,n(x) = r 

j=1 (x−nj −αj), then 

an,j = qr(−|n|−β)qr(nj +αj) 

Qr,n(−|n|−β)Q ′ r,n (nj +αj) 

(nj +αj)(|n|+β) 

(|n|+nj +αj +β +1)(|n|+nj +αj +β)(|n|+nj +αj +β −1) 

bn,j = δn −δn+ej , δn = −(|n|+β) qr(−|n|−β) 

Qr,n(−|n|−β) . 

24 / 27

multiple orthogonal polynomials and modified Bessel 


∞ 

0 

∞ 

0 

x k Pn,m(x)x ν/2 Iν(2 √ x)e −cx dx = 0, k = 0,1,...,n−1 

x k Pn,m(x)x (ν+1)/2 Iν+1(2 √ x)e −cx dx = 0, k = 0,1,...,m−1 

withν > −1 andc > 0. Ifp2n(x) = Pn,n(x) andp2n+1(x) = Pn+1,n(x), 

then 

with 

bn = 

xpn(x) = pn+1(x)+bnpn(x)+cnpn−1(x)+dnpn−2(x) 

c(2n+ν +1))+1 

c 2 

andy = pn(x) satisfies 

, cn = n(2+c(n+ν)) 

c3 , dn = n(n−1) 

c4 xy ′′′ +(−2cx+ν +2)y ′′ +(c 2 x+c(n−ν −2)−1)y ′ −c 2 ny = 0. 

25 / 27

multiple orthogonal polynomials and modified Bessel 


∞ 

0 

∞ 

0 

x k Pn,m(x)x α+ν/2 Kν(2 √ x)dx = 0, k = 0,1,...,n−1 

x k Pn,m(x)x α+(ν+1)/2 Kν+1(2 √ x)dx = 0, k = 0,1,...,m−1 

withα > −1 andν ≥ 0. Letp2n(x) = Pn,n(x) andp2n+1(x) = Pn+1,n(x) 

xpn(x) = pn+1(x)+bnpn(x)+cnpn−1(x)+dnpn−2(x) 

bn = (n+α+1)(3n+α+2ν)−(α+1)(ν −1), 

cn = n(n+α)(n+α+ν)(3n+2α+ν), 

dn = n(n−1)(n+α−1)(n+α)(n+α+ν −1)(n+α+ν) 

andy = pn(x) satisfies 

x 2 y ′′′ +x(2α+ν +3)y ′′ +[(α+1)(α+ν +1)−x]y ′ +ny = 0. 

26 / 27

eferences 

■ Chapter 23: Multiple Orthogonal Polynomials in “Classical and Quantum 

Orthogonal Polynomials of one Variable” (M.E.H. Ismail), Cambridge 

University Press, 2005. 

■ A.I. Aptekarev, A. Branquinho, W. Van Assche, Multiple orthogonal 

polynomials for classical weights, Trans. Amer. Math. Soc. 355 ((2003) 

■ E. Coussement, W. Van Assche, Multiple orthogonal polynomials associated 

with the modified Bessel functions of the first kind, Constr. Approx. 19 (2003) 

■ J. Coussement, W. Van Assche, Differential equations for multiple orthogonal 

polynomials with respect to classical weights: raising and lowering operators, 

J. Phys. A: Math. Gen. 39 (2006) 

■ W. Van Assche, Nearest neighbor recurrence relations for multiple orthogonal 

polynomials, manuscript 

■ A.B.J. Kuijlaars, Multiple orthogonal polynomial ensembles, Contemporary 

Mathematics 507 (2010) 

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Multiple orthogonal polynomials as special functions

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