for modified Bessel weights KU Leuven, Belgium Madison WI, December 7, 2013
Classical orthogonal polynomials The (very) classical orthogonal polynomials are those of Jacobi, Laguerre and Hermite.
Classical orthogonal polynomials The (very) classical orthogonal polynomials are those of Jacobi, Laguerre and Hermite. Their weight functions satisfy the so-called Pearson equation [σ(x)w(x)] = τ(x)w(x) where σ is a polynomial of degree 2 and τ a polynomial of degree 1 and boundary conditions σw(a) = 0 = σw(b).
Classical orthogonal polynomials The (very) classical orthogonal polynomials are those of Jacobi, Laguerre and Hermite. Their weight functions satisfy the so-called Pearson equation [σ(x)w(x)] = τ(x)w(x) where σ is a polynomial of degree 2 and τ a polynomial of degree 1 and boundary conditions σw(a) = 0 = σw(b). w σ τ Hermite e x2 1 2x Laguerre x α e x x x + α + 1 Jacobi (1 x) α (1 + x) β 1 x 2 x(α + β + 2) α + β
Generalizations If one replaces the derivative by a difference, q-difference, divided difference (Askey-Wilson) operator then one gets all the orthogonal polynomials in the Askey table.
Generalizations If one replaces the derivative by a difference, q-difference, divided difference (Askey-Wilson) operator then one gets all the orthogonal polynomials in the Askey table. (not the ASCII table!)
Generalizations If one replaces the derivative by a difference, q-difference, divided difference (Askey-Wilson) operator then one gets all the orthogonal polynomials in the Askey table. (not the ASCII table!) What about replacing the first derivative in the Pearson equation by a second derivative?
Generalizations If one replaces the derivative by a difference, q-difference, divided difference (Askey-Wilson) operator then one gets all the orthogonal polynomials in the Askey table. (not the ASCII table!) What about replacing the first derivative in the Pearson equation by a second derivative? Problem (A.P. Prudnikov, 1992) Construct the orthogonal polynomials for the weight ρ ν = 2x ν/2 K ν (2 x) on [0, ) for ν 0, where K ν is the modified Bessel function of the second kind, satisfying x 2 y (x) + xy (x) (x 2 + ν 2 )y(x) = 0.
The problem statement is not natural. It is really a problem for a system of first order differential equations (x ν ρ ν ) = x ν 1 ρ ν+1 ρ ν+1 = ρ ν and involves two weight functions on [0, ) ρ ν (x) = 2x ν/2 K ν (2 x), ρ ν+1 (x) = 2x (ν+1)/2 K ν+1 (2 x).
The problem statement is not natural. It is really a problem for a system of first order differential equations (x ν ρ ν ) = x ν 1 ρ ν+1 ρ ν+1 = ρ ν and involves two weight functions on [0, ) ρ ν (x) = 2x ν/2 K ν (2 x), ρ ν+1 (x) = 2x (ν+1)/2 K ν+1 (2 x). The natural question is to ask for the multiple orthogonal polynomials for the two weights (x α ρ ν, x α ρ ν+1 ) on [0, ), with α > 1 and ν 0.
of type 2 satisfy 0 0 P n,m (x)x k x α ρ ν (x) dx = 0, k = 0, 1,..., n 1, P n,m (x)x k x α ρ ν+1 (x) dx = 0, k = 0, 1,..., m 1, where P n,m is a monic polynomial of degree m + n.
of type 2 satisfy 0 0 P n,m (x)x k x α ρ ν (x) dx = 0, k = 0, 1,..., n 1, P n,m (x)x k x α ρ ν+1 (x) dx = 0, k = 0, 1,..., m 1, where P n,m is a monic polynomial of degree m + n. Type 1 multiple orthogonal polynomials are the pair (A n,m, B n,m ), with deg A n,m = n and deg B n,m = m, satisfying 0 [A n,m (x)ρ ν (x)+b n,m (x)ρ ν+1 (x)]x α x k dx = 0, k = 0, 1,..., n+m. and we often write Q n,m (x) = A n,m (x)ρ ν (x) + B n,m (x)ρ ν+1 (x).
MOPS for modified Bessel functions K ν Theorem (VA + Yakubovich, 2000; Ben Cheikh and Douak, 2000) The type 1 multiple orthogonal polynomials for the weights (x α ρ ν, x α ρ ν+1 ), where are given by x α Q n,n 1 (x) = d 2n ρ ν (x) = 2x ν/2 K ν (2 x), x [0, ) dx 2n [x 2n+α ρ ν (x)], x α Q n,n (x) = d 2n+1 dx 2n+1 [x 2n+α+1 ρ ν (x)],
MOPS for modified Bessel functions K ν Theorem (VA + Yakubovich, 2000; Ben Cheikh and Douak, 2000) The type 1 multiple orthogonal polynomials for the weights (x α ρ ν, x α ρ ν+1 ), where are given by x α Q n,n 1 (x) = d 2n ρ ν (x) = 2x ν/2 K ν (2 x), x [0, ) dx 2n [x 2n+α ρ ν (x)], x α Q n,n (x) = d 2n+1 and the type 2 multiple orthogonal polynomials are dx 2n+1 [x 2n+α+1 ρ ν (x)], P n,n (x) = A n,n (x)b n,n 1 (x) A n,n 1 (x)b n,n (x) P n+1,n (x) = A n+1,n (x)b n,n (x) A n,n (x)b n+1,n (x)
Properties Let then p 2n (x) = P n,n (x), p n (x) = ( 1) n (α + 1) n (α + ν + 1) n 1 F 2 ( p 2n+1 (x) = P n+1,n (x), ) n α + 1, α + ν + 1 ; x
Properties Let then p 2n (x) = P n,n (x), p n (x) = ( 1) n (α + 1) n (α + ν + 1) n 1 F 2 ( p 2n+1 (x) = P n+1,n (x), ) n α + 1, α + ν + 1 ; x Recurrence relation xp n (x) = p n+1 (x) + b n p n (x) + c n p n 1 (x) + d n p n 2 (x) with b n = (n + α + 1)(3n + α + 2ν) (α + 1)(ν 1), c n = n(n + α)(n + α + ν)(3n + 2α + ν), d n = n(n 1)(n + α)(n + α 1)(n + α + ν)(n + α + ν 1).
Properties One has d dx pα n (x) = np α+1 n 1 (x), d dx x α q α n (x) = x α 1 q α 1 n+1 (x).
Properties One has d dx pα n (x) = np α+1 n 1 (x), d dx x α q α n (x) = x α 1 q α 1 n+1 (x). Differential equation: if y(x) = p n (x), then x 2 y (x)+x(2α+ν+3)y (x)+[(α+1)(α+ν+1) x]y (x)+ny(x) = 0.
Modified Bessel function of the first kind What about the modified Bessel function I ν (x)?
Modified Bessel function of the first kind What about the modified Bessel function I ν (x)? It has no moments, but we can introduce an exponential factor and consider ω ν (x) = x ν/2 I ν (2 x)e c x, x [0, ), with c > 0 and ν 0. Then xω ν(x) = (ν cx)ω ν (x) + ω ν+1 (x) ω ν+1(x) = ω ν (x) cω ν+1 (x).
Modified Bessel function of the first kind What about the modified Bessel function I ν (x)? It has no moments, but we can introduce an exponential factor and consider ω ν (x) = x ν/2 I ν (2 x)e c x, x [0, ), with c > 0 and ν 0. Then xω ν(x) = (ν cx)ω ν (x) + ω ν+1 (x) ω ν+1(x) = ω ν (x) cω ν+1 (x). The weight ω ν is known as the non-central χ 2 distribution when 2ν is a positive integer.
MOPS for modified Bessel functions I ν Theorem (Douak 1999; Coussement + VA 2003) q 2n(x) = Q n,n(x), q 2n+1(x) = Q n+1,n(x), p 2n(x) = P n,n(x), p 2n+1(x) = P n+1,n(x) The type 1 multiple orthogonal polynomials for the weights (ω ν, ω ν+1 ) are given by n+1 ( ) n + 1 q n (x) = ( c) k x (ν+k)/2 I ν+k (2 x)e cx, k k=0
MOPS for modified Bessel functions I ν Theorem (Douak 1999; Coussement + VA 2003) q 2n(x) = Q n,n(x), q 2n+1(x) = Q n+1,n(x), p 2n(x) = P n,n(x), p 2n+1(x) = P n+1,n(x) The type 1 multiple orthogonal polynomials for the weights (ω ν, ω ν+1 ) are given by n+1 ( ) n + 1 q n (x) = ( c) k x (ν+k)/2 I ν+k (2 x)e cx, k k=0 The type 2 multiple orthogonal polynomials are p n (x) = ( 1)n c 2n n k=0 ( ) n c k k!l ν k k (cx), where L ν k are the Laguerre polynomials.
Properties The type 2 multiple orthogonal polynomials satisfy the recurrence relation with xp n (x) = p n+1 (x) + b n p n (x) + c n p n 1 (x) + d n p n 2 (x), b n = c n = d n = (ν + 2n + 1)c + 1 c 2, n((ν + n)c + 2) c 3, n(n 1) c 4.
Properties One has d dx qν+1 n (x) = q ν n+1(x), d dx pν n(x) = np ν+1 n 1 (x).
Properties One has d dx qν+1 n (x) = q ν n+1(x), d dx pν n(x) = np ν+1 n 1 (x). Differential equation: if y(x) = p n (x), then xy (x)+( 2cx+ν+2)y (x)+(c 2 x+c(n ν 2) 1)y (x) c 2 ny(x) = 0.
Distribution of zeros For the multiple orthogonal polynomials related to the modified Bessel functions K ν we have for n/n t > 0 b n lim n,n N 2 = 3t2, lim n,n c n N 4 = 3t4, d n lim n,n N 6 = t6.
Distribution of zeros For the multiple orthogonal polynomials related to the modified Bessel functions K ν we have for n/n t > 0 b n lim n,n N 2 = 3t2, lim n,n c n N 4 = 3t4, d n lim n,n N 6 = t6. Theorem (E+J Coussement + VA, 2008) The asymptotic distribution of the (scaled) zeros of p n is given by 1 lim n n n f j=1 ( xj,n n 2 ) = 4 27 for every f C([0, 4 27 ]) and 27/4 0 f (x)h(4x/27) dx, h(y) = 3 3 (1 + 1 y) 1/3 (1 1 y) 1/3 4π y 2/3, y [0, 1].
5 4 3 y 2 1 0 0.2 0.4 0.6 0.8 1 x
Distribution of zeros For the multiple orthogonal polynomials related to the modified Bessel functions I ν we have for n/n t > 0 b n lim n,n N = 2t c, lim n,n c n N 2 = t2 c 2, lim d n n,n N 3 = 0.
Distribution of zeros For the multiple orthogonal polynomials related to the modified Bessel functions I ν we have for n/n t > 0 b n lim n,n N = 2t c, lim n,n c n N 2 = t2 c 2, lim d n n,n N 3 = 0. Theorem (E Coussement + VA, 2003) The asymptotic distribution of the (scaled) zeros of p n is given by 1 lim n n n f j=1 ( xj,n n ) = c 4/c 4 cx f (x) dx, 2π 0 cx for every f C([0, 4/c]).
Distribution of zeros For the multiple orthogonal polynomials related to the modified Bessel functions I ν we have for n/n t > 0 b n lim n,n N = 2t c, lim n,n c n N 2 = t2 c 2, lim d n n,n N 3 = 0. Theorem (E Coussement + VA, 2003) The asymptotic distribution of the (scaled) zeros of p n is given by 1 lim n n n f j=1 ( xj,n n ) = c 4/c 4 cx f (x) dx, 2π 0 cx for every f C([0, 4/c]). Furthermore ( c) n p n (x/n) lim n n ν = e 1/c (cx) ν/2 J ν (2 cx). n!
5 4 3 y 2 1 0 0.2 0.4 0.6 0.8 1 x
Tools looking for an application We now have found multiple orthogonal polynomials for two weights involving modified Bessel functions x α( x ν/2 K ν (2 x), x (ν+1)/2 K ν+1 (2 ) x) and e cx( x ν/2 I ν (2 x), x (ν+1)/2 I ν+1 (2 ) x).
Tools looking for an application We now have found multiple orthogonal polynomials for two weights involving modified Bessel functions x α( x ν/2 K ν (2 x), x (ν+1)/2 K ν+1 (2 ) x) and e cx( x ν/2 I ν (2 x), x (ν+1)/2 I ν+1 (2 ) x). Are they useful?
Non-intersecting squared Bessel paths Let {X 1 (t),..., X d (t) : t 0} be d independent Brownian motions, then R(t) = X1 2(t) + X 2 2(t) + + X d 2(t), t 0 is the Bessel process: distance of the d dimensional Brownian motion in R d to the origin.
Non-intersecting squared Bessel paths Let {X 1 (t),..., X d (t) : t 0} be d independent Brownian motions, then R(t) = X1 2(t) + X 2 2(t) + + X d 2(t), t 0 is the Bessel process: distance of the d dimensional Brownian motion in R d to the origin. R 2 (t) has a non-central χ 2 distribution with d degrees of freedom.
Non-intersecting squared Bessel paths Let {X 1 (t),..., X d (t) : t 0} be d independent Brownian motions, then R(t) = X1 2(t) + X 2 2(t) + + X d 2(t), t 0 is the Bessel process: distance of the d dimensional Brownian motion in R d to the origin. R 2 (t) has a non-central χ 2 distribution with d degrees of freedom. Consider n squared Bessel processes R 2 1 (t),..., R2 n(t) such that R 2 1 (0) = R2 2 (0) = = R2 n(0) = a > 0, R 2 1 (1) = R2 2 (1) = = R2 n(1) = 0 The paths do not intersect
Non-Intersecting Squared Bessel Processes 227 A 2.5 2 1.5 x 1 0.5 0 0 0.2 0.4 0.6 0.8 1 t B 7
Kuijlaars, Martínez-Finkelshtein, Wielonsky, 2009 Let x 1, x 2,..., x n be the points on the vertical line at time t, then these points form a determinantal point process: P n,t (x 1, x 2,..., x n ) = 1 Z n,t det ( K n (x i, x j ) ) i,j=1,...,n with n 1 K n (x, y) = p j (2nax)q j (2nay), j=0 where q j are the type 1 and p j are the type 2 multiple orthogonal polynomials for the modified Bessel functions (I ν, I ν+1 ), where ν = d 2 1.
Kuijlaars, Martínez-Finkelshtein, Wielonsky, 2009 Let x 1, x 2,..., x n be the points on the vertical line at time t, then these points form a determinantal point process: P n,t (x 1, x 2,..., x n ) = 1 Z n,t det ( K n (x i, x j ) ) i,j=1,...,n with n 1 K n (x, y) = p j (2nax)q j (2nay), j=0 where q j are the type 1 and p j are the type 2 multiple orthogonal polynomials for the modified Bessel functions (I ν, I ν+1 ), where ν = d 2 1. ( n ) E (z x j ) = (2an) n p n (2anz). i=1
Kuijlaars, Martínez-Finkelshtein, Wielonsky, 2009 Let x 1, x 2,..., x n be the points on the vertical line at time t, then these points form a determinantal point process: P n,t (x 1, x 2,..., x n ) = 1 Z n,t det ( K n (x i, x j ) ) i,j=1,...,n with n 1 K n (x, y) = p j (2nax)q j (2nay), j=0 where q j are the type 1 and p j are the type 2 multiple orthogonal polynomials for the modified Bessel functions (I ν, I ν+1 ), where ν = d 2 1. ( n ) E (z x j ) = (2an) n p n (2anz). i=1 The points (x 1,..., x n ) have for n the same asymptotic distribution as the zeros of p n (2anx)
Products of random matrices Akeman, Kieburg, Wei, J. Phys. A: Math. Theor. 46 (2013).
Products of random matrices Akeman, Kieburg, Wei, J. Phys. A: Math. Theor. 46 (2013). Let X 1 and X 2 be independent complex matrices of size n with independent and identically distributed normal random variables a j,k + ib j,k, (Ginibre random matrix) then the density is, up to a normalization constant exp[ Tr (X j X j )], j = 1, 2
Products of random matrices Akeman, Kieburg, Wei, J. Phys. A: Math. Theor. 46 (2013). Let X 1 and X 2 be independent complex matrices of size n with independent and identically distributed normal random variables a j,k + ib j,k, (Ginibre random matrix) then the density is, up to a normalization constant exp[ Tr (X j X j )], j = 1, 2 The squared singular values of X 1 (the eigenvalues of X 1 X 1) follow a determinantal process in terms of Laguerre polynomials (Wishart-Laguerre ensemble).
Products of random matrices Akeman, Kieburg, Wei, J. Phys. A: Math. Theor. 46 (2013). Let X 1 and X 2 be independent complex matrices of size n with independent and identically distributed normal random variables a j,k + ib j,k, (Ginibre random matrix) then the density is, up to a normalization constant exp[ Tr (X j X j )], j = 1, 2 The squared singular values of X 1 (the eigenvalues of X 1 X 1) follow a determinantal process in terms of Laguerre polynomials (Wishart-Laguerre ensemble). We are interested in the squared singular values of the product X 2 X 1, i.e, the eigenvalues of X 1 X 2 X 2X 1.
Products of random matrices Theorem (Lun Zhang, 2013) The squared singular values of X 2 X 1 form a determinantal process with kernel n 1 K n (x, y) = p j (n 2 x)q j (n 2 x) j=0 where q j are the type 1 and p j the type 2 multiple orthogonal polynomials for the weights (K 0 (2 x), K 1 (2 x)).
Products of random matrices Theorem (Lun Zhang, 2013) The squared singular values of X 2 X 1 form a determinantal process with kernel n 1 K n (x, y) = p j (n 2 x)q j (n 2 x) j=0 where q j are the type 1 and p j the type 2 multiple orthogonal polynomials for the weights (K 0 (2 x), K 1 (2 x)). Furthermore ) E (z X1 X2 X 2 X 1 = n 2n p n (n 2 z) and the squared singular values are asymptotically distributed as the zeros of p n (n 2 z).
Longer products: Kuijlaars and Zhang, 2013 The squared singular values of X M X M 1 X 1 also form a determinantal process with multiple orthogonal polynomials, but now for the M weights (w 0 (x), w 1 (x),..., w M 1 ), where ( w k (x) = G M,0 0,M 0, 0,..., 0, k ) x, with G M,0 0,M a Meijer G-function
Longer products: Kuijlaars and Zhang, 2013 The squared singular values of X M X M 1 X 1 also form a determinantal process with multiple orthogonal polynomials, but now for the M weights (w 0 (x), w 1 (x),..., w M 1 ), where with G M,0 0,M ( w k (x) = G M,0 0,M a Meijer G-function ( Gp,q m,n a1,..., a p b 1,..., b q = 1 2πi γ 0, 0,..., 0, k ) x, ) x m j=1 Γ(b j + u) n j=1 Γ(1 a j u) q j=m+1 Γ(1 b j u) p j=n+1 Γ(a j + u) z u du.
Longer products: Kuijlaars and Zhang, 2013 The squared singular values of X M X M 1 X 1 also form a determinantal process with multiple orthogonal polynomials, but now for the M weights (w 0 (x), w 1 (x),..., w M 1 ), where with G M,0 0,M ( w k (x) = G M,0 0,M a Meijer G-function ( Gp,q m,n a1,..., a p b 1,..., b q = 1 2πi γ 0, 0,..., 0, k ) x, ) x m j=1 Γ(b j + u) n j=1 Γ(1 a j u) q j=m+1 Γ(1 b j u) p j=n+1 Γ(a j + u) z u du. (see, e.g., R. Beals and J. Szmigielski, Meijer G-functions: a gentle introduction, Notices AMS 60 (2013), 866 872.)
Products of rectangular matrices Let Y M = X M X M 1 X 1, where X j is a N j N j 1 random matrix with independent (complex) Gaussian entries and suppose N 0 = min{n 0, N 1,... N M }, ν j = N j N 0.
Products of rectangular matrices Let Y M = X M X M 1 X 1, where X j is a N j N j 1 random matrix with independent (complex) Gaussian entries and suppose N 0 = min{n 0, N 1,... N M }, ν j = N j N 0. Then the squared singular values of Y M form a determinantal process with multiple orthogonal polynomials for the M weights (w 0, w 1,..., w M 1 ), where w k (x) = G M,0 0,M ( ν M, ν M 1,..., ν 2, ν 1 + k ) x.
Products of rectangular matrices Let Y M = X M X M 1 X 1, where X j is a N j N j 1 random matrix with independent (complex) Gaussian entries and suppose N 0 = min{n 0, N 1,... N M }, ν j = N j N 0. Then the squared singular values of Y M form a determinantal process with multiple orthogonal polynomials for the M weights (w 0, w 1,..., w M 1 ), where w k (x) = G M,0 0,M ( ν M, ν M 1,..., ν 2, ν 1 + k ) x. If M = 1 then w 0 (x) = x ν 1 e x on [0, ): Laguerre polynomials
Products of rectangular matrices Let Y M = X M X M 1 X 1, where X j is a N j N j 1 random matrix with independent (complex) Gaussian entries and suppose N 0 = min{n 0, N 1,... N M }, ν j = N j N 0. Then the squared singular values of Y M form a determinantal process with multiple orthogonal polynomials for the M weights (w 0, w 1,..., w M 1 ), where w k (x) = G M,0 0,M ( ν M, ν M 1,..., ν 2, ν 1 + k ) x. If M = 1 then w 0 (x) = x ν 1 e x on [0, ): Laguerre polynomials If M = 2 then w 0 (x) = 2x (ν1+ν2)/2 K ν1 ν 2 (2 x), w 1 (x) = 2x (ν1+ν2+1)/2 K ν1 ν 2+1(2 x) multiple orthogonal polynomials with respect to modified Bessel functions of the second kind.
References W. Van Assche, S.B. Yakubovich, associated with Macdonald functions, Integral Transform and Special Functions 9, nr. 3 (2000), 229 244. E. Coussement, W. Van Assche, associated with the modified Bessel functions of the first kind, Constructive Approximation 19 (2003), 237 263. E. Coussement, J. Coussement, W. Van Assche, Asymptotic zero distribution for a class of multiple orthogonal polynomials, Trans. Amer. Math. Soc. 360, nr. 10 (2008), 5571 5588. A.B.J. Kuijlaars, A. Martínez-Finkelshtein, F. Wielonsky, Non-intersecting squared Bessel paths and multiple orthogonal polynomials for modified Bessel weights, Commun. Math. Phys. 286 (2009), 217 275. Lun Zhang, A note on the limiting mean distribution of singular values for products of two Wishart random matrices, arxiv:1305.0726v2 [math-ph] (May 2013) A.B.J. Kuijlaars, Lun Zhang, Singular values of products of Ginibre random variables, multiple orthogonal polynomials and hard edge scalings, arxiv:1308.1003v1 [math-ph] (August 2013)