Jacobi-Angelesco multiple orthogonal polynomials on an r-star
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1 M. Leurs Jacobi-Angelesco m.o.p. 1/19 Jacobi-Angelesco multiple orthogonal polynomials on an r-star Marjolein Leurs, (joint work with Walter Van Assche) Conference on Orthogonal Polynomials and Holomorphic Dynamics, Carlsberg Academy, Copenhagen, Denmark August 16, 018
2 Overview Legendre polynomials on [0, 1] Legendre-Angelesco polynomials on [ 1, 0] [0, 1] Jacobi-Angelesco polynomials on [ 1, 0] [0, 1] Jacobi-Angelesco polynomials on an r-star M. Leurs Jacobi-Angelesco m.o.p. /19
3 M. Leurs Jacobi-Angelesco m.o.p. 3/19 Legendre polynomials on [0, 1] The shifted Legendre polynomials are orthogonal polynomials for the constant weight function on [0, 1] 1 0 P n (x)x k dx = 0, 0 k n 1.
4 M. Leurs Jacobi-Angelesco m.o.p. 3/19 Legendre polynomials on [0, 1] The shifted Legendre polynomials are orthogonal polynomials for the constant weight function on [0, 1] 1 0 P n (x)x k dx = 0, 0 k n 1. They are given by the Rodrigues formula P n (x) = ( 1)n n! d n dx n x n (1 x) n.
5 M. Leurs Jacobi-Angelesco m.o.p. 3/19 Legendre polynomials on [0, 1] The shifted Legendre polynomials are orthogonal polynomials for the constant weight function on [0, 1] 1 0 P n (x)x k dx = 0, 0 k n 1. They are given by the Rodrigues formula An explicit expression is P n (x) = ( 1)n n! P n (x) = n k=0 ( n k d n dx n x n (1 x) n. )( n + k n ) ( 1) n k x k.
6 M. Leurs Jacobi-Angelesco m.o.p. 3/19 Legendre polynomials on [0, 1] The shifted Legendre polynomials are orthogonal polynomials for the constant weight function on [0, 1] 1 0 P n (x)x k dx = 0, 0 k n 1. They are given by the Rodrigues formula An explicit expression is P n (x) = ( 1)n n! P n (x) = The differential equation is n k=0 ( n k d n dx n x n (1 x) n. )( n + k n ) ( 1) n k x k. x(1 x)y (x 1)y + n(n + 1)y = 0.
7 M. Leurs Jacobi-Angelesco m.o.p. 4/19 Type I Legendre-Angelesco polynomials on [ 1, 0] [0, 1] Let (n, m) N
8 M. Leurs Jacobi-Angelesco m.o.p. 4/19 Type I Legendre-Angelesco polynomials on [ 1, 0] [0, 1] Let (n, m) N. 1 A n,m B n,m 0 1 Definition The type I Legendre-Angelesco polynomial (A n,m, B n,m ) is such that deg A n,m = n 1, deg B n,m = m 1 and 1 1 ) (A n,m (x)χ [ 1,0] + B n,m (x)χ [0,1] x k dx = 0, 0 k n + m, and 1 ) (A n,m (x)χ [ 1,0] + B n,m (x)χ [0,1] x n+m 1 dx = 1. 1
9 M. Leurs Jacobi-Angelesco m.o.p. 5/19 Type I Legendre-Angelesco polynomials on [ 1, 0] [0, 1] Introduce two families of polynomials p n (x) = q n (x) = n ( n k n ( n k k=0 k=0 )( n + k n )( n + k 1 n ) ( 1) n k x k, ) ( 1) n k x k.
10 M. Leurs Jacobi-Angelesco m.o.p. 5/19 Type I Legendre-Angelesco polynomials on [ 1, 0] [0, 1] Introduce two families of polynomials p n (x) = q n (x) = n ( n k n ( n k k=0 k=0 Their Mellin transforms are )( n + k n )( n + k 1 n ) ( 1) n k x k, ) ( 1) n k x k. p n (x)x s dx = ( 1) n ( 1 s ) n (s + 1) n+1, q n (x)x s dx = ( 1) n ( s ) n. (s + 1) n+1
11 M. Leurs Jacobi-Angelesco m.o.p. 6/19 Type I Legendre-Angelesco polynomials on [ 1, 0] [0, 1] Theorem (Geronimo, Iliev, Van Assche, 017) The type I Legendre-Angelesco polynomials on the diagonal are given by B n+1,n+1(x) = (n + )n+1 p n(x), n! A n+1,n+1(x) = B n+1,n+1( x), 1 A n,m B n,m 0 1
12 M. Leurs Jacobi-Angelesco m.o.p. 6/19 Type I Legendre-Angelesco polynomials on [ 1, 0] [0, 1] Theorem (Geronimo, Iliev, Van Assche, 017) The type I Legendre-Angelesco polynomials on the diagonal are given by B n+1,n+1(x) = (n + )n+1 p n(x), n! and near the diagonal they are given by ( ) ( ) n + n n + n 1 b nb n+1,n(x) = q n(x) p n(x), n n ( ) ( ) n + n n + n 1 b nb n,n+1(x) = q n(x) + p n(x), n n for some constant b n. A n+1,n+1(x) = B n+1,n+1( x), A n+1,n(x) = B n,n+1( x), A n,n+1(x) = B n+1,n( x), 1 A n,m B n,m 0 1
13 M. Leurs Jacobi-Angelesco m.o.p. 7/19 Type I Jacobi-Angelesco polynomials on [ 1, 0] [0, 1] A n,m B n,m 1 0 1
14 M. Leurs Jacobi-Angelesco m.o.p. 7/19 Type I Jacobi-Angelesco polynomials on [ 1, 0] [0, 1] α 1 A n,m B n,m β α 0 1
15 M. Leurs Jacobi-Angelesco m.o.p. 7/19 Type I Jacobi-Angelesco polynomials on [ 1, 0] [0, 1] α 1 A n,m B n,m β α 0 1 Definition The type I Jacobi-Angelesco polynomial (A n,m, B n,m ) is such that deg A n,m = n 1, deg B n,m = m 1 and 1 1 ) (A n,m(x)χ [ 1,0] + B n,m(x)χ [0,1] x β (1 x ) α x k dx = 0, 0 k n + m, 1 1 where α, β > 1. ) (A n,m (x)χ [ 1,0] + B n,m (x)χ [0,1] x β (1 x ) α x n+m 1 dx = 1,
16 M. Leurs Jacobi-Angelesco m.o.p. 8/19 Type I Jacobi-Angelesco polynomials on [ 1, 0] [0, 1] Define p n (x) = n k=0 ( ) n Γ(n + α + β+k + 1) k Γ(n + α + 1)Γ( β+k + 1) ( 1)n k x k,
17 M. Leurs Jacobi-Angelesco m.o.p. 8/19 Type I Jacobi-Angelesco polynomials on [ 1, 0] [0, 1] Define p n (x) = n k=0 ( ) n Γ(n + α + β+k + 1) k Γ(n + α + 1)Γ( β+k + 1) ( 1)n k x k, Theorem (Van Assche, L.) The type I Jacobi-Angelesco polynomials on the diagonal are given by B n+1,n+1 (x) = (α + β + n + ) n+1 p n (x), n! A n+1,n+1 (x) = B n+1,n+1 ( x).
18 M. Leurs Jacobi-Angelesco m.o.p. 8/19 Type I Jacobi-Angelesco polynomials on [ 1, 0] [0, 1] Define p n (x) = n k=0 ( ) n Γ(n + α + β+k + 1) k Γ(n + α + 1)Γ( β+k + 1) ( 1)n k x k, Theorem (Van Assche, L.) The type I Jacobi-Angelesco polynomials on the diagonal are given by B n+1,n+1 (x) = (α + β + n + ) n+1 p n (x), n! A n+1,n+1 (x) = B n+1,n+1 ( x). If α = 0 then p n (x) = n k=0 ( n k )( n + β+k n ) ( 1) n k x k.
19 M. Leurs Jacobi-Angelesco m.o.p. 9/19 Type I Jacobi-Angelesco polynomials on [ 1, 0] [0, 1] Define two families of polynomials n ( ) n Γ(n + α + β+k + 1) p n (x) = k Γ(n + α + 1)Γ( β+k k=0 + 1) ( 1)n k x k, n ( ) n Γ(n + α + β+k 1 + 1) q n (x) = k Γ(n + α + 1)Γ( β+k 1 + 1) ( 1)n k x k. k=0
20 M. Leurs Jacobi-Angelesco m.o.p. 9/19 Type I Jacobi-Angelesco polynomials on [ 1, 0] [0, 1] Define two families of polynomials n ( ) n Γ(n + α + β+k + 1) p n (x) = k Γ(n + α + 1)Γ( β+k k=0 + 1) ( 1)n k x k, n ( ) n Γ(n + α + β+k 1 + 1) q n (x) = k Γ(n + α + 1)Γ( β+k 1 + 1) ( 1)n k x k. k=0 Theorem (Van Assche, L.) The type I Jacobi-Angelesco polynomials near the diagonal are given by γ n B n+1,n (x) = c n q n (x) d n p n (x), γ n B n,n+1 (x) = c n q n (x) + d n p n (x), A n,n+1 (x) = B n+1,n ( x), A n+1,n (x) = B n,n+1 ( x), c n = β+n Γ(n + α + + 1) Γ(n + α +, Γ(n + α + 1)Γ( β+n dn = + 1) β+n+1 ) Γ(n + α + 1)Γ( β+n+1 ).
21 M. Leurs Jacobi-Angelesco m.o.p. 10/19 r-star For r N 0, define an r-star in C r j=1 [0, ω j 1 ], ω = e πi r. ω ω ω 0 1 ω ω ω 4 ω 5
22 M. Leurs Jacobi-Angelesco m.o.p. 11/19 Type I Jacobi-Angelesco polynomials on an r-star Multi-index n = (n 1, n,..., n r ) of size n = n 1 + n n r.
23 M. Leurs Jacobi-Angelesco m.o.p. 11/19 Type I Jacobi-Angelesco polynomials on an r-star Multi-index n = (n 1, n,..., n r ) of size n = n 1 + n n r. Definition The type I Jacobi-Angelesco polynomial (A n,1,..., A n,r ) is such that for 1 j r: deg A n,j = n j 1 and r j=1 ω j 1 0 A n,j (x) x β (1 x r ) α x k dx = 0, 0 k n, r j=1 0 ω j 1 where α, β > 1. A n,j (x) x β (1 x r ) α x n 1 dx = 1,
24 M. Leurs Jacobi-Angelesco m.o.p. 11/19 Type I Jacobi-Angelesco polynomials on an r-star Multi-index n = (n 1, n,..., n r ) of size n = n 1 + n n r. Definition The type I Jacobi-Angelesco polynomial (A n,1,..., A n,r ) is such that for 1 j r: deg A n,j = n j 1 and r j=1 ω j 1 0 A n,j (x) x β (1 x r ) α x k dx = 0, 0 k n, ω r j=1 0 ω j 1 A n,j (x) x β (1 x r ) α x n 1 dx = 1, A n, 0 A n,1 1 where α, β > 1. ω A n,3
25 M. Leurs Jacobi-Angelesco m.o.p. 1/19 Type I Jacobi-Angelesco polynomials on an r-star Define the polynomials p n (x) = n k=0 ( ) n Γ(n + α + β+k r + 1) k Γ(n + α + 1)Γ( β+k r + 1) ( 1)n k x k.
26 M. Leurs Jacobi-Angelesco m.o.p. 1/19 Type I Jacobi-Angelesco polynomials on an r-star Define the polynomials p n (x) = n k=0 ( ) n Γ(n + α + β+k r + 1) k Γ(n + α + 1)Γ( β+k r + 1) ( 1)n k x k. Theorem (Van Assche, L.) The type I Jacobi-Angelesco polynomials on the diagonal n = (n + 1,..., n + 1) are for 1 j r A n,j (x) = 1 r (rn + r + rα + β) n+1 p n (ω j+1 x), ω = e πi/r. n!
27 M. Leurs Jacobi-Angelesco m.o.p. 13/19 Type I Jacobi-Angelesco polynomials on an r-star Define for j = 0, 1,..., r 1 the polynomials p n (x; j) = n k=0 ( ) n Γ(n + α + β+k j r + 1) k Γ(n + α + 1)Γ( β+k j r + 1) ( 1)n k x k.
28 M. Leurs Jacobi-Angelesco m.o.p. 13/19 Type I Jacobi-Angelesco polynomials on an r-star Define for j = 0, 1,..., r 1 the polynomials p n (x; j) = n k=0 ( ) n Γ(n + α + β+k j r + 1) k Γ(n + α + 1)Γ( β+k j r + 1) ( 1)n k x k. Theorem (Van Assche, L.) The type I Jacobi-Angelesco polynomials below the diagonal, i.e. for n e k and n = (n,..., n) where 1 j, k r, are γ n,r A n ek,j(x) =ω j 1 ν n 1,0 p n 1 (ω j+1 x; 1) ω k 1 ν n 1,1 p n 1 (ω j+1 x; 0), with γ n,r a normalizing constant and ν n,j the leading coefficient of p n (x; j).
29 M. Leurs Jacobi-Angelesco m.o.p. 14/19 Type I Jacobi-Angelesco polynomials on an r-star Theorem (Van Assche, L.) The type I Jacobi-Angelesco polynomials above the diagonal, i.e., for n + e k and n = (n,..., n) where 1 j, k r, are r 1 τ n,r A n+ ek,j(x) = ω k+1 l=0 ω (j k)l ν n,l p n (ω j+1 x; l), ω = e πi/r, with τ n,r a normalizing constant and ν n,j the leading coefficient of p n (x; j).
30 M. Leurs Jacobi-Angelesco m.o.p. 15/19 Type I Jacobi-Angelesco polynomials on an r-star The polynomial p n (x) = n k=0 satisfies the differential equation ( ) n Γ(n + α + β+k r + 1) k Γ(n + α + 1)Γ( β+k r + 1) ( 1)n k x k x(1 x r )y (r+1) + (r + β)y (r) + where for l = 0, 1,..., r [( ) r + l c l (n) = ( 1) r+l+1 (n r + 1) r l l r c l (n)x l y (l) = 0, l=0 (rn + r ln) + ( ) ] r (rα + β). l
31 M. Leurs Jacobi-Angelesco m.o.p. 16/19 Asymptotic zero distribution Let 0 < x 1,n < x,n <... < x n,n < 1 be the zeros of p n and µ n = 1 n 1 dµ n (x) δ xj,n S n (z) = n z x = 1 p n(z) n p n (z) j=1 0
32 M. Leurs Jacobi-Angelesco m.o.p. 16/19 Asymptotic zero distribution Let 0 < x 1,n < x,n <... < x n,n < 1 be the zeros of p n and µ n = 1 n 1 dµ n (x) δ xj,n S n (z) = n z x = 1 p n(z) n p n (z) j=1 One can compute the asymptotic zero distribution with the Stieltjes transform µ n n µ 0
33 M. Leurs Jacobi-Angelesco m.o.p. 16/19 Asymptotic zero distribution Let 0 < x 1,n < x,n <... < x n,n < 1 be the zeros of p n and µ n = 1 n 1 dµ n (x) δ xj,n S n (z) = n z x = 1 p n(z) n p n (z) j=1 One can compute the asymptotic zero distribution with the Stieltjes transform µ n n µ 0 Stieltjes transform S n S Inverse Stieltjes transform
34 M. Leurs Jacobi-Angelesco m.o.p. 16/19 Asymptotic zero distribution Let 0 < x 1,n < x,n <... < x n,n < 1 be the zeros of p n and µ n = 1 n 1 dµ n (x) δ xj,n S n (z) = n z x = 1 p n(z) n p n (z) j=1 One can compute the asymptotic zero distribution with the Stieltjes transform µ n n µ 0 S(z) = lim n S n(z) Stieltjes transform S n S Inverse Stieltjes transform
35 M. Leurs Jacobi-Angelesco m.o.p. 16/19 Asymptotic zero distribution Let 0 < x 1,n < x,n <... < x n,n < 1 be the zeros of p n and µ n = 1 n 1 dµ n (x) δ xj,n S n (z) = n z x = 1 p n(z) n p n (z) j=1 One can compute the asymptotic zero distribution with the Stieltjes transform µ n n µ 0 S(z) = lim n S n(z) Stieltjes transform Inverse Stieltjes transform S n S p n(z) = np n(z)s n(z) ( ) p n (z) = n p n(z) Sn + 1 S n n... ( p n (r+1) (z) = n r+1 p n(z) S r+1 n ) + O(1/n)
36 M. Leurs Jacobi-Angelesco m.o.p. 17/19 Asymptotic zero distribution Insert this in the differential equation of p n and let n to find an algebraic equation for S r 1 ( ) r + 1 z(1 z r )S r+1 + ( 1) r+l+1 (zs) l = 0, l l=0 We need the solution that satisfies zs(z) 1 as z.
37 M. Leurs Jacobi-Angelesco m.o.p. 17/19 Asymptotic zero distribution Insert this in the differential equation of p n and let n to find an algebraic equation for S r 1 ( ) r + 1 z(1 z r )S r+1 + ( 1) r+l+1 (zs) l = 0, l l=0 We need the solution that satisfies zs(z) 1 as z. The Stieltjes inversion formula gives dµ(x) = u r (x)dx where u r (x) = rx r 1 w r (x r ), x [0, 1].
38 Asymptotic zero distribution Insert this in the differential equation of p n and let n to find an algebraic equation for S r 1 ( ) r + 1 z(1 z r )S r+1 + ( 1) r+l+1 (zs) l = 0, l l=0 We need the solution that satisfies zs(z) 1 as z. The Stieltjes inversion formula gives dµ(x) = u r (x)dx where With the change of variables u r (x) = rx r 1 w r (x r ), x [0, 1]. ˆx = x r = 1 c r (sin(r + 1)θ) r+1 sin θ(sin rθ) r, 0 < θ < π r + 1, and c r := (r+1)r+1 r, we have r w r (ˆx) = r + 1 sin θ sin rθ sin(r + 1)θ πˆx (r + 1) sin rθ e iθ r sin(r + 1)θ. M. Leurs Jacobi-Angelesco m.o.p. 17/19
39 M. Leurs Jacobi-Angelesco m.o.p. 18/19 Asymptotic zero distribution Figure: The asymptotic densities u r for p n : r = 1 (solid), r = (dash), r = 3 (dash dot), r = 4 (longdash) and r = 5 (dots)
40 M. Leurs Jacobi-Angelesco m.o.p. 19/19 References Jeffrey S. Geronimo, Plamen Iliev, Walter Van Assche, Alpert multiwavelets and Legendre-Angelesco multiple orthogonal polynomials, SIAM J. Math. Anal. 49 (017), no. 1, M.E.H. Ismail, Classical and Quantum Orthogonal Polynomials in One Variable, Encyclopedia of Mathematics and it Applications 98, Cambridge University Press, 005. Marjolein Leurs, Walter Van Assche, Jacobi-Angelesco polynomials on an r-star, arxiv:
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