Limits for Korteweg-de Vries Institute, University of Amsterdam T.H.Koornwinder@uva.nl http://www.science.uva.nl/~thk/ Lecture on September 10, 2012 at the Conference on Harmonic Analysis, Convolution Algebras, and Special Functions, Technische Universität München last modified : April 18, 2013
Contents 1 Partitions 2 Jacobi polynomials 3 Orthogonal symmetric polynomials 4 BC type Jacobi polynomials 5 The second order differential operator 6 Jack polynomials 7 Expansion of BC Jacobi in Jack polynomials 8 Two possible limit cases of BC n Jacobi polynomials 9 Generalized binomial coefficients 10 Solution of the first limit case 11 Very partial solution of the second limit case 12 The two-variable case 13 Some references
Partitions = ( 1,..., n ) Z n such that 1 2... n 0. := n i=1 i l() := max{j j > 0} is called a partition of weight and of length l() (with maximal length n). A partition is often diagramatically displayed by a Young tableau, consisting of boxes (i, j) with i = 1,..., l() and j = 1,..., i for a given i. Its conjugate is obtained by reflection of the tableau of with respect to the diagonal. For instance, = (4, 4, 2, 2, 1) and = (5, 4, 2, 2) are displayed by and
Partitions (cntd.) Dominance partial order: µ m i=1 i m i=1 µ i (m = 1,..., n). Inclusion partial order: µ i µ i (i = 1,..., n) (so the tableau of is a subset of the tableau of µ). Note that µ µ. Symmetrized monomials: m (x) := x µ (x µ := x µ 1 1... x n µn, S n is symmetric group). µ S n The m form a basis of the space of symmetric polynomials in n variables.
Jacobi polynomials Jacobi polynomials P (α,β) n 1 0 (α, β > 1) satisfy the orthogonality P (α,β) n (1 2x) P (α,β) m (1 2x) x α (1 x) β dx = 0 (n m). They are 2 F 1 hypergeometric polynomials: P (α,β) n (1 2x) = (α + 1) n n! ( ) n, n + α + β + 1 2F 1 ; x, α + 1 where (a) k := a(a + 1)... (a + k 1) is the Pochhammer symbol and ( ) a, b 2F 1 c ; z := k=0 (a) k (b) k (c) k k! z k, terminating after term with k = n if a = n (n = 0, 1,...).
Orthogonal symmetric polynomials Let {p n } n=0 be a system of monic polynomials p n of degree n satisfying the orthogonality 1 0 p n (x) p m (x) w(x) dx = 0 (n m) for some integrable weight function w 0 on [0, 1]. For = ( 1,..., n ) a partition define p 0 := p µ1 (x 1 )... p µn (x n ). Then p 0 = µ; µ Orthogonality relation: µ S n c,µ m µ with c, = 1. [0,1] n p 0 (x) p0 µ(x) w(x 1 )... w(x n ) dx 1... dx n = 0 ( µ).
Orthogonal symmetric polynomials (cntd.) ρ := (n 1, n 2,..., 1, 0); ε(σ) is the sign of σ S n. (x) := (x i x j ) = ε(σ) x σρ (Vandermonde). σ S n 1 i<j n s (x) := 1 ε(σ)x σ(+ρ) (Schur polynomial). (x) σ S n Then s = c,µ m µ with c, = 1. µ; µ p 1 1 (x) := ε(σ) p µ1 (x 1 )... p µn (x n ). (x) µ=σ(+ρ); σ S n Then p 1 = a,µ s µ = b,µ m µ with a, = b, = 1. µ; µ Orthogonality relation: µ; µ [0,1] n p 1 (x) p1 µ(x) w(x 1 )... w(x n ) (x) 2 dx 1... dx n = 0 ( µ).
Orthogonal symmetric polynomials (cntd.) The cases p 0 and p1 suggest to define for a coupling constant 0 a system {p } such that (i) p = m + c,µ m µ ; µ; µ< (ii) p (x) m µ(x) w(x 1 )... w(x n ) (x) 2 dx 1... dx n = 0 [0,1] n if µ <. Then p (x) p µ(x) w(x 1 )... w(x n ) (x) 2 dx 1... dx n = 0 [0,1] n if µ <, but generally not if µ but unrelated by. One might also consider this definition with < being a lexicographic ordering. Then, of course, full orthogonality is achieved, but then the question is whether (i) already holds for some partial ordering with which the chosen lexicographic ordering is compatible. For general weight functions the answer is negative.
BC type Jacobi polynomials Definition For w(x) := x α (1 x) β let p α,β, be equal to p as just defined. This is called a BC type Jacobi polynomial. These polynomials, when rewritten as trigonometric polynomials and when reparametrized, become the Heckman-Opdam Jacobi polynomials for root system BC n. Let {ε i } n i=1 be the standard basis of Rn. The root system BC n is the subset {±ε i } n i=1 {±2ε i} n i=1 {±ε i ± ε j } 1 i<j n of R n. Let the Weyl group invariant multiplicity function be such that ε 1, 2ε 1, ε 1 + ε 2 have multiplicities k 1, k 2, k 3 respectively. Then for x i = sin 2 ( 1 2 θ i), α = k 1 + k 2 1 2, β = k 2 1 2, = k 3, p α,β, (x) is equal to the Heckman-Opdam Jacobi polynomial for root system BC n and multiplicities k 1, k 2, k 3, living on the torus {(e iθ 1,..., e iθn )}.
BC type Jacobi polynomials (cntd.) The motivation for introducing these polynomials came from the theory of spherical functions on compact symmetric spaces. For each root system the corresponding Heckman-Opdam polynomials can be interpreted as spherical functions for some very special values of the parameters. Then full orthogonality is evident by Schur orthogonality of matrix elements of irreducible representations.
BC type Jacobi polynomials (cntd.) In the general case full orthogonality was first proved by Heckman in 1987 by using deep results by Deligne. Later, in 1991, Heckman could give a much quicker proof of the full orthogonality by starting with non-symmetric Heckman-Opdam polynomials, where he extended the theory of Dunkl operators. For the root system BC 2 the corresponding Jacobi polynomials were already introduced by K (1974), using lexicographic ordering. It was next proved by Sprinkhuizen in 1976 that then the expansion in symmetrized monomials only needed the dominance partial ordering. Almost immediately after Heckman s paper of 1987, Macdonald already proved full orthogonality in the BC n case as a limit case of a similar result in the q-case.
The second order differential operator Recall that, for w(x) := x α (1 x) β, the BC n type Jacobi polynomial p α,β, is defined by the two properties: (i) p α,β, = m + c,µ m µ ; µ; µ< (ii) p α,β, (x) m µ (x) w(x 1 )... w(x n ) (x) 2 dx 1... dx n = 0 [0,1] n if µ <. We get an equivalent definition if we replace (ii) by: (ii) D α,β, p α,β, = d α,β, p α,β,, where (with ( i := / x i ) D α,β, = n i=1 (x 2 i x i ) 2 i + n ((2+α+β)x i (α+1)) i +2 i=1 i,j;i j x 2 i x i i x i x j and d α,β, = n i=1 i( i + α + β + 1 + 2(n i)).
The second order differential operator (cntd.) That (i) and (ii) imply (ii) follows because D α,β, m = d α,β, m + a,µ m µ and µ:µ< (D α,β, f )(x) g(x) w(x 1 )... w(x n ) (x) 2 dx 1... dx n [0,1] n = f (x) (D α,β, g)(x) w(x 1 )... w(x n ) (x) 2 dx 1... dx n [0,1] n for all symmetric polynomials f, g. That (i) and (ii) imply (ii) follows because moreover dµ α,β, if > µ. d α,β,
Jack polynomials For = ( 1,..., n ) a partition and 0 the Jack polynomial j is defined as a symmetric polynomial in x 1,..., x n, homogeneous of degree, such that (i) j = m + c,µ m µ ; µ;µ< (ii) D j = d j, where n D = xi 2 i 2 + 2 xi 2 i x i x j d = i=1 i,j;i j n i ( i 1 + 2(n i)). i=1 and Jack polynomials were introduced by Jack in 1969. They were taken up later by Macdonald and Stanley. They become A n 1 type Heckman-Opdam Jacobi polynomials when their homogeneity is divided out.
Jack polynomials (cntd.) Jack polynomials degenerate for n = 1 to monomials x l (no coupling constant for n = 1). Elementary cases for n > 1 are j 0 = m ; j 1 = s, j := lim j = e, where e := en n e n 1 n n 1... e 1 2 1 with e i the i-th elementary symmetric polynomial. Jack polynomials can be considered as multivariate symmetric -dependent analogues of the monomials in one variable. For c R write c n := (c, c,..., c) R n. Put j (x) := j (x)/j (1n ) ( j (1n ) is explicitly known). There is a generalized binomial formula of the form j (x + 1 n ) = ( ) jµ (x), which defines the µ µ: µ ( ) generalized binomial coefficients. µ
Expansion of BC Jacobi in Jack polynomials Theorem (Macdonald, unpublished manuscript) There are coefficients c α,β,,µ such that p α,β, = Of course, c α,β,, = 1. Also write p α,β, = n C α,β,,µ ( ) µ + εi µ µ;µ µ;µ ( i µ i )( i + µ i + α + β + 1 + 2(n i)) c α,β,,µ j µ. C α,β,,µ j µ. Then i=1 = ( ) C α,β, µ + εi,µ+ε i (µ i + α + 1 + (n i)) with µ i; µ+ε i = (µ i +1+(l(µ+ε i ) i)) µ i µ j + 1 + (j i 1). µ i µ j + 1 + (j i) j; j i Since the coefficient of C α,β,,µ in the above recurrence is nonzero if µ, the recurrence determines the C α,β,,µ uniquely, up to a constant factor fixed by C α,β,, = j (1n ).
Expansion of BC Jacobi in Jack polynomials (cntd.) The recurrence follows by applying D α,β, to both sides of p α,β, = C α,β,,µ j µ, and next by writing µ;µ D α,β, = D D + (2 + α + β) D := i=1 n i jµ = i=1 D j µ = i,j;i j n x i i (α + 1) i=1 n x i i 2 + 2 x i i, x i x j i; µ ε i partition i; µ ε i partition ( µ µ ε i ( ) µ µ ε i see Hallnäss, IMRN, 2009. ) j µ ε i, n i, i=1 and by using that where n x i i jµ = µ jµ, i=1 (µ i 1 + (n i)) jµ ε i,
Two possible limit cases of BC n Jacobi polynomials For n = 1 we have p α,β l (x) = ( 1)l (α + 1) l (l + α + β + 1) l l k=0 ( l) k (l + α + β + 1) k (α + 1) k k! Let pl a α (x) be the limit of pα,β l (x) as α, β such that α+β a (a 0). Then p a l (x) = ( a)l l k=0 ( l) k k! (a 1 x) k = (x a) l. Problem 1. Compute the limit p a, (x) of pα,β, (x) as α, β α such that α+β a. Problem 2. Compute the limit p α,β, (x) of p α,β, (x) as. x k.
Generalized binomial coefficients Recall j (x + 1 n ) = ( ) jµ (x). µ µ: µ By homogeneity: j (x + a n ) = ( ) a µ jµ (x). µ κ; κ =k, µ κ µ: µ There is a deeper result (see Lassalle, CRAS, 1990): ( ) ( ) ( )( ) κ µ = κ µ k µ µ In particular, for k = µ + 1 : ( ) ( ) ( ) µ + εi = ( µ ) µ + ε i µ µ i; µ+ε i..
Solution of Problem 1 Theorem (Beerends & K, unpublished; Rösler, K & Voit, 2012) The limit p a, α (x) of pα,β, (x) as α, β such that α+β a, is equal to j (x an ). Proof (two different proofs are given in the cited papers). Recall that p α,β, = C α,β,,µ j µ with µ;µ n C α,β,,µ i=1 Hence p a, ( i µ i )( i + µ i + α + β + 1 + 2(n i)) = i; µ+ε i = µ;µ C a,,µ = a µ ( ) C α,β, µ + εi,µ+ε i (µ i + α + 1 + (n i)). µ C a,,µ j µ exists with ( ) C a, µ + εi,µ+ε i µ i; µ+ε i
Solution of Problem 1 (cntd.) A solution of the recurrence C a,,µ = a ( ) C a, µ + εi µ,µ+ε i µ i; µ+ε i is given by C a,,µ = ( a) µ ( µ), since this reduces to the true identity ( ) ( ) ( µ + εi = ( µ ) µ + ε i i; µ+ε i µ µ Hence p a, (x) = const. C a,,µ j µ (x) µ;µ = const. ( ( a) µ µ µ;µ ) ). j µ (x) = const. j (x a n ) = j (x an ).
Problem 1 (cntd.) The just proved theorem is used in Rösler, K & Voit (2012) to obtain a similar limit result in the non-terminating case (by analytic continuation, using Carlson s theorem and suitable estimates). For = 1 2, 1, 2 and α, β suitably restricted this has an interpretation as limit of spherical functions on noncompact finite dimensional Grassmann manifolds to spherical functions on infinite dimensional Grassmann manifolds.
The case = l n Theorem (Macdonald, unpublished) p α,β, l (x) n p α,β, l (0) = n µ;µ c α,β, l n,µ j µ (x) with c α,β, l n,µ = ( l; ) µ(l + α + β + (n 1) + 1; ) µ (α + (n 1) + 1; ) µ h (µ) µ. Here (a; ) µ := n (a (i 1)) µi i=1 (generalized Pochhammer symbol) and h (µ) := (µ j i + 1 (µ i j + 1)) (i,j) µ (product of upper hook lengths). We can compute the limit for of c α,β, l n,µ.
The case = l n for Recall: c α,β, l n,µ = ( l) µ(l + α + β + (n 1) + 1) µ (α + (n 1) + 1) µ h (µ) µ. Use that (l + α + β + (n 1) + 1; ) µ lim = (n + α + β + 1) µ n, (α + (n 1) + 1; ) µ (α + 1) µn lim lim ( l; ) µ µ 2+ +µ n = ( l) µ1 ( 1) µ 2+ +µ n 2 µ 3... (n 1) µn, 1 h (µ) µ 1 = 2 µ33 µ4... (n 1) µn (µ 1 µ 2 )!... (µ n 1 µ n )! µ n!. Altogether, c α,β, l n,µ := lim c α,β, l n,µ = (l + α + β + 1) µ n (α + 1) µn ( l) µ1 ( 1) µ2+ +µn (µ 1 µ 2 )!... (µ n 1 µ n )! µ n!. Recall: jµ := lim j µ = e µ 1 µ 2 1... e µ n 1 µ n n 1 e n µn.
The case = l n for (cntd.) lim = = p α,β, l (x) n p α,β, l (0) = n l µ n=0 µ;µ c α,β, l n,µ j µ (x) = µ;µ (l + α + β + 1) µn (α + 1) µn ( l) µ1 ( 1) µ 2+ +µ n e µ 1 µ 2 1... e µ n 1 µ n n 1 e n µn (µ 1 µ 2 )!... (µ n 1 µ n )! µ n! ( l) µn (l + α + β + 1) µn (α + 1) µn µ n! ( 1) µ 1+ +µ n 1 (l µ n )! (l µ 1 )! l µ n=0 ( l) µn (l + α + β + 1) µn (α + 1) µn µ n! e µn n µ 1,...,µ n 1 ; µ n µ n 1... µ 1 l e µ 1 µ 2 1 (µ 1 µ 2 )!... e (( 1) n e n ) µn (1 e 1 + e 2 ± e n 1 ) l µn. µ n 1 µ n n 1 (µ n 1 µ n )!
The case = l n for (cntd.) Theorem lim p α,β, l (x) n p α,β, l n (0) = (1 e 1 + e 2 ± e n 1 ) l ( l, l + α + β + 1 2 F 1 ; α + 1 ( 1) n e n 1 e 1 + e 2 ± e n 1 ).
The case n = 2 For n = 2 many things can be computed much more explicitly (K & Sprinkhuizen, 1978), with heavy use of the shift operators (which exist also for general n, see Opdam, 1988 and Heckman, 1991, but are explicitly given for n = 2). In particular: j 1, 2 (x 1, x 2 ) = ( 1 2 )! () 1 2 1 2 i=0 = 2 1 2 ( 1 2 )! (x 1 x 2 ) 1 2 ( 1+ 2 ) P ( 1 (2 + 1 2 ) 1 2 1 2 p α,β, lim 1, 2 (x 1, x 2 ) p α,β, = (1 x 1 x 2 ) 1 1, 2 (0, 0) () i () 1 2 i i! ( 1 2 i)! x 1 i 1 x 2+i 2 ( 2, 1 2 ) x 1 + x 2 2(x 1 x 2 ) 1 2 ( 2, 2 + α + β + 1 x 1 x 2 2 F 1 ; α + 1 1 x 1 x 2 There are strong indications that the case n = 2 is not yet typical for the case of general n. ), ).
Some references R. J. Beerends & E. M. Opdam, Certain hypergeometric series related to the root system BC, Trans. AMS 339 (1993), 581 609. M. Hallnäs, Multivariable Bessel polynomials related to the hyperbolic Sutherland model with external Morse potential, IMRN (2009), 1573 1611. G. J. Heckman, An elementary approach to the hypergeometric shift operators of Opdam, Invent. Math. 103 (1991), 341 350. Jack, Hall-Littlewood and Macdonald polynomials, Contemporary Math. 417, 2006. T. Koornwinder & I. Sprinkhuizen-Kuyper, Generalized power series expansions for a class of orthogonal polynomials in two variables, SIAM J. Math. Anal. 9 (1978), 457 483.
Some references (cntd.) M. Lassalle, Une formule du binôme généralisée pour les polynômes de Jack, CRAS Paris 310 (1990), 253 256. M. Lassalle, Polynômes de Jacobi généralisés, CRAS Paris 312 (1991), 425 428. I. G. Macdonald, Symmetric functions and Hall polynomials, second ed., Oxford University Press, 1995. I. G. Macdonald, Hypergeometric functions, unpublished manuscript. M. Rösler, T. Koornwinder & M. Voit, Limit transition between hypergeometric functions of type BC and type A, arxiv:1207.0487, 2012. R. P. Stanley, Some combinatorial properties of Jack symmetric functions, Adv. Math. 77 (1989), 76 115.