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arXiv:math/0511138v3 [math.QA] 22 Apr 2007

DIFFERENTIAL EQUATIONS FOR JACOBI-PI˜NEIRO POLYNOMIALS

E. MUKHIN AND A. VARCHENKO

Abstract. For r ∈ Z≥0, we present a Fuchsian linear differential operator of order

r + 1 with three singular points at 0,1,∞. This operator annihilates the r-multiple

Jacobi-Pi˜ neiro polynomial.

1. Introduction

Let r be a natural number. Consider a Fuchsian differential operator

D =

r+1

?

i=0

ci(x)di

dxi

with singular points at z1,...,zn,∞ and with kernel consisting of polynomials only. An

interest to such operators had arisen recently in relation with the Bethe ansatz method

in the Gaudin model, where such operators were used to construct eigenvectors of the

Gaudin Hamiltonians, see [ScV], [MV1]-[MV3], [MTV1], [MTV2].

In the Gaudin model, one considers the tensor product M = M1⊗ ··· ⊗ Mnof finite

dimensional irreducible glr+1-modules, located respectively at z1,...,zn. The module

Ms, sitting at zs, is determined by the exponents of D at zs. One constructs r + 1 one-

parameter families of commuting linear operators Hi(x) : M → M, i = 1,...,r+1, acting

on M and called the Gaudin Hamiltonians. The problem is to construct eigenvectors and

eigenvalues of the Gaudin Hamiltonians.

It turns out, that having the kernel of the differential operator D, i.e. the r + 1-

dimensional vector space of polynomials, one constructs (under certain conditions) an

eigenvector vD ∈ M of the Gaudin Hamiltonians with corresponding eigenvalues being

the coefficients of D,

Hi(x)vD = ci(x)vD,i = 1,...,r + 1 .

The Bethe ansatz idea is to construct all eigenvectors of the Gaudin Hamiltonians by

choosing different operators D with the same singular points and exponents.

This philosophy motivates the detailed study of Fuchsian operators with prescribed

singular points, exponents, and polynomial kernels.

The important model case is the study of operators with three singular points 0,1,∞.

The operators with special exponents 0,k + 1,k + 2,...,k + r at x = 1 and arbitrary

exponents at x = 0,∞ were studied in [MV2]. It was discovered in [MV2] that the kernel

Research of A.V. is supported in part by NSF grant DMS-0244579.

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2 E. MUKHIN AND A. VARCHENKO

of such a differential operator consists of Jacobi-Pi˜ neiro polynomials, a special type of

multiple orthogonal polynomials, see Lemma 4.4 in [MV2].

This appearance of orthogonal polynomials in the Bethe ansatz constructions helped

us in [MV2] study eigenvectors of the Gaudin Hamiltonians.

In this short paper, we give an example of a reverse implication, namely, that the Bethe

ansatz considerations may be useful in studying orthogonal polynomials. We construct

a Fuchsian differential operator with singular points at 0,1,∞ annihilating the Jacobi-

Pi˜ neiro polynomial, see the precise statement and the discussion of the result in Section

5. Such an operator can be used in studying the Jacobi-Pi˜ neiro polynomials.

We thank referees for helping to improve the exposition.

2. Jacobi-Pi˜ neiro polynomials

Let l1,...,lrbe integers such that l1≥ ··· ≥ lr≥ 0. Let m1,...,mrand k be negative

real numbers. We use the notation m = (m1,...,mr), l = (l1,...,lr).

The Jacobi-Pi˜ neiro polynomial [P] is the unique monic polynomial of degree l1whose

coefficients are rational functions of m,l,k and which is orthogonal to functions

1,x,...,xl1−l2−1

?

with respect to the scalar product given by

???

l1−l2

,x−m2−1,x−m2,...,x−m2+l2−l3−2

?

???

l2−l3

,...,x−Pr

?

i=2mi−r+1,...,x−Pr

i=2mi−r+lr

???

lr

(f(x),g(x)) =

?1

0

f(x)g(x)(x − 1)−k−1x−m1−1dx.

We denote the Jacobi-Pi˜ neiro polynomial by Pm,l,k(x).

If l2 = l3= ··· = lr = 0, then the Jacobi-Pi˜ neiro polynomial is the classical Jacobi

polynomial P(α,β)

l

(x) on interval [0,1] with l = l1, α = −k − 1, β = −m1− 1.

The Jacobi-Pi˜ neiro polynomial may be given by the Rodrigues-type formula, see [ABV]:

P(m,l,k) = c(x − 1)k+1x

Pr

i=1mi−r× (2.1)

×

dlr−lr+1

dxlr−lr+1xlr−lr+1−mr−1dlr−1−lr

where c is a nonzero constant.

The coefficients of the Jacobi-Pi˜ neiro polynomial Pm,l,k(x) are rational functions of

m,l,k and therefore the polynomial Pm,l,k(x) is well defined for almost all complex

m1,...,mr,k.

dxlr+1−lr... xl2−l3−m2−1dl1−l2

dxl1−l2

?xl1−l2−m1−1(x − 1)l1−k−1?,

3. Spaces of polynomials the first and second type

We describe remarkable spaces of polynomials which contain Jacobi-Pi˜ neiro polynomi-

als. See [MV2] for the relation of these spaces to the Bethe Ansatz method.

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DIFFERENTIAL EQUATIONS FOR JACOBI-PI˜NEIRO POLYNOMIALS9

However, the characteristic equations for exponents of Dm,l,kat x = 0,1,∞ do not de-

termine the coefficients of Dm,l,k. The triviality of the monodromy of Dm,l,kis essential

for the uniqueness of the operator Dm,l,k, in contrast with the situation for the operator

D∨

equations for exponents of D∨

m,l,k, where the uniqueness of the operator D∨

m,l,kis determined by the characteristic

m,l,konly.

References

[ABV] A. I. Aptekarev, A. Branquinho, and W. Van Assche Multiple orthogonal polyno-

mials for classical weights, Trans. Amer. Math. Soc. 355 (2003), no. 10, 3887–3914

[CV]J. Coussement and 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), 3311-3318

[MTV1] E. Mukhin, V. Tarasov, and A. Varchenko, The B. and M. Shapiro conjecture in

real algebraic geometry and the Bethe ansatz, math.AG/0512299, 1–17

[MTV2] E. Mukhin, V. Tarasov, and A. Varchenko, Bethe Eigenvectors of Higher Transfer

Matrices, math.QA/0605015, 1–48

[MV1] E. Mukhin and A. Varchenko, Critical Points of Master Functions and Flag Vari-

eties, Communications in Contemporary Mathematics (2004), vol. 6, no. 1, 111-163

[MV2] E. Mukhin and A. Varchenko, Multiple orthogonal polynomials and a counterex-

ample to Gaudin Bethe Ansatz Conjecture, math.QA/0501144, 1–40. To appear in

Transactions of AMS

[MV3] E. Mukhin and A. Varchenko, Spaces of quasi-polynomials and the Bethe Ansatz,

math.QA/0604048, 1–29

[P] L. R. Pi˜ neiro, On simultaneous Pade approximants for a collection of Markov

functions, Vestnik Mosk. Univ. Ser., I, no. 2 (1987), 52–55 (in Russian); translated

in Moscow Univ. Math. Bull. 42, no. 2 (1987), 52–55

[ScV] I. Scherbak and A. Varchenko, Critical points of functions, sl2 representations

and Fuchsian differential equations with only univalued solutions, Dedicated to

Vladimir I. Arnold on the occasion of his 65th birthday. Mosc. Math. J. 3 (2003),

no. 2, 621–645, 745

E.M.: Department of Mathematical Sciences, Indiana University - Purdue University

Indianapolis, 402 North Blackford St, Indianapolis, IN 46202-3216, USA,

mukhin@math.iupui.edu

A.V.: Department of Mathematics, University of North Carolina at Chapel Hill,

Chapel Hill, NC 27599-3250, USA, anv@email.unc.edu