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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 POLYNOMIALS3

Parameters (m,l,k) are called consistent if all mi,liand k are nonnegative integers

satisfying

k ≥ l1≥ l2≥ ··· ≥ lr≥ 0,ls− ls+1≤ ms

(s = 1,...,r) .

(3.1)

Let (m,l,k) be consistent. We use the convention:

l0= k,lr+1= 0 .

We call a complex r + 1-dimensional vector space of polynomials V (m,l,k) ⊂ C[x]

the space of polynomials of the first type associated to (m,l,k) if the space satisfies the

following two conditions:

• The space V (m,l,k) has a basis of the form

{v0(m,l,k), v1(m,l,k)xm1+1, v2(m,l,k)xm1+m2+2, ..., vr(m,l,k)x

Pr

i=1mi+r}, (3.2)

where for i = 0,...,r, the polynomial vi(m,l,k) ∈ C[x] is a monic polynomial of

degree k − li+ li+1.

• If a polynomial p ∈ V (m,l,k) vanishes at x = 1, then the multiplicity of zero at

x = 1 is at least k + 1.

It is easy to see that the basis polynomials in (3.2) have increasing degrees.

Below we will show that for any consistent parameters (m,l,k), there exists a unique

space of polynomials of the first type associated to (m,l,k), see Theorem 5.2. Moreover,

we will show that this space contains the Jacobi-Pi˜ neiro polynomial P(m,l,k), see Lemma

4.4.

We call a complex r + 1-dimensional vector space of polynomials U(m,l,k) ⊂ C[x]

the space of polynomials of the second type associated to (m,l,k) if the space satisfies the

following two conditions:

• The space U(m,l,k) has a basis of the form

{u0(m,l,k), u1(m,l,k)xmr+1, u2(m,l,k)xmr+mr−1+2,...,ur(m,l,k)x

Pr

i=1mi+r}, (3.3)

where for i = 0,...,r, the polynomial ui(m,l,k) ∈ C[x] is a monic polynomial of

degree lr−i− lr−i+1.

• There exists a nonzero polynomial p ∈ U(m,l,k) which has zero at x = 1 of order

k + r.

It is easy to see that the basis polynomials in (3.3) have increasing degrees.

Below we will show that for any consistent parameters (m,l,k), there exists a unique

space of polynomials of the second type associated to (m,l,k), see Theorem 5.2.

The spaces of the first type and of the second type are dual in the sense of [MV1] which

we now describe.

Define an r-tuple T = (T1,...,Tr) of polynomials in x by

T1 = (x − 1)kxm1,Ti = xmi

(i = 2,...,r) .

(3.4)

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4 E. MUKHIN AND A. VARCHENKO

For functions f1,...,fsof x, the Wronskian W(f1,...,fs) is defined by

W(f1,...,fs) = det

?di

dxifj

?

i,j=1,...,s

.

For functions f1,...,fs of x, define the divided Wronskians W†

W†

U(f1,...,fs) by

W†

W†

V(f1,...,fs) and

V(f1,...,fs) = W(f1,...,fs) T1−s

U(f1,...,fs) = W(f1,...,fs) T1−s

1

T2−s

2

T2−s

...T−1

s−1,

rr−1...T−1

r−s+2.

Lemma 3.1. Let (m,l,k) be consistent parameters.

Let V be a space of the first type associated to (m,l,k). Then the space

U = {W†

V(f1,...,fr), f1,...,fr∈ V }

is a space of polynomials of the second type associated to (m,l,k).

Let U be a space of the second type associated to (m,l,k). Then the space

V = {W†

U(f1,...,fr), f1,...,fr∈ U }

is a space of polynomials of the first type associated to (m,l,k).

Proof. The lemma follows from the definitions.

?

4. Recursion for spaces V (m,l,k)

We show the existence of spaces V (m,l,k) of the first type by constructing them

recursively as follows.

Let m1,...,mrbe nonnegative numbers. Let 0 = (0,...,0). Then clearly the parame-

ters (m,0,k = 0) are consistent.

Introduce the numbers

ei = i +

i?

j=1

mj,

(i = 0,...,r) .

In particular, e0= 0.

Lemma 4.1. The space

V (m,0,0) = span?1 = xe0, xe1, ..., xer?

is a space of the first type associated to (m,0,0).

Proof. The lemma is proved by direct verification.

?

For i = 0,1,...,r, introduce the first order linear differential operators

Di(m,l,k) = x(x − 1)d

dx

− (k +

i?

s=1

ms− li+ li+1+ i)(x − 1) − k − 1 .

(4.1)

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DIFFERENTIAL EQUATIONS FOR JACOBI-PI˜NEIRO POLYNOMIALS5

For i = 1,...,r, let 1i = (1,...,1,1,0,...,0) be the r-tuple where we have i ones and

r − i zeros. Let 10 = 0 = (0,...,0).

For all i,j ∈ {0,1,...,r}, we have

Dj(m,l + 1i,k + 1) Di(m,l,k) = Di(m,l + 1j,k + 1) Dj(m,l,k) .

Lemma 4.2. Suppose (m,l,k) and (m,l + 1i,k + 1) are consistent parameters. Let

V (m,l,k) be a space of the first type associated to (m,l,k). Then the space

V (m,l + 1i,k + 1) = {Di(m,l,k)v , v ∈ V (m,l,k)}

is a space of the first type associated to (m,l + 1i,k + 1).

Proof. The proof is straightforward.

?

Theorem 4.3. Let (m,l,k) be consistent parameters. Then there exist a space V (m,l,k)

of the first type and a space U(m,l,k) of the second type associated to (m,l,k).

Proof. Let i(1), ... , i(k) ∈ {0,...,r} be any sequence of indices such that 0 occurs in

the sequence exactly k − l1times and every number i = 1,...,r occurs in the sequence

exactly li− li+1 times. Then l =

?k

(m,?j

Introduce the linear differential operator

s=11i(s)and for every for j = 0,...,k, the tuple

s=11i(s),j) forms a consistent set of parameters.

Dm,l,k = Dj(k)(m,

k−1

?

s=1

1j(s),k − 1) ... Dj(2)(m,1j(1),1) Dj(1)(m,0,0) (4.2)

of order k.

By Lemma 4.1, a space V (m,l,k) of the first type associated to (m,l,k) can be con-

structed by application of the operator Dm,l,kto the space V (m,0,0) of Lemma 4.1.

A space U(m,l,k) of the second type associated to (m,l,k) can be constructed from

the space of the first type by the construction of Lemma 3.1.

?

Let (m,l,k) be consistent parameters. Let V (m,l,k) be the space of the first type

associated to (m,l,k). Let v0(m,l,k) ∈ V (m,l,k) be the monic polynomial of degree l1.

Such a polynomial in V (m,l,k) is unique according to the definition of the space of the

first type associated to (m,l,k).

Lemma 4.4. The polynomial v0(m,l,k) is the Jacobi-Pi˜ neiro polynomial P(m,l,k).

Proof. The polynomial v0(m,l,k) is obtained by application of the operator Dm,l,kto the

function 1. It is a straightforward calculation to check that this formula for v0(m,l,k)

coincides with the Rodrigues-type formula for the Jacobi-Pi˜ neiro polynomial P(m,l,k)

in formula (2.1).

?