Contents
Abstract

The paper investigates the properties of certain biorthogonal polynomials appearing in a specific simultaneous Hermite-Padé approximation scheme. Associated to any totally positive kernel and a pair of positive measures on the positive axis we define biorthogonal polynomials and prove that their zeroes are simple and positive. We then specialize the kernel to the Cauchy kernel and show that the ensuing biorthogonal polynomials solve a four-term recurrence relation, have relevant Christoffel-Darboux generalized formulæ  and their zeroes are interlaced. In addition, these polynomial solve a combination of Hermite-Padé approximation problems to a Nikishin system of order . The motivation arises from two distant areas; on one side, in the study of the inverse spectral problem for the peakon solution of the Degasperis-Procesi equation; on the other side, from a random matrix model involving two positive definite random Hermitian matrices. Finally, we show how to characterize these polynomials in term of a Riemann–Hilbert problem.

Cauchy Biorthogonal Polynomials

M. Bertola 111Work supported in part by the Natural Sciences and Engineering Research Council of Canada (NSERC), Grant. No. 261229-03 and by the Fonds FCAR du Québec No. 88353., M. Gekhtman  222Work supported in part by NSF Grant DMD-0400484., J. Szmigielski  333Work supported in part by the Natural Sciences and Engineering Research Council of Canada (NSERC), Grant. No. 138591-04

Centre de recherches mathématiques, Université de Montréal

C. P. 6128, succ. centre ville, Montréal, Québec, Canada H3C 3J7

E-mail: bertola@crm.umontreal.ca

Department of Mathematics and Statistics, Concordia University

1455 de Maisonneuve W., Montréal, Québec, Canada H3G 1M8

Department of Mathematics 255 Hurley Hall, Notre Dame, IN 46556-4618, USA

E-mail: Michael.Gekhtman.1@nd.edu

Department of Mathematics and Statistics, University of Saskatchewan

106 Wiggins Road, Saskatoon, Saskatchewan, S7N 5E6, Canada

E-mail: szmigiel@math.usask.ca


1 Introduction and motivations

This paper mainly deals with a class of biorthogonal polynomials of degree satisfying the biorthogonality relations

(1-1)

where are positive measures supported on with finite bimoments. These polynomials will be introduced in Sec. 2 in a more general context of polynomials associated to general totally positive kernels (Def. 2.1) with which they share some general properties in regard to their zeroes.

While these properties are interesting in their own right, we wish to put the work in a more general context and explain the two main motivations behind it. They fall within two different and rather distant areas of mathematics : peakon solutions to nonlinear PDEs and Random Matrix theory.

Peakons for the Degasperis-Procesi equation.

In the early 1990’s, Camassa and Holm [11] introduced the (CH) equation to model (weakly) dispersive shallow wave propagation. More generally, the CH equation belongs to the so-called b-family of PDEs

(1-2)

Two cases, and within this family are now known to be integrable: the case is the original CH equation whereas the case is the Degasperis-Procesi [14] (DP) equation, which is more directly related to the present paper.

In all cases the b-family admits weak (distributional) solutions of the form:

(1-3)

if and only if the positions and the heights satisfy the system of nonlinear ODEs:

(1-4)

for . The non-smooth character of the solution manifests itself by the presence of sharp peaks at , hence the name peakons. For the CH equation the peakons solution were studied in [2, 1], while for the DP equation in [20, 21]; in both cases the solution is related to the isospectral evolution of an associated linear boundary-value problem

(1-5)

The variables and the quantities are related by

(1-6)

Because of the similarity to the equation of an inhomogeneous classical string (after a separation of variables) we refer to the two linear ODEs as the quadratic and cubic string, respectively. The case of peakons corresponds to the choice

(1-7)

The remarkable fact is that in both cases the associated spectral problems have a finite positive spectrum; this is not so surprising in the case of the quadratic string which is a self-adjoint problem, but it is quite unexpected for the cubic string, since the problem is not self-adjoint and there is no a priori reason for the spectrum to even be real [21].

As it is natural within the Lax approach to integrable PDEs, the spectral map linearizes the evolution of the isospectral evolution: if are the eigenvalues of the respective boundary value problems and one introduces the appropriate spectral residues

(1-8)

then one can show [20] that the evolution linearizes as follows (with the dot representing the time evolution)

(1-9)

Since this is not the main focus of the paper, we are deliberately glossing over several interesting points; the interested reader is referred to [21] and our recent work [8] for further details. In short, the solution method for the DP equation can by illustrated by the diagram

In the inverse spectral map resides the rôle of the biorthogonal polynomials to be studied here, as we briefly sketch below. The inverse problem for the ordinary string with finitely many point masses is solved by the method of continued fractions of Stieltjes’ type as was pointed out by M.G. Krein ([17]). The inverse problem for the cubic string with finitely many masses is solved with the help of the following simultaneous Hermite-Padé type approximation ([21])

Definition 1.1 (Padé-like approximation problem).

Let denote the spectral measure associated with the cubic string boundary value problem and , denote the Weyl functions introduced in [21]. Then, given an integer , we seek three polynomials of degree satisfying the following conditions:

  1. [Approximation]:

  2. [Symmetry]: with , .

  3. [Normalization]:

This approximation problem has a unique solution ([21]) which, in turn, is used to solve the inverse problem for the cubic string. We point out that it is here in this approximation problem that the Cauchy kernel makes its, somewhat unexpected, appearance through the spectral representation of the second Weyl function.

Random Matrix Theory

The other source of our interest in biorthogonal polynomials comes from random matrix theory. It is well known [22] that the Hermitean matrix model is intimately related to (in fact, solved by) orthogonal polynomials (OPs). Not so much is known about the role of biorthogonal polynomials (BOPs). However, certain biorthogonal polynomials somewhat similar to the ones in the present paper appear prominently in the analysis of “the” two–matrix model after reduction to the spectrum of eigenvalues [5, 7, 6, 15]; in that case the pairing is of the form

(1-10)

and the associated biorthogonal polynomials are sometimes called the Itzykson–Zuber BOPs, in short, the IZBOPs.

Several algebraic structural properties of these polynomials and their recurrence relation (both multiplicative and differential) have been thoroughly analyzed in the previously cited papers for densities of the form for polynomials potentials and for potentials with rational derivative (and hard–edges) in [3].

We recall that while ordinary OPs satisfy a multiplicative three–term recurrence relation, the BOPs defined by (1-10) solve a longer recurrence relation of length related to the degree of the differential over the Riemann sphere [3]; a direct (although not immediate) consequence of the finiteness of the recurrence relation is the fact that these BOPs (and certain integral transforms of them) are characterized by a Riemann–Hilbert problem for a matrix of size equal to the length of the recurrence relation (minus one). The BOPs introduced in this paper share all these features, although in some respects they are closer to the ordinary orthogonal polynomials than to the IZBOPs.

The relevant two–matrix model our polynomials are related to was introduced in [10]. We now give a brief summary of that work. Consider the set of pairs of Hermitean positive-definite matrices endowed with the (–invariant) Lebesgue measure denoted by . Define then the probability measure on this space by the formula:

(1-11)

where (the partition function) is a normalization constant, while stand for the product of the densities (the Radon–Nikodym derivatives of the measures with respect to the Lebesgue measure) over the (positive) eigenvalues of .

This probability space is similar to the two–matrix model discussed briefly above for which the coupling between matrices is [16] instead of . The connection with our BOPs (1-1) is analogous to the connection between ordinary orthogonal polynomials and the Hermitean Random matrix model [22], whose probability space is the set of Hermitean matrices equipped with the measure In particular, we show in [10] how the statistics of the eigenvalues of the two matrices can be described in terms of the biorthogonal polynomials we are introducing in the present work. A prominent role in the description of that statistics is played by the generalized Christoffel–Darboux identities we develop in Section 4.

We now summarize the main results of the paper:

  • for an arbitrary totally positive kernel and arbitrary positive measures on we prove that the matrix of bimoments is totally positive (Thm. 2.1);

  • this implies that there exist, unique, sequences of monic polynomials of degree , biorthogonal to each other as in (2.1); we prove that they have positive and simple zeroes (Thm. 2.5);

  • we then specialize to the kernel ; in this case the zeroes of () are interlaced with the zeroes of the neighboring polynomials (Thm. 3.2 );

  • they solve a four–term recurrence relation as specified after (1-1) (Cor. 4.2);

  • they satisfy Christoffel–Darboux identities (Prop. 4.3, Cor. 4.3, Thms. 5.3, 5.5)

  • they solve a Hermite-Padé approximation problem to a novel type of Nikishin systems (Sec. 5, Thms. 5.1, 5.2);

  • they can be characterized by a Riemann–Hilbert problems, (Props. 6.1, 6.2) ;

In the follow-up paper we will explain the relation of the asymptotics of the BOPs introduced in this paper with a rigorous asymptotic analysis for continuous (varying) measures using the nonlinear steepest descent method [9].

2 Biorthogonal polynomials associated to a totally positive kernel

As one can see from the last section the kernel , which we will refer to as the Cauchy kernel, plays a significant, albeit mysterious, role. We now turn to explaining the role of this kernel. We recall, following [19], the definition of the totally positive kernel.

Definition 2.1.

A real function of two variables ranging over linearly ordered sets and , respectively, is said to be totally positive (TP) if for all

(2-1)

we have

(2-2)

We will also use a discrete version of the same concept.

Definition 2.2.

A matrix is said to be totally positive (TP) if all its minors are strictly positive. A matrix is said to be totally nonnegative (TN) if all its minors are nonnegative. A TN matrix is said to be oscillatory if some positive integer power of is TP.

Since we will be working with matrices of infinite size we introduce a concept of the principal truncation.

Definition 2.3.

A finite by matrix is said to be the principal truncation of an infinite matrix if . In such a case will be denoted .

Finally,

Definition 2.4.

An infinite matrix is said to be TP (TN) if is TP (TN) for every .

Definition 2.5.

Basic Setup

Let be a totally positive kernel on and let be two Stieltjes measures on . We make two simplifying assumptions to avoid degenerate cases:

  1. is not an atom of either of the measures (i.e. has zero measure).

  2. and have infinitely many points of increase.

We furthermore assume:

  1. the polynomials are dense in the corresponding Hilbert spaces , ,

  2. the map , is bounded, injective and has a dense range in .

Under these assumptions provides a non-degenerate pairing between and :

(2-3)
Remark 2.1.

Assumptions  3 and  4 could be weakened, especially the density assumption, but we believe the last two assumptions are the most natural to work with in the Hilbert space set-up of the theory.

Now, let us consider the matrix of generalized bimoments

(2-4)
Theorem 2.1.

The semiinfinite matrix is TP.

Proof.

According to a theorem of Fekete, (see Chapter 2, Theorem 3.3 in [19] ), we only need to consider minors of consecutive rows/columns. Writing out the determinant,

we find

Since our intervals are subsets of we can absorb the powers of into the measures to simplify the notation. Moreover, the function enjoys the following simple property

for any . Finally, the product measures are clearly permutation invariant.

Thus, without any loss of generality, we only need to show that

which is tantamount to showing positivity for . First, we symmetrize with respect to the variables ; this produces

Subsequent symmetrization over the variables does not change the value of the integral and we obtain (after restoring the definition of )

Finally, since is permutation invariant, it suffices to integrate over the region , and, as a result

(2-5)

Due to the total positivity of the kernel the integrand is a positive function of all variables and so the integral must be strictly positive. ∎

To simplify future computations we define so that the matrix of generalized bimoments (2-4) is simply given by: Now, let denote the semi-infinite upper shift matrix. Then we observe that multiplying the measure by or, multiplying by , is tantamount to multiplying on the left by , or on the right by respectively, which gives us a whole family of bimoment matrices associated with the same but different measures. Thus we have

Corollary 2.1.

For any nonnegative integers the matrix of generalized bimoments is TP.

We conclude this section with a few comments about the scope of Theorem 2.1.

Remark 2.2.

Provided that the negative moments are well defined, the theorem then applies to the doubly infinite matrix , .

Remark 2.3.

If the intervals are and then the proof above fails because we cannot re-define the measures by multiplying by powers of the variables, since they become then signed measures, so in general the matrix of bimoments is not totally positive. Nevertheless the proof above shows (with or ) that the matrix of bimoments is positive definite and –in particular– the biorthogonal polynomials always exist, which is known and proved in [15].

2.1 Biorthogonal polynomials

Due to the total positivity of the matrix of bimoments in our setting, there exist uniquely defined two sequences of monic polynomials

such that

Standard considerations (Cramer’s Rule) show that they are provided by the following formulæ

(2-6)
(2-7)

where by equation (2-5). For convenience we re-define the sequence in such a way that they are also normalized (instead of monic), by dividing them by the square root of ;

(2-8)
(2-9)

Thus .

We note also that the BOPs can be obtained by triangular transformations of

(2-10)

where are (formally) invertible lower triangular matrices such that , where, we recall, is the generalized bimoment matrix. Moreover, our BOPs satisfy, by construction, the recursion relations:

which will be abbreviated as

(2-11)

where and are Hessenberg matrices with positive entries on the supradiagonal, and are infinite column vectors respectively.

The biorthogonality can now be written as where denotes the semi-infinite identity matrix. Moreover

(2-12)
Remark 2.4.

The significance of the last two formulas lies in the fact that the operator of multiplication is no longer symmetric with respect to the pairing and as a result the matrices and are distinct.

2.2 Simplicity of the zeroes

In this section we will use the concept of a Chebyshev system of order and a closely related concept of a Markov sequence. We refer to [23] and [17] for more information. The following theorem is a convenient restatement of Lemma 2 in [17], p.137. For easy display we replace determinants with wedge products.

Theorem 2.2.

Given a system of continuous functions let us define the vector field

(2-13)

Then is a Chebyshev system of order on iff the top exterior power

(2-14)

for all in . Furthermore, for , if we denote the truncation of to the first components by , then is a Markov system iff the top exterior power

(2-15)

for all in and all .

The following well known theorem is now immediate

Theorem 2.3.

Suppose is a Chebyshev system of order on , and suppose we are given distinct points in . Then, up to a multiplicative factor, the only generalized polynomial , which vanishes precisely at in is given by

(2-16)
Theorem 2.4.

Denote by . Then is a Chebyshev system of order on . Moreover, as defined in Theorem 2.3 changes sign each time passes through any of the zeros .

Proof.

It is instructive to look at the computation. Let , then using multi-linearity of the exterior product,

where Thus . The rest of the proof is the argument about the sign of the integrand. To see how sign changes we observe that the sign of depends only on the ordering of , in view of the total positivity of the kernel. In other words, the sign of is where is the permutation rearranging in an increasing sequence. ∎

Corollary 2.2.

Let . Then is a Markov sequence on ,

Proof.

Indeed, Theorem 2.2 implies that the group acts on the set of Chebyshev systems of order . It suffices now to observe that are obtained from by an invertible transformation. ∎

Remark 2.5.

Observe that is a Markov sequence regardless of biorthogonality.

Biorthogonality enters however in the main theorem

Theorem 2.5.

The zeroes of are all simple and positive. They fall within the convex hull of the support of the measure (for ’s) and (for the ’s).

Proof.

We give first a proof for . The theorem is trivial for . For , let us suppose has zeros of odd order in the convex full of . In full analogy with the classical case, , since

by biorthogonality, forcing, in view of positivity of , to change sign in the convex hull of . In the general case, denote the zeros by . Using a Chebyshev system on we can construct a unique, up to a multiplicative constant, generalized polynomial which vanishes exactly at those points, namely

(2-17)

where

It follows then directly from biorthogonality that

(2-18)

On the other hand, is proportional to in Theorem 2.3 which, by Theorem 2.4, changes sign at each of its zeroes,� so the product is nonzero and of fixed sign over . Consequently, the integral is nonzero, since is assumed to have infinitely many points of increase. Thus, in view of the contradiction, , hence , for is a polynomial of degree . The case of follows by observing that the adjoint is also a TP kernel and hence it suffices to switch with throughout the argument given above. ∎

Lemma 2.1.

In the notation of Corollary 2.2 has zeros and sign changes in the convex hull of .

Proof.

Clearly, since is a Chebyshev system of order on , the number of zeros of cannot be greater than . Again, from

we conclude that changes sign at least once within the convex hull of . Let then , be all zeros of within the convex hull of at which changes its sign. Thus, on one hand,

while, on the other hand, using biorthogonality we get

which shows that . ∎

In view of Theorem 2.3 the statement about the zeros of has the following corollary

Corollary 2.3.

Heine-like representation for

(2-19)

where are the zeros of and is a constant.

3 Cauchy BOPs

From now on we restrict our attention to the particular case of the totally positive kernel, namely, the Cauchy kernel

(3-1)

whose associated biorthogonal polynomials will be called Cauchy BOPs . Thus, from this point onward, we will be studying the general properties of BOPs for the pairing

(3-2)

Until further notice, we do not assume anything about the relationship between the two measures , other than what is in the basic setup of Definition 2.5.

3.1 Rank One Shift Condition

It follows immediately from equation (3-1) that

(3-3)

which, with the help of the shift matrix and the matrix of bimoments , can be written as:

Moreover, by linearity and equation (2-12), we have

(3-4)

which connects the multiplication operators in and . Before we elaborate on the nature of this connection we need to clarify one aspect of equation (3-4).

Remark 3.1.

One needs to exercise a great deal of caution using the matrix relation given by equation (3-4). Its only rigorous meaning is in action on vectors with finitely many nonzero entries or, equivalently, this equation holds for all principal truncations.

Proposition 3.1.

The vectors are strictly positive (have nonvanishing positive coefficients).

Proof.

We prove the assertion only for , the one for being obtained by interchanging the roles of and .

From the expressions (2-9) for we immediately have

(3-5)

Since we know that for any we need to prove the positivity of the other determinant. Determinants of this type were studied in Lemma 4.10 in [21].

We nevertheless give a complete proof of positivity. First, we observe that

(3-7)

Here the symbol is to remind that the vector consists of entries (whereas consists of entries) and that the Vandermonde determinant is taken accordingly. Note also that the variable never appears in the product in the denominator. Symmetrizing the integral in the ’s with respect to labels , but leaving fixed, gives

(3-8)

Symmetrizing now with respect to the whole set we obtain

(3-9)

Moreover, since the integrand is permutation invariant, it suffices to integrate over the region