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The multicomponent 2D Toda hierarchy: generalized matrix orthogonal polynomials, multiple orthogonal polynomials and Riemann--Hilbert problems

We consider the relation of the multi-component 2D Toda hierarchy with matrix orthogonal and biorthogonal polynomials. The multi-graded Hankel reduction of this hierarchy is considered and the corresponding generalized matrix orthogonal polynomials are studied. In particular for these polynomials we consider the recursion relations, and for rank one weights its relation with multiple orthogonal polynomials of mixed type with a type II normalization and the corresponding link with a Riemann--Hilbert problem.


Introduction
Recently, multiple orthogonal polynomials, the related Riemann-Hilbert problems and its application to different areas, for example Brownian motions have deserved much attention [1]. The field of matrix orthogonal polynomials has been also a growing area of research with some similarities with the scalar case but much more richness, see [2]. The relation of multiple orthogonal polynomials with multicomponent KP hierarchies has been noticed in [3] and the string equation formalism of integrable systems has been applied in [4]. Ten years ago Adler and van Moerbeke [5] -in the context of the so called discrete KP hierarchy-introduced what they named as generalized orthogonal polynomials, and what they claimed to be the corresponding Riemann-Hilbert problem, while later on they studied the related Darboux transformations [6]. Recently, and for the Toeplitz case Cafasso [7] extended this work in order to consider block matrices within the non-Abelian Ablotiwz-Ladick latttice.
Following Ueno and Takasaki [8] and the seminal paper of Mulase [9] in [10] we gave a description of the infinite multicomponent 2D Toda lattice hierarchy in terms of a Gaussian factorization (also known as Borel factorization) in an infinite dimensional Lie group, while later on [11] we analyzed the dispersionless limit of the hierarchy with the aid of the factorization problem. (See [12] for a discussion on different cases where this factorization makes sense.) Following [5] we could argue as follows: i) in the one hand the multicomponent Toda hierarchy may be viewed as an LU factorization of certain deformed infinite-dimensional matrix and ii) on the other hand, the same matrix could be thought as a moment matrix and the corresponding LU factorization should give us the corresponding generalized matrix orthogonal polynomials. In this manner, we would be able to build a bridge between multicomponent Toda hierarchy and matrix orthogonal polynomials. This is the main idea developed in this paper. First, we connect matrix orthogonal and biorthogonal polynomials with the multi-component 2DToda lattice hierarchy, focusing in particular on the Hänkel reduction. Second, we generalize the band condition of [5] to the multicomponent case and consider what we refer as multi-graded Hänkel. This leads to a multi-component extension of generalized orthogonal polynomials which in some cases can be described in terms of multiple orthogonal polynomials of mixed type with a type II normalization. This connection allows us to give an appropriate Riemann-Hilbert problem for this generalized orthogonal polynomials (notice that the one discussed in [5] is not correct).
The layout of the paper is as follows. In this introduction we give a overview of the Gaussian factorization and the semi-infinite multi-component 2D Toda lattice hierarchy. Then, in §2 we discuss how matrix orthogonal polynomials and the multi-component 2D Toda lattice hierarchy are connected. Finally, in §3 we consider multi-graded Hänkel reduction, extended generalized orthogonal polynomials, mixed multiple orthogonal polynomials and corresponding Riemann-Hilbert problems.

Gaussian factorization and the semi-infinite multi-component 2D Toda lattice hierarchy
For the construction of a Lie group theoretical setting we denote by Λ the shift operator for matrix valued sequences. The associative algebra of linear operators on these sequences can be identified with the associative algebra of semi-infinite matrices with entries taking values in C N ×N , the set of N × N complex matrices. With the usual commutator for linear operators this algebra is also a Lie algebra denoted by g whose Lie group G is the group of invertible linear operators in g. Let's take g ∈ G and consider the following Gaussian factorization problem g = S −1S where S is a block lower triangular matrix, with S ii = I N , being I N ∈ C N ×N the identity matrix, andS is an block upper triangular matrix. In [12] it is proven that the Borel decomposition holds if all the principal minors do not vanish. Thus, the factorization holds under "small" continuous deformations and we can consider the factorization g(t) = S(t) −1S (t) where t stands for a set of complex variables. As was discussed in [10] this factorization problem leads to an integrable hierarchy of nonlinear PDE known as multicomponent 2D Toda lattice hierarchy. Let us discuss these issues in more depth. Observe that the matrix associated with the shift operator is the block matrix (I N δ i,i+1 ) and Λ t is the operator associated with the transposed matrix (I N δ i+1,i ). If E ab , a, b = 1, . . . , N is the canonical basis of C N ×N and t = ({t ja }, {t ja }), j = 1, 2, . . . and a = 1, . . . , N , is a collection of complex parameters we introduce and consider Gaussian factorization of g(t) := W 0 (t)gW 0 (t) −1 .
Following [10] we define the Lax operators where all the coefficients in the Λ-expansions belong to C N ×N . The multi-component 2D Toda hierarchy has the following Lax representation where the sub-indices + and − denote the block upper triangular, strictly block lower triangular projections, respectively.
2 Matrix orthogonal polynomials and the multi-component 2DToda lattice hierarchy Following Adler and van Moerbeke [5] we construct families of matrix orthogonal and biorthogonal polynomials associated with the 2D Toda lattice hierarchy.
In the first place, we define the following families of (time-dependent) matrix polynomials where χ(z) := (I N , zI N , z 2 I N , . . .) t and the symbol † denotes Hermitian conjugation.Next we consider a matrix-valued bilinear pairing between matrix polynomials. Given matrix polynomials P (z) = i k=0 P k z k and Q(z) = j l=0 Q l z l (of degrees i, j, respectively) we have where z k I N , z l I N denotes the matrix for the bilinear pairing in the canonical basis and for each (k, l) is an N × N complex matrix. This pairing has the following properties 1. Is linear in the first component: Is skew-linear in the second component: , then the families p(z) andp(z) are biorthogonal matrix polynomials for the linear pairing, i.e. Moreover, 2. In addition, if the time-dependent initial condition g(t) is Hermitian for all t then p(z) andp(z) are two families of matrix orthogonal polynomials, moreover, the two families are proportional. Proof.
2. Let's study the conditions under which p(z) is a family of matrix orthogonal polynomials. If we take two polynomials in the family, such as p i (z) = i k=1 S ik z k and p j (z) = j l=1 S jl z l we have Observe thatSS † is clearly block upper-diagonal with its Hermitian conjugate given by Therefore,SS † is Hermitian and block upper-diagonal, which implies thatSS † is a block diagonal matrix and the blocks in the diagonal are C N ×N Hermitian matrices.

The Hänkel case
We choose g to be a block Hänkel matrix so that Λg = gΛ t or In this case we have for the blocks of the moment matrix g we easily deduce that g(t) is block Hänkel if g is. We have Proposition 2. Assume that g is block Hänkel then with γ (j) a block moment matrix; i.e. γ (j) = R x j ρ(x)dx then the pairing can be viewed as a scalar product in the real line whose matrix moment is g ; i.e., the Hänkel symmetry ensures that there is only dependence in j + k) we have: In general arbitrary continuous deformations do not preserve the Hermitian character of g. If we look for families of matrix orthogonal polynomials on the real line we should make restricted deformations. Let's make this point clear. In the following z * denotes the complex conjugate of z ∈ C. 2. Moreover, if the continuous deformation parameters satisfy one of the two following conditions (a) t ja ,t ja ∈ R and satisfy t ja = t j ,t ja =t j , a = 1, . . . , N .
the result holds for the time dependent moment matrix. Proof.
1. If the matrix g is block Hänkel and the blocks are Hermitian the matrix g is itself Hermitian. Given i, j, pair of indices for an element of g, there exist four integer indices (k, l),(m, n) with k, l 0 y m, n = 1, . . . , N that satisfy a ij = (A kl ) mn = (A kl ) * nm = (A lk ) * nm = a * ji (we use A ∈ C N×N for a block of g) as a consequence g = g † and by Proposition 1 the first part of the result holds.
2. Let us compute the time evolution for g. We have is the j-th Schur 1 polynomial for the component a. Consequently and taking so it is straight forward to see the block structure of W 0 , whose blocks are given by j is the j-th Schur polynomial for the component a but now in the variables −t ja ).
We are now ready to compute Then (a) If the first condition holds then all matrices s j ands j are real and scalar so (b) If the second condition holds s j =s † j for all j so 1 We remind the reader that the Schur polynomials are given by exp( j Λ j so s P j 1 +···+jp=j tj 1 a · · · tj p a = tja + · · · where the missing term is a polynomial of utmost degree j in the variables t1a, t2a, . . . , tj−1,a.

Under any of the two conditions g(t) is block-Hänkel with Hermitian blocks, so is itself
Hermitian. That proves the second part of the proposition.
We now discuss the recursion formulae. Using the Hänkel condition Λg = gΛ t and taking the usual definition for the Lax operator we conclude L =L and hence we have L = Λ+u(n)+v(n)Λ t . Proposition 4. The polynomials p(z),p(z) satisfy a three term recurrence law given by p n+1 (z) = zp n (z) − u(n)p n (z) − v(n)p n−1 (z) Proof. From the definition of L we have Lp(z) = LSχ(z) = zp(z) = (Λ + u(n) + v(n)Λ t )p(z). If we take the sequence terms {p n (z)} n we conclude zp n (z) = p n+1 (z) + u(n)p n (z) + v(n)p n−1 . For the polynomialsp(z) we haveL †p (z) = zp(z) = (Λv † (n) + u † (n) + Λ t )p(z), and using the sequence {p n (z)} n we obtain another recurrence law.

Multi-graded Hänkel reduction, generalized orthogonality, multiple orthogonal polynomials and Riemann-Hilbert problems
In this section we will study generalized Hänkel type conditions. Given a multi-indexn = (n 1 , . . . , n N ) with n a non-negative integers we define for A ∈ g the power An = N a=1 A na E aa . For two multi-indicesn andm a matrix g is said to be a (n,m) multi-graded Hänkel if If as before g ij ∈ C N ×N denotes a block in g then we can write g ij = (g ij,ab ) 1 a,b N and the multi-graded Hänkel condition reads g i+na j,ab = g i j+m b ,ab . An ample family of multi-graded Hänkel matrices can be constructed in terms of weights ρ j,ab as the moments where the weights satisfy a generalized periodicity condition of the form Thus, given the weights ρ 0,ab , . . . , ρ m b −1,ab , all the others are fixed by (5). From now on, we concentrate only in these cases of multigraded Hänkel matrices. We notice that for the 1component case and for n 1 = m 1 case these moments matrices were studied in [5]- [6].
Proposition 5. For multi-graded Hänkel matrices the matrix polynomials p i satisfy the following generalized orthogonality conditions Proof. From Sg(t)S −1 = 1 G we get Using the Euclidean division i = θ c m c + σ c , with θ c 0, 0 σ c < m c we get a better insight of the orthogonality relations (6), for a, c = 1, . . . , N ,

Evolution
From (3) we conclude that g(t) is of (n,m) multi-graded Hänkel type if g is. In general, the evolution of g is given in terms of Schur polynomials by which recalling (4) leads to the following evolution of the weights where t(x) = N a=1 t a (x)E aa and t a (x) := j 1 t ja x j . It can be easily checked that ρ j (t) satisfies the periodicity condition (5) if ρ j does. From (8) we infer that . . . where and therefore Finally, we deduce where we have used the discrete Fourier transform of the weightŝ

Recursion relations and symmetries
In terms of Lax operators the multi-graded Hänkel reduction reads [10] L : Within this subsection we assume that n 1 · · · n N 1, m 1 · · · m N 1, and suppose that n 1 = · · · = n r and n r > n r+1 . Given (1) from (12) we deduce that while we also have L = N b=1L m bC b and therefore We conclude the block band structure Proposition 6. The polynomials p i (z) are subject to Proof. We only have to show that L p = p( n a=1 z na E aa ). But this follows from Similarly, from L †p (z) =p(z)( N b=1 z m b E bb ) a recursion relation follows for thep i . Finally we notice that in this case the following symmetry conditions hold [10] for i 1 and b = 1, . . . , N .

Relation with multiple orthogonal polynomials
We will see that when the weights ρ j are particular rank one matrices there is a nice correspondence with multiple orthogonal polynomial of mixed type with a normalization of type II. Following [1] we take two sets of non negative multi-indicesν = (ν 1 , . . . , ν p ),μ = (µ 1 , . . . , µ q ) and write |ν| = p J=1 ν J and |µ| = q K=1 µ K . We also take weights {w 1J } p J=1 and {w 2K } q K=1 which are assumed to be non-negative functions on the real line. For a fixed pairν,μ we say that {Aν ,μ,J } J=1,...,p is a set of multiple orthogonal polynomials of mixed type if deg Aν ,μ,J ν J − 1 and the following orthogonality relations are satisfied. Alternatively defining the following linear forms Qν ,μ (x) := n J=1 Aν ,μ,J (x)w 1J (x) the orthogonality relations can be written in the following way For multiple orthogonal polynomials as described above we will take |ν| = |µ| and assume that the following conditions hold 1. For each K = 1, . . . , p the orthogonality relations for the multi-indicesν + e K ,μ have a unique solution with Aν ,μ,K monic and deg Aν ,μ,K = ν K − 1, with deg Aν ,μ,J < ν J − 1 if J = K. We will call it type II normalization to the K-th component and write that normalized solution as {A (II,K) ν,μ,J } J=1,...,p . 2. For each K = 1, . . . , q the orthogonality relations for the multi-indicesν,μ − e K have a unique solution with the following normalization: deg Aν ,μ,J = ν J −1 and R Qν ,μ (x)w 2K x µ K dx = 1. We will call it normalization of type I to the K-th component and write that normalized solution as {A (I,K) ν,μ,J } J=1,...,p . In order to connect (7) with multiple orthogonal polynomials we consider the Euclidean division i = q a n a + r a ,with q a 0 and 0 r a < n a , and write where Π ij,ab (z n b ) are polynomials in z n b such that deg Π ij,ab q b , j r b or j = r b + 1 and b = a, q b − 1, j > r b + 1 or j = r b + 1 and b = a.
Notice that the monic character of p i gives the normalization of Π i ra+1,aa which happens to be a monic polynomial with deg Π i ra+1,aa = q a .
The inversion formula for (15) can be deduced as follows. If we denote by ǫ b := exp(2πi/n b ), a primitive n b -th root of the unity, and evaluate at ǫ k b z, k = 0, . . . , n b − 1 we get the following system of equations that we solve in order to obtain the polynomials Π ik,ab in terms of a discrete Fourier transform of the polynomial p i through the formula Then, (7) can be written as These equations strongly suggest to perform a change of variables in each integrand of the type y = x n b . For that aim, it is relevant that when n b is an even number then supp(ρ j,bc ) ⊂ R + , otherwise the change of variables is ill defined. In fact, for these even cases, one easily see that the weights must be supported on the positive axis or uniqueness of the orthogonal polynomials is not ensured. Moreover, for n b odd it is also necessary to assume that the weight is supported either only in the positive real numbers or only in the negative real line, this requirement comes from the positivity condition on the weights and the use of (19). Hereon we will assume that the all weights are supported on the positive real semiline. When the mentioned change of variable is performed in (16) we get We are ready to describe the relation among multiple orthogonal polynomials of type II and generalized matrix polynomials. First, given (a, j) with a = 1, . . . , N and j ∈ N we make the definitions N (a, j) := n 1 + · · · + n a−1 + j and M(a, j) := m 1 + · · · + n a−1 + j + 1.