Ladder relations for a class of matrix valued orthogonal polynomials

In this paper we study algebraic and differential relations for matrix valued orthogonal polynomials (MVOPs) defined on $\mathbb{R}$. Using recent results by Casper and Yakimov, we investigate MVOPs with respect to a matrix weight of the form $W(x)=e^{-v(x)}e^{xA} T e^{xA^\ast}$, where $v$ is a scalar polynomial of even degree with positive leading coefficient and $A$ and $T$ are constant matrices. We obtain ladder operators, discrete string equations for the recurrence coefficients and multi-time Toda equations for deformations with respect to parameters in the weight, and we show that the Lie algebra generated by the ladder operators is finite dimensional. Hermite-type matrix valued weights are studied in detail: in this case the weight is characterized by the ladder operators, and the Lie algebra generated by them can be extended to a Lie algebra that is isomorphic to the standard Harmonic oscillator algebra. Freud-type matrix weights are also discussed. Finally, we establish the link between these ladder relations and those considered previously by A. Dur\'{a}n and M. Ismail.


Introduction
Matrix valued orthogonal polynomials (MVOPs) were introduced by Krein in the 1940's and they appear in different areas of mathematics and mathematical physics, including spectral theory [22], scattering theory [21], tiling problems [14], integrable systems [6,2,28] and stochastic processes [24,12,13]. There is also a fruitful interaction between harmonic analysis of matrix valued functions on compact symmetric pairs and matrix valued orthogonal polynomials. The first example of such an interaction is a family of matrix valued orthogonal polynomials related with the spherical functions of the compact symmetric pair (SU(3), S(U(2) × U(1)), which appeared in [23]. Inspired by [37], the case of (SU(2) × SU (2), diag) gave a direct approach [34,35] leading to a general set-up in the context of multiplicity free pairs [27,41]. In this context, some properties of the orthogonal polynomials such as orthogonality, recurrence relations and differential equations are understood in terms of the representation theory of the corresponding symmetric spaces, see also [1] for the quantum group case and [36] for multivariable matrix orthogonal polynomials.
The interpretation of matrix valued orthogonal polynomials in terms of the representation theory of a certain symmetric pair is typically only for a limited (discrete) number of the parameters involved. It is then necessary to develop analytic tools to extend to a general set of parameters. In this context, shift operators for matrix valued orthogonal polynomials turned out to be very useful [7,8,33,31,28].
In the last two decades, there has been significant progress in understanding how the differential and algebraic properties of the classical scalar orthogonal polynomials can be extended to the matrix valued setting. A. Durán and M. Ismail [19] introduced first order lowering and raising operators for MVOPs, and these results were rederived later on using the Riemann-Hilbert formulation by Grünbaum and coauthors [25]. There is also extensive work on orthogonal polynomial solutions of matrix valued differential equations of second order from an analytic point of view, we refer the reader for instance to [15,18,16,17]. Recently, Casper and Yakimov [9] proposed a general framework to investigate the matrix Bochner problem, that is, the classification of N × N weight matrices W (x) whose associated MVOPs are eigenfunctions of a second order differential operator.
The purpose of this paper is to apply the setup of [9] to MVOPs defined on the real line, in particular for exponential-type weights.
Given N ∈ N, we denote by M N (C) the space of all N × N matrices with complex entries. Let W : R → M N (C) be a positive definite matrix weight supported in the (possibly infinite) interval [a, b]. For M N (C)-valued functions H, G, we define the matrix valued inner product (1.1) H, G = b a H(y)W (y)G(y) * dy ∈ M N (C).
Using standard arguments it can be shown that there exists a unique sequence (P (x, n)) n of monic matrix valued orthogonal polynomials (MVOPs) with respect to W , in the following sense: P (x, n), P (x, m) = H(n)δ n,m , where the squared norm H(n) is a positive definite matrix, see for instance [11,26]. As a direct consequence of orthogonality, the polynomials P (x, n) satisfy the following three-term recurrence relation (1.2) xP (x, n) = P (x, n + 1) + B(n)P (x, n) + C(n)P (x, n − 1), where B(n), C(n) ∈ M N (C). Note that these matrix coefficients multiply the MVOPs from the left. From the orthogonality relations, we also obtain that where X(n) is the one-but-leading coefficient of P (x, n), i.e. P (x, n) = x n + x n−1 X(n) + · · · . The structure of this paper is the following: in Section 2, following the approach of Casper and Yakimov in [9], we discuss differential and difference operators for these MVOPs. In this noncommutative setting operators can act both from the right and from the left. We consider two isomorphic algebras of operators acting on MVOPs, one algebra of matrix valued differential operators acting from the right, F R (P ), and a second algebra of matrix valued discrete operators acting from the left, F L (P ). In this construction, a differential operator D ∈ F R (P ) acts naturally on the variable of the MVOPs, whereas a difference operator M ∈ F L (P ) acts on its degree.
The approach proposed in [9] is particularly explicit in the case of exponential weights defined on the real line; these weights are studied in Section 3 and written in the form W (x) = e −v(x) e xA T e xA * , with x ∈ R, where the potential v is an even polynomial with positive leading coefficient and A and T are constant matrices. In this case, the differential operator D = ∂ x + A has a simple adjoint D † = −D + v ′ (x), with respect to the matrix valued inner product given by W . The actions of D and D † on the MVOPs are which will imply that D, D † ∈ F R (P ). Our first result states that D, D † induce ladder relations: where k = deg v, with some matrix coefficients A j (n) and A j (n). These operators are closely related to the creation and annihilation operators given in [19], with the advantage that D and D † are each other's adjoint. This property is crucial to show that the Lie algebra generated by the operators D and D † is finite dimensional and it is isomorphic to the algebra generated by the ladder operators for the scalar weight w(x) = e −v(x) , see for instance [10], [29,Chapter 3]. From the ladder relations, we obtain nonlinear algebraic equations for the coefficients of the recurrence relation (1.2). In the literature these identities are often called discrete (or Freud) string equations. We include two examples: Hermite-type weights with v(x) = x 2 + tx and t ∈ R, and Freud-type weights with v(x) = x 4 + tx 2 , and in this last case the discrete string equations are in fact a matrix analogue of the discrete Painlevé I equation [40]. We remark that this kind of identity, which is very relevant in integrable systems, is obtained here as a result of the relation between the two Fourier algebras of operators and in particular from the fact that F L (P ) and F R (P ) are isomorphic. Section 4 is devoted to the detailed study of Hermite-type matrix valued weights. In this setting, we show that the ladder relations in fact characterize the matrix valued weight. We establish the link between the Lie algebra generated by D and D † and the standard Harmonic oscillator algebra. An important feature of this case is that the matrix valued Hermite polynomials are related to a degenerate Hamiltonian which is a composition of the quantum harmonic oscillator with a non-relativistic spin Hamiltonian. This generalizes the well known property of scalar Hermite polynomials being related to the Schrödinger Hamiltonian.
Complementing the previous results, in Section 5, we investigate similar identities of differential and algebraic type for a deformation of the matrix weight with respect to extra parameters. Examples include the non-Abelian Toda and Langmuir lattice equations. For the particular case of a multi-time Toda deformation, we give a Lax pair formulation, analogous to [29, (2.8.5)] for the scalar case.
In the appendix we establish the link between the ladder relations obtained with this methodology and the ladder operators previously considered by A. Durán and M. Ismail in [19].
Acknowledgements. The authors would like to thank Erik Koelink and Mourad E. H. Ismail for fruitful discussions about the content and scope of this paper. Bruno Eijsvoogel thanks Koen Reijnders, John van de Wetering and Walter Van Assche for useful discussions as well. The support of an Erasmus+ travel grant and EPSRC grant "Painlevé equations: analytical properties and numerical computation", reference EP/P026532/1 is gratefully acknowledged. The work of Pablo Román was supported a FONCyT grant PICT 2014-3452 and by SeCyTUNC and the work of Bruno Eijsvoogel is supported by FWO research project G0C9819N.

Preliminaries
In this section we introduce the left and right Fourier algebras related to the sequence of monic MVOPs, following a recent work of Casper and Yakimov [9]. For this, we view the sequence P (x, n) as a function P : C × N 0 → M N (C). It is, therefore, natural to consider the space of functions P = {Q : C × N 0 → M N (C) : Q(x, n) is rational in x for fixed n}.
A differential operator of the form where F j : C → M N (C) is a rational function of x, acts on an element Q ∈ P from the right by We denote the algebra of all differential operators of the form (2.1) by M N . Now we consider a left action on P by discrete operators. For j ∈ Z, let δ j be the discrete operator which acts on a sequence A : where we take the value of a sequence at a negative integer to be equal to 0. A discrete operator where A −ℓ , . . . , A k are sequences, acts on elements of P from the left by We denote the algebra of all discrete operators of the form (2.2) by N N , and we adapt the construction given in [9, Definition 2.20] to our setting: Using these Fourier algebras, we prove the following uniqueness result: Lemma 2.2. Given D ∈ F R (P ), there exists a unique M ∈ F L (P ) such that M · P = P · D. Conversely, given M ∈ F L (P ), there exists a unique D ∈ F R (P ) such that M · P = P · D.
Proof. Let us assume that there exist M 1 , M 2 ∈ F L (P ) such that By taking the leading coefficient of (2.4) we obtain that A k (n) = 0. Proceeding recursively we conclude that A j (n) = 0 for all j = −ℓ, . . . , k. The converse is proven in a similar way.
It follows directly from the definition that the elements of F L (P ) are related to the elements of F R (P ). Lemma 2.2 shows that the map is in fact a bijection.
which implies that M 1 M 2 ∈ F L (P ). Therefore the linear space F L (P ) is a subalgebra of N N . A similar computation shows that F R (P ) is an algebra. We shall refer to F L (P ) and F R (P ) as the left and right Fourier algebras respectively. Now it follows from (2.5) that M 1 M 2 · P = P · ϕ(M 1 )ϕ(M 2 ) for all M 1 , M 2 ∈ F L (P ). On the other hand, by the definition of ϕ, we have that M 1 M 2 · P = P · ϕ(M 1 M 2 ) and, since ϕ is bijective, we conclude that ϕ is an isomorphism of algebras.
Remark 2.4. We can write the three-term recurrence relation (1.2) as The main result from [9] that we use in this paper is the existence of an adjoint operation † in the Fourier algebras F L (P ) and F R (P ), see [9, §3.1]. In order to introduce the adjoint on F L (P ), we first note that the algebra of discrete operators N N has a * -operation given by where the squared norm H(n) is viewed as a sequence. The following relation holds: On the other hand, we say that D † is the adjoint of an operator D ∈ F R (P ) if . It follows from [9, Theorem 3.7] that ϕ(M † ) = ϕ(M ) † for all M ∈ F L (P ), so that the Fourier algebras F L (P ) and F R (P ) are closed under the the adjoint operation †.
Definition 2.5. Given a pair (M, D) with M ∈ F L (P ) and D ∈ F R (P ), a relation of the form is called a ladder relation. If the operator M only contains nonpositive (nonnegative) powers of δ, we say that it is a lowering (raising) relation.
Observe that if a pair (M, D) gives a raising relation, then it follows from (2.7) and (2.6) that (M † , D † ) gives a lowering relation and viceversa.

Ladder relations for exponential weights
In this section we investigate the existence of lowering and raising relations for a class of matrix valued weights. Assume that (a, b) = (−∞, ∞) and that where k is even, A and T are constant matrices and T is invertible and self-adjoint. In the approach of [19,10], the lowering and raising operators depend on both the degree of the polynomials and the variable x. In our case, we split the dependence on the variables x and n. We start with the matrix valued differential operator and we compute the following explicit form of the adjoint: Proof. For polynomials P, Q ∈ M N (C)[x], we have from (1.1) and (3.2) that P · D, Q = P ′ , Q + P A, Q .
Integrating P ′ , Q by parts and using that W is invertible and self-adjoint, we obtain Observe that the boundary terms on the right hand side of (3.3) vanish because of the exponential decay of the matrix weight at ±∞. Since P A, Q = P, QW A * W −1 , we have Using the explicit expression for W we complete the proof of the proposition.
Remark 3.2. For a given polynomial q, we denote by (q(L)) j (n) the coefficient of the difference operator q(L) of order j in δ. In other words, we have The calculation of (q(L)) j (n) can be carried out following the scheme shown in Figure 1: (q(L)) j (n) is equal to the sum over all possible paths from P (x, n) to P (x, n + j) in deg q steps, where in each path we multiply the coefficients corresponding to each arrow. For example if q(x) = x 3 we have q(L) = L 3 , and in order to compute L 3 −1 (n) we have a total of six paths from P (x, n) to P (x, n − 1) in three steps: Figure 1. Scheme for the calculation of (q(L)) j (n).
Proposition 3.3. Let W be a matrix weight as in (3.1). Then the monic polynomials P (x, n) satisfy the lowering relation where k = deg v(x) and, using the notation (3.4), we have is a polynomial of degree n with A as its leading coefficient. Therefore Moreover for j < 0, we have In the third equality we have used that D † = −D + v ′ (x) and that, for j < 0, δ j · P · D has a lower degree than P . In the fourth equality we use the fact that v ′ (x) is a scalar function.
In order to complete the proof we note that (v ′ (L)) −j (n) = 0 for all j ≥ k. Proof. This is an immediate consequence of Definition 2.1 and Proposition 3.3.
This is a refinement for exponential weights of the general results in [25] where the authors use a Riemann-Hilbert formulation for matrix orthogonal polynomials. In particular, we give an elementary proof for the lowering relation and obtain the exact degree of the lowering operator M .
It follows from the discussion in Section 2 that there exists a unique M † ∈ F L (P ) such that Using the explicit expressions of D and D † , we find that The second equation of (3.6) and Remark 2.4 imply that we can write explicitly in terms of the difference operator coming from the three-term recurrence relation. Using the explicit formula for D in Proposition 3.3 and the definition of M † in (2.7), we verify Moreover we can use (3.5) to arrive at as well as similar formulas for higher order commutators. Note that each element in a Fourier algebra corresponds to a discrete-differential relation for the monic orthogonal polynomials. In particular, there is a distinguished subalgebra of F R (P ), namely the Lie algebra g generated by D and D † . In the following theorem we prove that g is independent of the matrix A and is isomorphic to the Lie algebra corresponding to the scalar case A = 0, N = 1 which was studied in [10, Theorem 3.1].
Theorem 3.5. The differential operators D and D † generate a Lie algebra g of dimension k + 1.
Proof. Let v (j) be the j-th derivative of v. Using that v (j) is scalar, and so it commutes with the matrix A, we first observe that [D, v (j) ] = −v (j+1) . Then for any M N (C)-valued smooth function , we obtain that the Lie algebra generated by D and D † is generated by {D, v ′ (x), . . . , v (k) }, and is, therefore, (k + 1)-dimensional.
From the previous results, we obtain nonlinear relations for the coefficients of the three-term recurrence relation. These identities can be seen as a non-Abelian analogue of the discrete string or Freud equations, see for instance [4, §4.1.1.5].
Theorem 3.6. Let W be a matrix weight with monic orthogonal polynomials P (x, n) such that and for some polynomial v(x) of degree k. Then the coefficients of the three term recurrence relation for P (x, n) satisfy the following commutation relations Proof. Consider (P · D) (x, n) = (M · P ) (x, n). In particular the coefficient of x n−1 gives (3.10) nI + X(n)A = AX(n) + A −1 (n).
Taking the difference of (3.10) for n + 1 and n we obtain which together with Proposition 3.3 gives the first desired result. For the second commutation relation we take (3.8) with n replaced by n − 1 and multiplied by C(n) from the left, and we subtract from it (3.8) with parameter n and multiplied by C(n) from the right. The result follows after cancellation of H(n)A * H(n − 1) −1 terms. Remark 3.7. Theorem 3.6 does not require the weight to be of exponential type as in (3.1). In the next section, however, we prove that for a polynomial v of degree two, the ladder relations (3.9), together with the moment of order zero, determine the weight to be of Hermite type.
Example 3.9. (Freud type weight) For a quartic even potential v(x) = x 4 + tx 2 we have the operators: The following relations hold true: The relation P · D = M · P in Proposition 3.3 is written explicitly as follows: where the coefficients are computed using that A −j (n) = (v ′ (L)) j (n) and the scheme in Figure 1: Furthermore, we use Theorem 3.6 to compute the discrete string equations − B(n)(C(n) + C(n + 1))C(n) − (C(n) + C(n + 1) + B(n) 2 + 2t)B(n)C(n).
On the other hand, we can sum the second identity from 0 to n − 1, to obtain since C(0) = 0. Also, because B(n) = X(n) − X(n + 1), in terms of the subleading coefficient of P (x, n), we obtain (3.10). If A = 0 then the weight is even and therefore X(n) = 0. In this case

Hermite-type weights
In this section we investigate further properties of Hermite-type matrix weights. First we will show that an arbitrary matrix weight W having operators D and D † as in Theorem 3.6 and with a specific moment of order zero is equivalent to a Hermite-type weight. Later we investigate a particular case of the matrix A that leads to the Harmonic oscillator algebra and a link with a quantum mechanical composite system.

4.1.
Characterization of Hermite-type weights with a ladder relation. The proof of this characterization follows the lines of the main result in [5], where the authors discuss a scalar Freud weight.
Theorem 4.1. Let W be a matrix weight, supported on R, and let (P (x, n)) n be the sequence of monic orthogonal polynomials . Let A be a matrix such that and Then W is equivalent to the Hermite-type weight Proof. In this proof, we first show that the string equations determine uniquely, up to the zeroth moment, the coefficients of the recurrence relation for the monic MVOPs with respect to W . Then the theorem follows by showing that the matrix weight (4.1) corresponds to a determinate moment problem, in the sense of [3]. We recall from Remark 3.7 that the string equations in Theorem 3.6 hold for the case of the matrix weight W . We need to write the string equations in terms of the squared norms. Using the isomorphism ϕ −1 and (3.12) we verify that 2L = M + M † − t and using (3.13) we obtain If we replace (4.2) in the first identity of (3.14), we obtain Therefore, the squared norms H(n) for n > 0 are completely determined by the choice of H(0). Furthermore, using the identities C(n) = H(n)H(n − 1) −1 and 2B(n) = A + H(n)A * H(n) −1 , we find that the coefficients of the recurrence relation and, thus, the monic orthogonal polynomials are completely determined as well.
Finally we need to prove that the moment problem for the Hermite weight W has a unique solution. By [3,Theorem 3.6], it suffices to show that the diagonal entries of the matrix valued measure W (x)dx are determinate. We follow the approach given by Freud in [20, Theorem 5.1. and 5.2]: We observe that for any β > 0. Therefore by [20,Theorem 5.2] the diagonal measures W (x) i,i dx are determinate and so is W .

4.2.
The harmonic oscillator algebra. In the special case of a Hermite-type weight of the form (3.1) with v(x) = x 2 and A = N j=2 µ j E j,j−1 , for some coefficients µ j , there exists a second order differential operator D which is symmetric with respect to W and so it has the orthogonal polynomials as eigenfunctions, see for instance [18,31]. The nilpotent matrix considered in [18] is upper triangular instead of lower triangular as in this paper so, in the rest of the section we will refer to [31]. It follows from [31,Proposition 3.5] that the monic orthogonal polynomials P (x, n) with respect to W satisfy where D and Γ(n) are given by and J is the diagonal matrix defined by J i,i = i. The expressions that we have are not identical to those in [31] because our weight W is related to theirs by a lower triangular matrix L(0): when k − ℓ is non-negative and even. Note that the matrices J and A satisfy the commutation relation [J, A] = A. By Definition 2.3, we have that D ∈ F R (P ), Γ ∈ F L (P ) and ϕ(Γ) = D. We consider the operators D and D † from (3.11) with t = 0 and normalized in the following way: Using the explicit expressions of the differential operators, we easily verify that Moreover, C ∈ F R (P ) and C is self-adjoint, i.e. C † = C. Using the explicit expressions of the operators, we obtain that C is the zeroth order operator Furthermore, if we denote M = 2 − 1 2 M and M † = 2 − 1 2 M † , using (3.13) we obtain which gives the following relation for the monic orthogonal polynomials: − A P (x, n + 1) + J + n + 1 2 I − 2C(n + 1) − B(n)A P (x, n) Observe that the Casimir operator ϕ −1 (C) of the Lie algebra ϕ −1 (h) is a second order difference operator having the sequence of monic MVOPs as eigenfunctions with a non-diagonal eigenvalue acting on P (x, n) from the right.
The fact that the operator C commutes with D, D † and D is translated via the isomorphism ϕ into the following relations Writing the last equation explicitly, we obtain two new commutation relations for the coefficients of the recurrence relation: Note that these commutation relations involve the diagonal matrix J which does not appear in the matrix valued string equations.

4.2.1.
Vector-valued eigenfunctions of D. For every λ ∈ C we define V λ to be the vector space of all vector-valued eigenfunctions of the differential operator D which are in L 2 (W ), i.e.
Note that from (4.3), the k-th row of P (x, n) is an eigenfunction of D with eigenvalue n + k and, thus, it is in V n+k . Since C commutes with D, we have that F · C ℓ ∈ V λ for all F ∈ V λ and ℓ ∈ N.
From the first commutation relation in (4.4), we obtain that in such a way that D : The operators D, D, D † and C act on the vector spaces V λ in the following way: In this sense, D and D † are lowering and raising operators respectively. If we conjugate the differential operators D, D and D † by Φ(x) = e − x 2 2 e xA , we obtain In this way the second order differential operator H splits into a scalar x-dependent part h plus a constant diagonal matrix ∆. Moreover, note that for N = 1, the H reduces to the operator h which is precisely the Hamiltonian for a spinless particle subject to a harmonic potential as in [39, Chapter 10, (1.1)] and the operators E and E † are, respectively, the operators J − and J + of [39,Chapter 10].
For β ∈ C, we consider the following vector space of eigenfunctions of H.
Observe that for every β ∈ C, the function is an isomorphism of vector spaces. The operators H, E, E † and ∆ act on the vector spaces V β in the following way: In order to find the eigenfunctions of H, we treat the operators h and ∆ separately. The scalar operator h is the well-known Schrödinger operator corresponding to the quantum harmonic oscillator. Its square-integrable eigenfunctions are the Hermite functions where H n (x) = (−1) n e x 2 d n dx n e −x 2 are the scalar Hermite polynomials. On the other hand, the left eigenvectors of the matrix ∆ are exactly the canonical vectors e k , k = 1, . . . , N . Therefore is an orthogonal basis for V λ for λ ∈ N >0 . We can however construct a different orthogonal basis using the MVOPs. The row vectors is also an orthogonal basis of V λ .

Remark 4.2.
We can interpret the differential operator H as a Hamiltonian of a quantum mechanical composite system. See [32, §11.8] for a treatment of composite quantum systems. The first part of the system would be the harmonic oscillator with h as its Hamiltonian. The second part would be a static spin s = (N − 1)/2 particle in a homogeneous magnetic field. ∆ corresponds to the Hamiltonian of the second system if we tune the magnetic field strength 1 such that the spacing of energy levels of the two separate systems, is equal. The combined system could then be seen as a particle with spin s and without electrical charge, that is constrained to move in one direction subject to a quadratic potential with an additional (tuned) magnetic field in a perpendicular direction.
Then the basis in eq. (4.5) corresponds to an orthogonal basis of separable states and the basis in eq. (4.6), to an orthogonal basis of non-separable (i.e. entangled) states.

Deformation of the weight and multi-time Toda lattice
In this section, we consider an arbitrary matrix weight of the form where v(x; t) is a polynomial of even degree with positive leading coefficient depending smoothly on a parameter t ≥ 0. In the following theorem we study the effect of differentiating the recurrence coefficients with respect to t, an idea that is natural when one considers orthogonal polynomials in the context of integrable systems.
Theorem 5.1. If we denote by . the derivative with respect to t, then the recurrence coefficients in (1.2) satisfy the following deformation equations: . .
where we use the same notation as in Remark 3.4.
Proof. Let P (x, n) be the monic orthogonal polynomials with W (x). Since P (x, n), P (x, m) = 0 for n > m, we have and then we can expand where we use the notation in Remark 3.4 for ( . v(L)) k (n). On the other hand, if we differentiate the three-term recurrence relation (1.2) with respect to t, we obtain (5.3) x .
P (x, n + 1) + B(n) . P (x, n) + C(n) . P (x, n − 1) while on the left hand side we get Combining (5.3) and (5.4), isolating the derivatives of the recurrence coefficients, we get Comparing coefficients of P (x, n) and P (x, n − 1) we obtain (5.2) for . B(n) and . C(n).
Note that for v(x) = x 2 + xt the relations (3.14) give  t j x j .
If we denote by . the derivative with respect to t j , then we have . v(L) = L j . Theorem 5.1 gives the expressions for the derivatives of the recurrence coefficients, but if j is large, then the coefficients ( . v(L)) k (n) in eq. (5.2) can be difficult to compute, and a much more convenient formulation is given as a Lax pair.
In the spirit of [29, §2.8], we identify the operator L with the block tridiagonal matrix with block entries (L nm ), L n,n+1 = I, L n,n = B(n), L n,n−1 = C(n), and L n,m = 0I if |n − m| ≥ 2. For a N × N -block semi-infinite matrix S = (S nm ), we define S + as the matrix obtained by replacing all the N × N blocks of S below the main diagonal by zero. Analogously, we let S − to be the matrix obtained by replacing all the N × N blocks above the subdiagonal by zero.
Then, we have the following result: Proof. We first observe that which proves the second equality. Using that . B(n) = ( . L) n,n and . C(n) = ( . L) n,n−1 , we will complete the proof by showing that L, (L j ) + n,n equals the right hand side of the first equation in (5.2), that L, (L j ) + n,n−1 equals the right hand side of the second equation of (5.2) and that L, (L j ) + n,m = 0 otherwise.

Appendix A. Comparison of ladder operators
In this article we have taken a different approach to ladder operators for matrix valued orthogonal polynomials than for example the one in [19]. Their approach is inspired by [10] and [30] for the scalar orthogonal polynomials. This appendix is meant to compare our approach with theirs for our class of weight functions, i.e. the exponential weights in (3.1).
The ladder relations for exponential weights are stated in Proposition 3.3: A −j (n)P (x, n − j) + [A, P (x, n)] , for monic polynomials P . For exponential weights on the real line, we have the following identity: where ρ(x) = W −1 (x)AW (x). The ladder relation given in [19] for exponential weights reads x − y P (y, n − 1) * dy (A.6) x − y P (y, n) * dy.
Comparing this last equation with (A.6), we find a relation between the two ladder operators, since F (x, n)H(n − 1) + E(x, n)H(n − 1) = P (x, n)A − AP (x, n) x − y P (y, n) * dy.