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Meixner matrix ensembles

We construct a family of matrix ensembles that fits Anshelevich's regression postulates for"Meixner laws on matrices". We show that the Laplace transform of a general n by n Meixner matrix ensemble satisfies a system of partial differential equations which is explicitly solvable for n=2. We rely on these solutions to identify the six types of 2 by 2 Meixner matrix ensembles.


Introduction
The classical Meixner laws are the one dimensional binomial, Poisson, negative binomial, gamma, normal, and hyperbolic Meixner laws that include hyperbolic secant law. These laws make their appearance as the orthogonality measures of a certain family of orthogonal polynomials [32], as the laws characterized by quadratic regression property [25], and as the exponential families corresponding to the quadratic variance functions [33], see also [19]. The Laha-Lukacs [23,25] characterization is the description of the possible distributions of a square-integrable random variable X such that there exist real numbers a, b, c satisfying E(X 2 |S) = aS 2 + bS + c where S = X + Y, where Y ∼ X and where Y is independent of X. Another convenient form of this condition is to postulate the existence of real numbers A, B, C such that (1.1) E((X − Y ) 2 |S) = AS 2 + BS + C (use (X − Y ) 2 = 2X 2 + 2Y 2 − S 2 to see that (A, B, C) = (4a − 1, 4b, 4c)).
In 2006 M. Anshelevich [4, pages 22 and 25] observes that (1.1) makes sense when the real random variable X is replaced by a random square matrix X. He then raises the question of defining analogues of Meixner distributions on matrices. In other words, he asks for the matrix version of Laha-Lukacs [25] result. This question on random matrices is so general that it is suitable to restrict this problem to random variables X valued in the space H n,1 of all symmetric matrices and such that such that UXU * ∼ X for matrices U in the orthogonal group O(n) = K n,1 . On the other hand, it is natural to extend this simpler framework to the space H n,2 of Hermitian complex matrices for random variables X such that UXU * ∼ X for matrices U in the unitary group U(n) = K n,2 and even to the space H n,4 of Hermitian-quaternionic matrices, again for random variables X such that UXU * ∼ X for matrices U in the symplectic group Sp(n) = K n,4 . There are also connections with free probability (see [9] and Section 6.3 below), and connections with Jordan algebras (see Section 6.4 below).
A probability law µ on H n,β (where β = 1, 2 or 4), or a random variable X with law µ, is said to be rotation invariant if the law of UXU * is also µ for all U in K n,β . In such case, following a physicists tradition we shall call µ an ensemble. Definition 1.1. We will say that the probability law µ on H n,β is a Meixner ensemble with parameters A, B, C ∈ R if µ is rotation invariant, the second moments exist, and µ has the following property: if X, Y are independent with the same law µ and S = X + Y, then This is a different concept of a Meixner matrix ensemble than the one introduced in [15] under the name "multivariate Meixner classes of invariant distributions of random matrices". In particular, we note that the oldest known ensemble, namely the Wishart distribution with mean proportional to I n is not a Meixner ensemble in the sense of Definition 1.1, since if X, Y are i.i.d. with Wishart distribution, then there exist two real numbers A, B such that which is quadratic in S but not of the desired form (1.2). For a proof of (1.3), see [27, page 582].
To complete this introduction, let us make a number of simple remarks about the Meixner ensembles. It is easy to check that if the law of X is a Meixner ensemble with parameters A, B, C, then an affine transformation X = αX + t with real α, t is also Meixner, with parameters A = A, B = αB − 4At, C = α 2 C + 4At 2 − 2Bαt.
In particular, since passing to −X changes only the sign of B, without loss of generality we will consider only B ≥ 0. Suppose now that E(X) and E(X 2 ) are given; in this case the constants A, B, C satisfy an equation with coefficients that depend on E(X), E(X 2 ). The equation is obtained from taking the expected values of both sides of (1.2). Finally, rotation invariance implies that both E(X) and E(X 2 ) are constant multiples of I n , so in the non-degenerate case E(X 2 ) = 0 we can normalize the matrices so that E( X) = 0 and E( X 2 ) = I n , and express relation (1.2) in terms of two real constants a, b as (1.4) E ( X − Y) 2 S = 2 1 + 2a In this normalized setting one can classify the Meixner ensembles into (i) the elliptic case b 2 > 4a, which in Laha-Lukacs theorem on R corresponds to negative binomial (a > 0), Poisson (a = 0), or binomial (a < 0) laws. (ii) the parabolic case b 2 = 4a, which corresponds to gamma law (a > 0) or Gaussian law (a = 0) (iii) the hyperbolic case b 2 < 4a, which corresponds to the family of hyperbolic Meixner laws that include hyperbolic secant law.
Of course, for non-degenerate ensembles (1.2) and (1.4) are equivalent. If (1.2) holds with constants A = 1, B, C, then the standardization X of X with mean µI n and variance Var(X) := E(X 2 ) − (E(X)) 2 = σ 2 I n satisfies (1.4) with What we call trivial Meixner ensembles are random multiples of the identity, X = ξI n where ξ is R-valued with one of the six Meixner laws. The aim of the paper is the study of non-trivial Meixner ensembles for all n. We will be able to find all of them for n = 2, we will give examples of Meixner ensembles for n ≥ 3 and we will write a system of n linear partial differential equations (PDEs) of the second order satisfied by the Laplace transform of a Meixner ensemble of given parameters (a, b) as defined by (1.4). Let us emphasize the fact that several different ensembles correspond to one set of parameters a, b in (1.4).
The zoo of the known Meixner ensembles includes obviously the Gaussian ensembles which are described by their cumulant transform c 1 tr θ + c 2 2 ( tr θ) 2 + c 3 2 tr (θ 2 ) with the familiar GOE, GUE and GSE (see [2] or [31]) corresponding to c 3 = 1 and c 1 = c 2 = 0 among them. The Gaussian ensembles yield a family of Meixner ensembles corresponding to a = b = 0 in (1.4), but the existence of non Gaussian ensembles for a = b = 0 and n ≥ 3 has not been proved or disproved. Another important example is a Bernoulli ensemble: if µ m is the distribution of the projection on a uniformly random m dimensional space of the Euclidean space of dimension n, a Bernoulli ensemble is any mixing of µ 0 , . . . , µ n ; for n = 1 we get back the ordinary Bernoulli distribution. The Bernoulli ensembles can be seen as the only distributions of random projections P such that UPU * ∼ P for U ∈ K n,β . The convolution of N identical Bernoulli ensembles provides the binomial ensemble, and randomizing N by Poisson or negative binomial distribution yields the Poisson and the negative binomial ensembles. The construction of the full family of Meixner ensembles with b 2 = 4a or with b 2 < 4a is available only for n = 2 and their extension to n ≥ 3 is a challenge.
The paper is organized as follows. In Section 2 we describe Laplace transforms of Meixner ensembles. In Section 3 we give examples of Meixner ensembles. In Section 4 we derive the system of PDEs that determine the Laplace transforms of the general n × n Meixner ensembles. The general solution of this system is elusive for n ≥ 3. In Section 5 we use the system of PDEs to show that under a natural integrability assumption the examples of Meixner ensembles from Section 3 exhaust all 2 × 2 Meixner ensembles. In Section 6 we collected additional material: Proposition 6.1 answers in negative a question raised in [9, Remark 5.8] for 2 × 2 random matrices with finite Laplace transform. Proposition 6.3 gives a series expansion for a Laplace transform of a random projection which can be used to derive (some) solutions to the system of PDEs. Section 6.4 makes some comments about the Jordan algebra context of the problem.

Laplace transforms
In this section we rewrite Meixner property (1.2) in terms of the Laplace transform. We will work simultaneously with symmetric, unitary and symplectic ensembles, thus with probability laws on H n,β which are invariant under the action X → UXU * where U is in the group K n,β . We call β = 1, 2, 4 the Peirce constant (see [16, page 97], where β is denoted as d and n is denoted by r). The space H n,β is a real vector space equipped with the Euclidean structure defined by the real inner product (x, y) → x|y := ℜ( tr (xy)) (the real part of the trace is needed for the symplectic case β = 4 only). For an H n,βvalued random variable X, the Laplace transform is a mapping from H n,β into (0, ∞], defined by L(θ) = E(exp( θ|X )).
(We will also use notation Θ µ when µ is the law of random variable X.) Throughout this paper we consider only random matrices X with values in H n,β with non-empty Θ X . We need a result of linear algebra that we are not going to prove. Consider the space L s (H n,β ) of symmetric endomorphisms of the Euclidean space H n,β . To each y ∈ H n,β we associate the elements y ⊗ y and P y of L s (H n,β ) defined respectively by h → (y ⊗ y)(h) = y tr (yh), h → P y (h) = yhy .
They provide important examples of elements of L s (H n,β ). Now, L s (H n,β ) is itself a linear space, and the result that we are going to admit as a black box is the following (see [14,Lemma 6.3] and [27, Proposition 3.1] for a proof): Proposition A. There exists a unique endomorphism Ψ of L s (H n,β ) such that for all y ∈ H n,β one has Ψ(y ⊗ y) = P y . Furthermore With this notation we have the simple but crucial result. (Note that this result could remove the hypothesis of invariance under rotations.) Proposition 2.1. Let µ be an ensemble on H n,β such that its Laplace transform L is finite on an open non empty set. The following are equivalent: (i) The ensemble µ is Meixner with real parameters A, B, C, i.e. (1.2) holds for X ∼ µ. (ii) With k(θ) = ln L(θ), we have Proof. For showing (i)⇒(ii) we first observe that To see this, note that since X, Y are i.i.d, the left hand side of (2.4) is The same reasoning gives Since E Se θ|S = 2L(θ)L ′ (θ), applying Ψ to (2.4) and (2.5), from (1.2) we get Since In the above proposition, the use of the endomorphism Ψ may seem surprising. For a random variable X valued in a linear space E, the link between E(X ⊗ X) and the Laplace transform L of X is easy through the second differential of L. However, if E is an algebra, like the space of square real matrices, relating E(X 2 ) to L is difficult. Here our reasoning was based on P X (A) = XAX and thus P X (I n ) = X 2 . Some other facts about Ψ are known; it is a symmetric operator on L s (H n,β ) with only two eigenspaces: the one generated by all the β 2 y ⊗ y + P y for the eigenvalue 1 and the one generated by all the β 2 y ⊗ y + P y for the eigenvalue −β/2. Details are in [29,Section 5] where the dimensions of these two subspaces are also computed.
We now show that Meixner ensembles are preserved under convolution power.
Definition 2.1. Let µ be a probability measure on a finite dimensional linear space such that its Laplace transform is finite on a non empty open set. The Jørgensen set Λ(µ) of µ is the set of α > 0 such that there exists a probability measure µ α with L µα = L α µ and Θ µα = Θ µ , see e.g. [14, page 767].
It will be convenient to have a version of Proposition 2.1 for standardized ensembles.
Conversely, if the logarithm of the Laplace transform of X satisfies (2.7) (together with the initial conditions) then (1.4) holds, and E( X) = 0, E( X 2 ) = I n .

Examples of Meixner Ensembles
In this section we give examples of Meixner ensembles. In Theorem 5.1 we will show that the examples exhaust the family of all Meixner ensembles on H 2,β with Θ µ = ∅.

3.1.
Bernoulli and binomial ensembles. The Bernoulli ensembles (which are different than the Bernoulli random matrices of independent two-valued entries) will be our basic building blocks for more complicated ensembles. Suppose P m is the orthogonal projection onto the random and uniformly distributed m-dimensional subspace, i.e. P m = UI m,n U * where I m,n is the n × n matrix with m ones on the diagonal followed by n − m zeroes, and U is uniformly distributed on the group K n,β . Definition 3.1. Denote by µ m the distribution of P m . A Bernoulli ensemble with parameters q 1 , . . . , q n ≥ 0 that add to at most one, is the law µ = q 0 δ 0 + q 1 µ 1 + · · · + q n−1 µ n−1 + q n δ In , In terms of random variables, Bernoulli ensemble can be realized by the random variable with probability q 0 = 1 − (q 1 + · · · + q n ) , P 1 with probability q 1 , . . .  Proof. We note that for any pair of projections Applying this algebraic identity to random projections X and Y, where X is given by (3.1) and Y is its independent copy, we see that (1.2) holds.
We note that for the Bernoulli ensemble formula (2.6) takes a particularly simple form: its Laplace transform L B (θ) = E(exp( θ|X )) satisfies . The corresponding version of (2.3) is ). The fact that all Bernoulli ensembles share the same A, B, C will be convenient for calculations. But it will be easier to classify the resulting Meixner laws into the familiar families from the standardized form that appears in formula (1.4). To do so, we find the moments of X. We first note that by rotation invariance, E(P k ) = cI n , so E tr (P k ) = nc. But tr (P k ) = tr (I k,n ) = k, so Therefore, withq = 1 n (q 1 + 2q 2 + · · · + nq n ), we have E(X 2 ) = E(X) =qI n . Ifq = 0, 1, the standardized matrix is X = 1 √q (1−q) (X −qI n ), and we get So (1.4) holds with parameters a = −1/4 and b = (1/2 −q)/ q(1 −q). An intrinsic characterization of Bernoulli ensembles with an elementary proof is as follows.
Proposition 3.2. If X with values in H n,β has a law invariant under rotations and X = X 2 then its law is a Bernoulli ensemble.
Proof. Since X is symmetric, it is an orthogonal projection. Denote by d(X) the dimension of its image. Invariance under rotations means that for any bounded measurable function f , we have E(f (UXU * ) − f (X)) = 0 for all U ∈ K n,β . If d = 0, 1, . . . , n and if f is zero outside of matrices of trace d, then This implies that the conditional law L(X|d(X) = d) is invariant under rotations, so it is a law of P d . Then X is a Bernoulli ensemble with parameters q m = Pr(d(X) = m).
This implies the following converse of Proposition 3.1. Proof. Let X, Y be independent random variables with law µ, and let S = X + Y. Since (X − Y) 2 = 2X 2 + 2Y 2 − S 2 , and since the law of (X, S) is the same as (Y, S) from (1.2) we get From this, we deduce that E(X) = E(X 2 ), and that which implies that E(X 3 ) = E(X). Using (3.7) again, we have By rotation invariance, the moments E(Y j ) = M j I n are given by real numbers, so E( tr (X r Y s )) = nM r M s = tr (E(X r )E(Y s )). Therefore, (3.8) implies that tr E(X 4 ) + 2 tr (EX 3 EX) + tr ((EX 2 ) 2 ) = tr E(X 3 ) + 2 tr (EX 2 ) 2 + tr (EX 2 EX).
Since we already proved that M 1 = M 2 = M 3 , we see that nM 4 = tr E(X 4 ) = tr E(X 3 ) = nM 1 . Let Λ 1 , Λ 2 , . . . Λ n be the (random) eigenvalues of X. From the equality of the first four moments of X, we see that Therefore, all eigenvalues of X are either 0 or 1, and X 2 = X. So by Proposition 3.2, the law µ of X is a Bernoulli ensemble.
The Jørgensen set of a Bernoulli ensemble is of interest due to connections with free probability which allows continuous values for the analog of parameter N. For n = 2, we will show that the Jørgensen set is N. The following sufficient condition shows that this is a "generic case". Proposition 3.4. If α ∈ Λ(µ) and µ is a Bernoulli ensemble with parameters q 1 , . . . , q n on H n,β , then (q 0 + q 1 z + · · · + q n z n ) α is a polynomial in variable z. In particular, nα is an integer.
Proof. For θ s = diag(s, s, . . . , s), the Laplace transform is L(θ s ) = n r=0 q r e rs = j (e s − z j ) m j , where z j ∈ C are the distinct roots of the polynomial q 0 + q 1 z + · · · + q n z n taken with their multiplicities m j . Since the Laplace transform must be an analytic function on its domain, we see that α must be rational, and that if α = p/q with relatively prime p, q ∈ N, then q must be a common divisor of all multiplicities m 1 , m 2 , . . . of the roots. So q divides also their sum m 1 + m 2 + · · · = n.
3.1.1. Binomial ensemble. The real, complex or quaternionic binomial ensembles are measures on H n,β which are parametrized by integer N and a discrete probability law on {0, 1, . . . , n}. We will follow the tradition of not listing the probability of 0 among the parameters.
Definition 3.2. The binomial ensemble Bin(N, q 1 , . . . , q n ) with parameters N = 1, 2, . . . and q 1 , . . . , q n , is the law of X = N j=1 X j . Proposition 3.5. A binomial ensemble with parameter N is a Meixner ensemble with Proof. This is a special case of Proposition 2.3, applied to the law µ of X 1 with parameters described in Proposition 3.1, and to α = N.
For standardization, we will need to know that
One can then check that (1.4) holds with a = 0 and b = 1 2 √λ . It is also easy to see that if the law of X is Poisson with parameters λ 1 , . . . , λ n > 0, then X = n m=1 X m is the sum of independent random variables X m from Poisson ensembles with parameters (0, . . . , 0, λ m , 0 . . . , 0). Furthermore, real random variable tr (X) has the compound Poisson law with the law of summands given as n k=0 λ k λ δ k . We remark that the Poisson model with exactly one parameter λ 1 = 0 appears explicitly in [12, page 638], as part of the construction of matrix models for all free-infinitely divisible laws; see also [7]. However, regression properties of the model were not analyzed.
3.3. Negative binomial ensemble. We now use Bernoulli ensembles to construct the negative binomial ensemble. Let N be a negative binomial random variable with parameters r > 0, 0 < p < 1, i.e.
The explicit construction is to start with auxiliary GUE/GOE/GSE matrix Z and independent real standard normal ζ.
Definition 3.5. For c 1 ∈ R, c 3 ≥ 0 and c 2 ∈ R such that nc 2 + c 3 ≥ 0, the Gaussian Meixner ensemble is the law of Proposition 3.8. The logarithm of the Laplace transform of the Gaussian Meixner ensemble is given by (3.14).
3.5. Gamma ensemble. The remaining types of Meixner ensembles will be constructed only for matrices of size n = 2. One difficulty we encounter is lack of continuity, which we now explain.
It is natural to expect that gamma ensemble arises as lim p→0 pX p of a sequence of negative binomial ensembles with varying parameter p while r and the ratios q j /(1 − p) are kept fixed. However from (3.12) we see that So this is a trivial ensemble of the form ξI n with real gamma-distributed ξ.
We now show that there are non-trivial gamma ensembles for n = 2. This construction is based on a more detailed analysis of the system of PDEs that arises from (2.7).
Definition 3.6. The Gamma ensemble on H 2,β with parameters p > β/2, c > 1 is defined by its Laplace transform Of course, this definition requires a proof that the required law on H 2,β exists. The Gamma ensemble on H 2,β is constructed by choosing the appropriate law on R β+2 , and arranging the corresponding real random variables into the random matrix. The construction is based on the laws analyzed by Letac and Weso lowski [28,Theorem 3.1]. According to this result, for p > β/2 and c > 1 there is a probability measure ν p,c (dx) on the open Proposition 3.9. Let (ξ 0 , . . . , ξ β+1 ) have joint distribution ν p,c . The gamma ensemble with parameters p, c is: where i, j, k are the standard quaternion basis.
Proof. We give the proof for β = 4. Writing The mean and the variance are E(X) = 2pc The proof relies on the system of PDEs derived in Theorem 4.1; it appears after Proposition 5.6.
Remark 3.11. Somewhat more generally, one can define gamma ensembles on H n,β as the ensembles with Laplace transform Such an ensemble is a Meixner ensemble, as it follows from Theorem 4.
Since this is only a one-parameter subset of the possible solutions, we do not pursue this construction further. .
Following [24], we work with differently normalized Bessel functions which are entire functions of z ∈ C for ν > −1.
where θ 1 , θ 2 are the eigenvalues of θ. The hyperbolic ensemble on H 2,β with parameters α, λ, ρ is defined by its Laplace transform: on the set Θ X . (When ρ = 0 we also allow λ = 1.) These ensembles are constructed again by finding an appropriate law on R β+2 and arranging the real random variables into a random matrix in H 2,β . We begin with an auxiliary fact about Laplace transforms of probability measures on product of Euclidean spaces.
Proposition 3.12. Let E and F two Euclidean spaces, and S ⊂ E, T ⊂ F be open non-empty sets. If f : S → R g : T → R are such that are Laplace transforms of probability measures for some α > 0 and also for α = 1, then for 0 < λ < 1, and (s, t) ∈ S × T , function Proof. We denote by γ α the gamma distribution on (0, ∞) with shape parameter 1 and scale parameter α > 0. Fix 0 < λ < 1, and let µ α and ν α be two probabilities on E and F such that the integrals are finite on the maximal open sets S ⊂ E and T ⊂ F respectively. (Denote µ = µ 1 and ν = ν 1 .) Consider the set of independent variables U, A, B, X 1 , are two standard Poisson processes. Define the random variables X and Y on E and F respectively by We have .
This shows that the distribution H α (dx, dy) of (X, Y ) satisfies on S × T We now apply this result to the following Laplace transform, with the notation of Definition 3.7.

The system of PDEs
In this section, we derive a system of PDEs for the Laplace transform of general Meixner ensembles. As previously, we consider simultaneously the real, complex, and quaternionic cases.
We introduce some notation. For θ ∈ H n,β and j = 0, 1, . . . , we consider σ j (θ) defined by Recall that for β = 4, the determinant if quaternionic matrices are defined only for hermitian matrices, see [16, page 29] and [13]. Note that σ 0 = 1, σ 1 (θ) = tr θ, σ 2 (θ) = 1 2 ( tr θ) 2 − 1 2 tr (θ 2 ), σ n (θ) = det θ, and that σ j = 0 for j > n. We also adopt the convention that σ j = 0 for j < 0. In general, σ j (θ) = e j (θ 1 , θ 2 , . . . ) is the j-th elementary symmetric function of the eigenvalues θ 1 , θ 2 , . . . of θ ∈ H n,β ; this notation and most of our calculations do not depend on the dimension n. Denote by U the open subset of R n such that if (σ 1 , . . . , σ n ) ∈ U then the polynomial in x defined by n i=0 (−1) n−i σ i x n−i has only distinct real roots. The main result of this section is the system of PDEs that determines the Laplace transform of a Meixner ensemble on a nonempty open set Θ X ⊂ H n,β . This system is written in terms of an auxiliary function g(σ 1 , . . . , σ n ) which is defined on an appropriate non-empty open set U X ⊂ U. The link between L and this auxiliary function g varies according to the fact that a or b are zero or not. To define the set U X , we need Θ X to be closed under conjugation (θ ∈ Θ X implies UθU * ∈ Θ X for all U ∈ K nβ ), and to have 0 in its closure. Consider the n-dimensional real subspace D n ⊂ H n,β consisting of diagonal matrices, and let D 0 n ⊂ D n denote the set of matrices with distinct eigenvalues. Since Θ X is invariant under rotations, Θ X ∩ D n is a nonempty open set with 0 in its closure. The set Θ X ∩ D 0 n is obtained from Θ X ∩ D n by removing a finite number of hyperplanes, so it is also non-empty, and has 0 in its closure.
Denote by σ : H n,β → R n the mapping θ → (σ 1 (θ), . . . , σ n (θ)). If θ ∈ D 0 n then by the Vandermonde's determinant, B θ = {I n , θ, θ 2 , . . . , θ n−1 } is a basis of D n . From formula (4.10) below, we see that in that basis, the matrix representation of the derivative of σ restricted to D n is triangular at θ ∈ D 0 n , with σ 0 = 1 on the diagonal, Therefore, the Jacobian of σ is 1 and σ : D 0 n → R n is an open mapping. Thus, the set is open, non-empty, and since σ(0) = 0, it has 0 ∈ R n in its closure.
To shorten the formulas, for i, j = 1, 2, . . . , n we write Recall the convention that σ j = 0 for j < 0 or j > n.
The left hand side D(g) of our systems of the PDE's resembles the system of PDE's for the Bessel function of matrix argument given in [21,Eqtn (18)] when β = 1, see also [34, page 275 Eqtn (31)]. However, as we will see in Proposition 5.6, in contrast to [21] our system has multiple analytic solutions.
We use also without proof the Newton formula we remark that under the convention that σ m = 0 for m > n, formula (4.8) holds true for all m ≥ 1, i.e. also for m > n. Since tr I n = n, it can be rewritten as We also need a differentiation formula.
The limits as θ → 0 in Θ X are: The remaining part of the proof consists of finding a suitable expression for f . Now the fact that the law of X is invariant under the rotations is equivalent to k being invariant under the substitution θ → UθU * for all U ∈ K n,β , and is equivalent to saying that in each of the three cases mentioned above, there exists a real analytic function g on U X such that f (θ) = g(σ 1 (θ), . . . , σ n (θ)) for all θ ∈ Θ X with distinct eigenvalues. (This is Weyl's formula, see [31, Lemma 2.6.1].) Using (4.10) we get The second derivative of f is trickier:

The calculation of Ψ(A)(I n ) is quick and gives
For calculating Ψ(B)(I n ) we prove the surprising formula:  Having in mind the calculation of Ψ(D m )(I n ) we write Thus we can compute We apply this to s = m − 1 − j and compute Ψ(D m )(I n ) : Note that we replaced m − 1 by m − 2 in the first sum of the right hand side. We obtain Since σ ′ 1 (θ) = I n , the three equations (4.12),(4.14) and (4.16) respectively become Noting that σ ′ 1 , σ ′ 2 , . . . , σ ′ n is a basis of the algebra generated by θ with distinct eigenvalues, the corresponding systems of PDEs arise by comparing the coefficients at σ ′ j in (4.27-4.29), respectively. To end the proof, we only need to find the coefficients P j (m, i) in the basis expansion This is accomplished by the following formula.
It is also clear that these calculations can be reversed. From the equations in listed in (i)-(iii), we see that (2.7) holds for all θ ∈ Θ X with distinct eigenvalues. By uniqueness of analytic extension, (2.7) extends to the whole open set Θ X . Remark 4.6. Particular Meixner ensemble can be found from the system of PDEs by restricting g to be a function of σ 1 , σ 2 only. In other worlds, we look for an ensemble such that the Laplace transform L(θ) is a function of tr θ, tr θ 2 . Such strategy leads us to the gamma ensemble on H n,β mentioned in Remark 3.11 and to Definition 3.5 of the Gaussian ensemble. This strategy leads only to trivial ensembles for b 2 = 4a.

Meixner ensembles on 2 × 2 matrices
The goal of this section is to use Theorem 4.1 to prove that the constructions from Section 3 exhaust all Meixner ensembles on H 2 , β. (The seventh case of non-random ensembles arises from degenerate affine transformations.) 5.1. Laplace transforms of binomial, negative binomial and Poisson ensembles. Lemma 5.2. If P is a random projection in Bin(1, 1, 0), then with σ j = σ j (θ), we have where I ν is a Bessel function (3.24).

Corollary 5.4.
(i) If X has Bin(N, q 1 , q 2 ) law, then its Laplace transform is well defined for all θ ∈ H β , and is given by If X has Poiss(λ 1 , λ 2 ) law, then its Laplace transform is well defined for all θ ∈ H β , and is given by where λ = λ 1 + λ 2 . (iii) If X has NB(r, q 1 , q 2 ) law, then its Laplace transform is well defined for θ in a neighborhood of 0 in H 2,β , and is given by Remark 5.5. A calculation shows that for a given b 2 > 4a > 0, parameter p = 1 − q 1 − q 2 in (5.5) can range over the set 2. Solutions of the system of PDEs for n = 2. We consider in more detail the case n = 2 with (5.6) U = {(σ 1 , σ 2 ) : 4σ 2 < σ 2 1 }. We require that the solutions extend by continuity to σ 1 = σ 2 = 0.
(i) For b 2 = 4a > 0 all solutions are where C is an arbitrary constant. Accordingly, (3.24), and C 1 , C 2 , C 3 are arbitrary real numbers such that C 1 + C 2 + C 3 = 1 and C 2 − C 1 = b/κ. Accordingly, , which can also be written as (5.13) where λ ∈ R is an arbitrary constant.
where λ is an arbitrary real constant. Accordingly, .
To conclude the proof of (i), we note that the general solution that is defined at The initial conditions determine C 1 = 1, C 2 = −b, which gives (5.10).
To conclude the proof of (ii), we note that the general solution bounded at the origin is (5.11), and the initial conditions are satisfied when C 1 + C 2 + C 3 = 1 and C 2 − C 1 = b/κ. This gives (5.12).
We are now ready to prove that gamma and hyperbolic Meixner ensembles are indeed Meixner.

5.2.1.
Poisson case: a = 0, b = 0. The system of PDEs simplifies to: We seek solutions such that Proposition 5.7 (Poisson ensemble). Consider the system (5.23), (5.24) with initial condition (5.25). The solution is where C is an arbitrary constant. Thus (5.26) Proof. The solution that depends on σ 1 only is in [25]; in our notation, To find the solutions that depend on both variables, we first note that (5.24) implies which we differentiate with respect to σ 1 to get (5.28) 2g 11 + g 2 + σ 1 g 12 − 2ag 1 = C ′ (σ 1 ).

5.2.2.
Gaussian case: a = 0, b = 0. The system of PDEs in Theorem 4.1(ii) simplifies to: Proposition 5.8. Consider the system (5.30), (5.31) with zero initial conditions. The solution is where C ∈ R is arbitrary. Thus Since the homogeneous system is the same as in the case b 2 − 4a = 0 of Proposition 5.6(i), we omit the details.
We recall the following facts about univariate Laplace transforms, which belong to the folklore: Proposition B. Suppose L(t) is a Laplace transform of a real random variable X. If L(t) is defined on an open interval Θ, then L has analytic extension to Θ + iR. If z 0 ∈ Θ + iR is a zero of L of order m ∈ N, then L α cannot be a Laplace transform of a probability measure unless mα ∈ N.
Proof of Theorem 5.1. Without loss of generality, we may assume that X is non-degenerate, and therefore that it is standardized with mean 0 and variance I n . Therefore, (1.4) holds with some constants a, b, as explained in the introduction, replacing X by −X if needed, without loss of generality we may assume b ≥ 0. Therefore the Laplace transform of X must be given by one of the six formulas from Proposition 5.6(i-iii), Proposition 5.7 or Proposition 5.8. It remains therefore to show that these functions are the Laplace transforms of probability measures only when the parameters satisfy the conditions listed in the corresponding definitions of our Meixner ensembles. The solutions, and our definitions, depend on the constraints satisfied by parameters a, b, so we need to consider each case separately.
In the proof, we repeatedly compute L(θ) on diagonal matrices (i) Case b 2 > 4a with a < 0. In this case, all potential Laplace transforms for the standardized ensembles are given by formula (5.12), and (5.3) shows that L(θ) is indeed the Laplace transform if a = −1/(4N) and C 1 , C 2 , C 3 ≥ 0. It remains to show that L(θ) is not a Laplace transform if 1/(4a) ∈ N or if one of the constants is negative. Denote α = −1/(4a) > 0. If C 3 = 0, then L( 1 κ θ + t ) = (C 1 e t + C 2 e −t ) α , and it is a classical fact that α ∈ N and C 1 , C 2 are the binomial probabilities, compare Proposition 3.4. Therefore, through the remainder of the proof we assume C 3 > 0. We first show that α ∈ N; the same reasoning shows that the Jørgensen set of a Bernoulli ensemble on To show that α ∈ N, we will apply Proposition B. We first observe that (3.24) implies where a n = 1 2 2n n!Γ(n + ν + 1) .
This shows that I ν ( √ z) is an entire function of order ρ = lim sup n→∞ log n 1 n log a n = 1/2.
Coming back to f (s), we see that the function s → f (s) has an infinite set Z 0 of roots. We claim that Z 0 has at least one simple root. If not, Z 0 would be included in the set Z 1 of roots of the derivative f ′ (s). Now from (5.35) we see that Z 1 is the set of roots of I (β+1)/2 which by Lommel's Theorem, lies on the imaginary axis ( [38, 15.25]). From [38, page 199] we see that lim t→±∞ I ν (it) = 0, therefore I ν (it) cannot be equal to (C 3 −1)/C 3 on an infinite subset of Z 1 . Therefore f has at least one simple zero, and from Proposition B we deduce that α must be an integer.
The final step is to show that C 1 , C 2 , C 3 ≥ 0. Since the second derivative of log L( 1 κ θ − t ) = α log(1 − C 3 + C 3 I (β−1)/2 (t)) is αC 3 /(β + 1), we see that C 3 ≥ 0. Next we compute the second derivative of 1 α log L( 1 κ θ + t ), which is Since α ∈ N, the Laplace transform L( 1 κ θ + t ) is well defined for all real t. The sign of the second derivative for large t is determined by C 1 C 3 , showing that C 1 ≥ 0. The sign of the second derivative as t → −∞ is determined by To verify that L(θ) fails to be a Laplace transform of an ensemble in all other cases, we compute Thus we must have C ≥ 1/2. Next, we note that in order for to be positive for large t, we must have C ≤ 1. This shows that Definition 3.3 covers all examples of Meixner ensembles on H 2,β with b = 0, a = 0 (iii) Case b 2 > 4a, a > 0. In this case, all potential Laplace transforms for the standardized ensembles are given by formula (5.13). To see which values of λ correspond to the Laplace transform (5.5) of the negative binomial ensemble, we take r = 1/(4a) and define parameters q 1 , q 2 and p = 1 − q 1 − q 2 as follows From the admissible range of p given in Remark 5.5 we see that (5.13) is a Laplace transform of an ensemble when 1 ≤ 1 − λ ≤ b/κ. It remains to show that no other values of λ are allowed. The second derivative of 4a log Next suppose 1 − λ > b/κ. Then, with ρ = b/(κ(1 − λ)) < 1 we see that is an increasing function of t > 0. So L(θ + t /κ) is well defined for all t > 0. We now compute the second derivative of (5.36) 4a log L(θ + t /κ) = − log(λ + (1 − λ) cosh(t) − b/κ sinh t). We get Since κ(λ − 1) + b < 0, this expression fails to be positive for large t, contradicting that L is a Laplace transform of a probability measure. Thus we must have (v) Case a > 0, b 2 = 4a (Proposition 5.6(i)). From [28] it follows that Definition 3. 6 covers all examples of Meixner ensembles on H 2,β with a > 0, b 2 = 4a. (vi) Case b 2 < 4a (Proposition 5.6(iii)). From Proposition 3.14 we see that (5.14) is a Laplace transform of a probability measure when 0 ≤ λ < 1, and also when λ = 1, b = 0. It remains to show that (3.25) fails to be a Laplace transform in the remaining cases.
So Definition 3.7 exhausts all the Meixner ensembles on H 2,β with b 2 < 4a.
6. Additional observations 6.1. Independence of S and S −1 X 2 S −1 . Bożejko and Bryc [9, Remark 5.8] raise the question whether there exists a non-trivial law on positive n × n matrices such that (6.1) Z := S −1 X 2 S −1 , S := X + Y are independent when X and Y are i. i. d. matrices with this law. Such laws could provide matrix models for the "free gamma" law.
Here we answer this question in negative, at least for the laws on positive 2 × 2 matrices which are invariant under orthogonal/unitary/symplectic group. We show that in the case of 2 × 2 positive random matrices, S −1 X 2 S −1 and S are independent only if X arises as a gamma random variable multiplied by I 2 .
Proof of Proposition 6.1. Denote Z = S −1 X 2 S −1 . Note that this is a positive-define matrix. First we note that for positive matrices the Laplace transform is defined on θ ∈ H when −θ is positive.
Since det(X)/ det(S) = √ det Z is independent of det S, by Lemma 6.2, there are only two non-degenerate choices for the law of X. It is known that (1.4) does not hold for the Wishart law, see [27, pg. 582], which excludes the law from Lemma 6.2(iii). Thus X is the identity matrix multiplied by a Gamma random variable.
Since P 2 = I 3 − P 1 for n = 3, we have the following.
It is convenient to set J κ (θ; β) = 0 if the partition κ has more than n parts. (The usual notation for J κ (x; β) is C κ (x; α) with α = 2/β.) We need several properties of Jack polynomials.
If d = β + 2 with β = 1, 2, 4 this algebra is H 2,β . For a Jordan algebra V, rotational invariance in H 2,β is generalized to the invariance by a certain compact group K acting on V , as described in Faraut Koranyi [16, page 6 and 55]. The Anshelevich problem can be solved for the Lorentz algebra for any d ≥ 1 since the rank is 2, and the preceding study for H 2,β extends easily to this case. Since the consideration of the Lorentz and Albert algebras can be seen as rather academic, we have refrained to place our paper in this framework. 6.2. The dimension of the set of solutions of the PDE systems from Theorem 4.1 is equal 4 if n = 2 and is unknown if n ≥ 3. The subset of solutions that are analytic at zero has dimension 3 for n = 2, and is unknown for n ≥ 3. A related system of PDEs in [21], has a one-dimensional set of solutions analytic at 0.

6.3.
Since the free binomial law is well defined for all real values of N ≥ 1, it would be interesting to determine the Jørgensen set of the Bernoulli ensembles for n > 2. Similarly, the converses to Propositions 3.6 and 3.7 are not known beyond dimension n = 2 which is known from Theorem 5.1.
6.5. For n ≥ 3, do we have non-Gaussian ensembles on H n,β such that if X, Y are i.i.d. then E((X − Y) 2 |X + Y) = 2I n ?
6.6. The distribution of eigenvalues as the dimension n → ∞ was studied in [12] for a special case of Poisson ensembles. It is interesting to study this limit for other Meixner ensembles.