Eigenvalue Theory for Beta-wishart Random Matirces

We introduce the class of Beta-Wishart random matrices and develop a comprehensive eigenvalue theory for it. The Beta-Wishart matrices generalize the classical real, complex, and quaternion Wishart matrices to a unform and continuous model valid for any β > 0. As most distributions are expressed in terms of the hypergeometric function of a matrix argument, many new identities for this function are established. Those may be of independent interest. 1. Definitions 1.1. Partitions and hook lengths. For an integer k ≥ 0 we say that κ = (κ1, κ2, . . .) is a partition of k (denoted κ ` k) if κ1 ≥ κ2 ≥ · · · ≥ 0 are integers such that κ1 + κ2 + · · · = k. The quantity |κ| = k is also called the size κ. We introduce a partial ordering of partitions and say that μ ⊆ λ if μi ≤ λi for all i = 1, 2, . . .. Then λ/μ is called a skew shape and consists of those boxes in the Young diagram of λ that do not belong to μ. Clearly, |λ/μ| = |λ| − |μ|. The skew shape κ/μ is a horizontal strip when κ1 ≥ μ1 ≥ κ2 ≥ μ2 ≥ · · · [25, p. 339]. The upper and lower hook lengths at a point (i, j) in a partition κ (i.e., i ≤ κj , j ≤ κi) are hκ(i, j) ≡ κj − i+ α(κi − j + 1); h∗(i, j) ≡ κj − i+ 1 + α(κi − j). The products of the upper and lower hook lengths are denoted, respectively, as H∗ κ = ∏ (i,j)∈κ hκ(i, j) and H κ ∗ = ∏ (i,j)∈κ h∗(i, j) and the product of the two plays an important role in what follows, thus we define (1.1) jκ ≡ H∗ κH ∗ . 1.2. Pochhammer symbol and multivariate Gamma function. For a partition κ = (κ1, κ2, . . . , κn) and α > 0, the Generalized Pochhammer symbol is defined as (1.2) (a) κ ≡ ∏ (i,j)∈κ ( a− i− 1 α + j − 1 ) = n ∏