Let e ⊂ R be a finite union of l + 1 disjoint closed intervals and denote by ωj the harmonic measure of the j leftmost bands. The frequency module for e is the set of all integral combinations of ω1, . . . , ωl. Let {ãn, b̃n}n=1 be a point in the isospectral torus for e and p̃n its orthogonal polynomials. Let {an, bn}n=1 be a half-line Jacobi matrix with an = ãn+δan, bn = b̃n+δbn. Suppose

Starting from Jacobi-Trudi’s type determinental expressions for the Schur functions corresponding to types B,C and D, we define a natural q-analogue of the multiplicity [V (λ) : M(μ)] when M(μ) is a tensor product of row or column shaped modules defined by μ. We prove that these q-multiplicities are equal to certain Kostka-Foulkes polynomials related to the root systems C or D. Finally we deriv...

Let e ⊂ R be a finite union of + 1 disjoint closed intervals, and denote by ωj the harmonic measure of the j left-most bands. The frequency module for e is the set of all integral combinations of ω1, . . . ,ω . Let {ãn, b̃n}∞n=−∞ be a point in the isospectral torus for e and p̃n its orthogonal polynomials. Let {an, bn}∞n=1 be a half-line Jacobi matrix with an = ãn + δan, bn = b̃n + δbn. Suppose ∞ ...

We exhibit 3 families of complete curvature homogeneous pseudo-Riemannian manifolds which are modeled on irreducible symmetric spaces and which are not locally homogeneous. All of the manifolds have nilpotent Jacobi operators; some of the manifolds are, in addition, Jordan Osserman and Jordan Ivanov-Petrova.

Using spectral theory of unitary operators and the theory of orthogonal polynomials on the unit circle, we propose a simple matrix model for the following circular analogue of the Jacobi ensemble:

We describe the asymptotic behavior of the multivariate BC–type Jacobi polynomials as the number of variables and the Young diagram indexing the polynomial go to infinity. In particular, our results describe the approximation of the spherical functions of the infinite-dimensional symmetric spaces of type B, C, D or BC by the spherical functions of the corresponding finite–dimensional symmetric ...

A refinement of a conjecture of Gasper concerning the values of (α, β), −1/2 < β < 0, −1 < α + β < 0, for which the inequalities

The multivariate orthogonal polynomials are related to a family of operators whose matrix representations are block Jacobi matrices. A sufficient condition is given so that these operators, in general unbounded, are commuting and selfadjoint. The spectral theorem for these operators is used to establish the existence of the measure of orthogonality in Favard's theorem.

The operator Lμ : f 7→ ∫ f(x)−f(y) x−y dμ(y) is, for a compactly supported measure μ with an L density, a closed, densely defined operator on L(μ). We show that the operator Q = pLμ−qLμ has polynomial eigenfunctions if and only if μ is a free Meixner distribution. The only time Q has orthogonal polynomial eigenfunctions is if μ is a semicircular distribution. More generally, the only time the o...

Given {Pn}n≥0 a sequence of monic orthogonal polynomials, we analyze their linear combinations with constant coefficients and fixed length, i.e., Qn(x) = Pn(x) + a1Pn−1(x) + · · ·+ akPn−k, ak 6= 0, n > k. Necessary and sufficient conditions are given for the orthogonality of the sequence {Qn}n≥0. An interesting interpretation in terms of the Jacobi matrices associated with {Pn}n≥0 and {Qn}n≥0 i...