The paper deals with two types of inverse spectral problems for the class of generalized Jacobi matrices introduced in [9]. Following the scheme proposed in [5], we deduce analogs of the Hochstadt–Lieberman theorem and the Borg theorem. Properties of a Weyl function of the generalized Jacobi matrix are systematically used to prove the uniqueness theorems. Trace formulas for the generalized Jaco...

For little q-Jacobi polynomials, q-Hahn polynomials and big q-Jacobi polynomials we give particular q-hypergeometric series representations in which the termwise q = 0 limit can be taken. When rewritten in matrix form, these series representations can be viewed as decompositions into a lower triangular matrix times upper triangular matrix. We develop a general theory of such decompositions rela...

For little q-Jacobi polynomials, q-Hahn polynomials and big q-Jacobi polynomials we give particular q-hypergeometric series representations in which the termwise q = 0 limit can be taken. When rewritten in matrix form, these series representations can be viewed as decompositions into a lower triangular matrix times upper triangular matrix. We develop a general theory of such decompositions rela...

In this article we identify several beautiful properties of Jacobi sums that become evident when these numbers are organized as a matrix and studied via the tools of linear algebra. In the process we reconsider a convention employed in computing Jacobi sum values by illustrating how these properties become less elegant or disappear entirely when the standard definition for Jacobi sums is utiliz...

We propose two ways for determining the Green’s matrix for problems admitting Hamiltonians that have infinite symmetric tridiagonal (i.e. Jacobi) matrix form on some basis representation. In addition to the recurrence relation comming from the Jacobi-matrix, the first approach also requires the matrix elements of the Green’s operator between the first elements of the basis. In the second approa...

Let L be a quasi-definite linear functional defined on the linear space of polynomials with real coefficients. In the literature, three canonical transformations of this functional are studied: xL, L + Cδ(x) and x L + Cδ(x) where δ(x) denotes the linear functional (δ(x))(xk) = δk,0, and δk,0 is the Kronecker symbol. Let us consider the sequence of monic polynomials orthogonal with respect to L....

The computation of a singular value decomposition of an m× n matrix A is certainly one of the most often demanded tasks in various applications. There are many algorithms for computing the full or partial singular value decomposition. Among them, the one-sided Jacobi method (coupled with some orderings) is reputable for its ability to compute the singular values as well as left and right singul...

We present explicit formulas for the distributions of the extreme eigenvalues of the β–Jacobi random matrix ensemble in terms of the hypergeometric function of a matrix argument. For β = 1, 2, 4, these formulas specialize to the well-known real, complex, and quaternion Jacobi ensembles, respectively.