Two-weight Hilbert transform and Lipschitz property of Jacobi matrices associated to hyperbolic polynomials

Two modified Jacobi methods for M-matrices

On a Spectral Property of Jacobi Matrices

Let J be a Jacobi matrix with elements bk on the main diagonal and elements ak on the auxiliary ones. We suppose that J is a compact perturbation of the free Jacobi matrix. In this case the essential spectrum of J coincides with [−2, 2], and its discrete spectrum is a union of two sequences {xj }, x + j > 2, x − j < −2, tending to ±2. We denote sequences {ak+1 − ak} and {ak+1 + ak−1 − 2ak} by ∂...

Quantum Hilbert matrices and orthogonal polynomials

Using the notion of quantum integers associated with a complex number q 6= 0, we define the quantum Hilbert matrix and various extensions. They are Hankel matrices corresponding to certain little q-Jacobi polynomials when |q| < 1, and for the special value q = (1 − √ 5)/(1 + √ 5) they are closely related to Hankel matrices of reciprocal Fibonacci numbers called Filbert matrices. We find a formu...

Quantum Hilbert matrices and orthogonal polynomials

Using the notion of quantum integers associated with a complex number q 6= 0, we define the quantum Hilbert matrix and various extensions. They are Hankel matrices corresponding to certain little q-Jacobi polynomials when |q| < 1, and for the special value q = (1 − √ 5)/(1 + √ 5) they are closely related to Hankel matrices of reciprocal Fibonacci numbers called Filbert matrices. We find a formu...

Strong asymptotics of the recurrence coefficients of orthogonal polynomials associated to the generalized Jacobi weight

We study asymptotics of the recurrence coefficients of orthogonal polynomials associated to the generalized Jacobi weight, which is a weight function with a finite number of algebraic singularities on [−1, 1]. The recurrence coefficients can be written in terms of the solution of the corresponding Riemann-Hilbert problem for orthogonal polynomials. Using the steepest descent method of Deift and...