František Štampach

The characteristic function for Jacobi matrices with applications

Abstract We introduce a class of Jacobi operators with discrete spectra which is characterized by a simple convergence condition. With any operator J from this class we associate a characteristic function as an analytic function on a suitable domain, and show that its zero set actually coincides with the set of eigenvalues of J in that domain. As an application we construct several examples of Jacobi matrices for which the characteristic function can be expressed in terms of special functions. In more detail we study the example where the diagonal sequence of J is linear while the neighboring parallels to the diagonal are constant.

The Nevanlinna parametrization for q -Lommel polynomials in the indeterminate case

The Hamburger moment problem for the q -Lommel polynomials which are related to the Hahn-Exton q -Bessel function is known to be indeterminate for a certain range of parameters. In this paper, the Nevanlinna parametrization for the indeterminate case is provided in an explicit form. This makes it possible to describe all N-extremal measures of orthogonality. Moreover, a linear and a quadratic recurrence relation are derived for the moment sequence, and the asymptotic behavior of the moments for large powers is obtained with the aid of appropriate estimates.

Nevanlinna extremal measures for polynomials related to q-1-Fibonacci polynomials

The aim of this paper is the study of q−1-Fibonacci polynomials with 0 < q < 1. First, the q−1-Fibonacci polynomials are related to a q-exponential function which allows an asymptotic analysis to be worked out. Second, related basic orthogonal polynomials are investigated with the emphasis on their orthogonality properties. In particular, a compact formula for the reproducing kernel is obtained that allows to describe all the N-extremal measures of orthogonality in terms of basic hypergeometric functions and their zeros. Two special cases involving q-sine and q-cosine are discussed in more detail.

On extremal properties of Jacobian elliptic functions with complex modulus

Abstract A thorough analysis of values of the function m ↦ sn ( K ( m ) u | m ) for complex parameter m and u ∈ ( 0 , 1 ) is given. First, it is proved that the absolute value of this function never exceeds 1 if m does not belong to the region in C determined by inequalities | z − 1 | 1 and | z | > 1 . The global maximum of the function under investigation is shown to be always located in this region. More precisely, it is proved that if u ≤ 1 / 2 , then the global maximum is located at m = 1 with the value equal to 1. While if u > 1 / 2 , then the global maximum is located in the interval ( 1 , 2 ) and its value exceeds 1. In addition, more subtle extremal properties are studied numerically. Finally, applications in a Laplace-type integral and spectral analysis of some complex Jacobi matrices are presented.

The Hahn-Exton q-Bessel function as the characteristic function of a Jacobi matrix

Abstract A family T(ν), ν ∈ ℝ, of semiinfinite positive Jacobi matrices is introduced with matrix entries taken from the Hahn-Exton q-difference equation. The corresponding matrix operators defined on the linear hull of the canonical basis in ℓ2(ℤ+) are essentially self-adjoint for |ν| ≥ 1 and have deficiency indices (1, 1) for |ν| < 1. A convenient description of all self-adjoint extensions is obtained and the spectral problem is analyzed in detail. The spectrum is discrete and the characteristic equation on eigenvalues is derived explicitly in all cases. Particularly, the Hahn-Exton q-Bessel function Jν(z; q) serves as the characteristic function of the Friedrichs extension. As a direct application one can reproduce, in an alternative way, some basic results about the q-Bessel function due to Koelink and Swarttouw.

The Hurwitz-type theorem for the regular Coulomb wave function via Hankel determinants

Abstract We derive a closed formula for the determinant of the Hankel matrix whose entries are given by sums of negative powers of the zeros of the regular Coulomb wave function. This new identity applied together with results of Grommer and Chebotarev allows us to prove a Hurwitz-type theorem about the zeros of the regular Coulomb wave function. As a particular case, we obtain a new proof of the classical Hurwitzs theorem from the theory of Bessel functions that is based on algebraic arguments. In addition, several Hankel determinants with entries given by the Rayleigh function and Bernoulli numbers are also evaluated.