arXiv subjects

Pavel Šťovíček

Publications and source records attributed to Pavel Šťovíček.

8 recordsLinked to original sources

arXiv2026

Matrices with cyclically monotone rows and Cantor numeration systems

We study a class of square matrices with non-negative elements which have cyclically monotone rows in the sense that each row of a matrix from the class consists of a cyclically non-increasing sequence of numbers starting from a maximal element on the diagonal. We prove that if every diagonal element is strictly larger than all other elements in the respective row, then the matrix is regular. This property enables us to solve an open problem that comes from the theory of non-standard numeration systems, also called Cantor numeration systems. The problem concerns a one-to-one relationship between Cantor real bases, which are supposed to be alternate, that is, periodic with a period p, and lists of p sequences of non-negative integers satisfying the so-called Parry condition.

math.NT↗

arXiv2026

A formula for the local Heun solution

The local Heun solution is the unique solution to Heun's equation which is analytic in the unit disk centered at $0\in\mathbb{C}$ and taking the value $1$ at the center of the disk. In this paper, as an application of the theory of orthogonal polynomials, we are able to express the coefficients in the corresponding power series as finite multiple sums. In addition, the obtained formula can be used to derive an explicit estimate on the coefficients giving a hint on their asymptotic behavior for large indices.

math.CV↗

arXiv2025

On positive Jacobi matrices with compact inverses

We consider positive Jacobi matrices $J$ with compact inverses and consequently with purely discrete spectra. A number of properties of the corresponding sequence of orthogonal polynomials is studied including the convergence of their zeros, the vague convergence of the zero counting measures and of the Christoffel--Darboux kernels on the diagonal. Particularly, if the inverse of $J$ belongs to some Schatten class, we identify the asymptotic behaviour of the sequence of orthogonal polynomials and express it in terms of its regularized characteristic function. In the even more special case when the inverse of $J$ belongs to the trace class we derive various formulas for the orthogonality measure, eigenvectors of $J$ as well as for the functions of the second kind and related objects. These general results are given a more explicit form in the case when $-J$ is a generator of a Birth--Death process. Among others we provide a formula for the trace of the inverse of $J$. We illustrate our results by introducing and studying a modification of $q$-Laguerre polynomials corresponding to a determinate moment problem.

math.SP↗

arXiv2024

Coulomb Green's function and an addition formula for the Whittaker functions

A series of the form $\sum_{\ell=0}^{\infty}c(κ,\ell)\,M_{κ,\ell+1/2}(r_{0})W_{κ,\ell+1/2}(r)P_{\ell}(\cos(γ))$ is evaluated explicitly where $c(κ,\ell)$ are suitable complex coefficients, $M_{κ,μ}$ and $W_{κ,μ}$ are the Whittaker functions, $P_{\ell}$ are the Legendre polynomials, $r_{0}<r$ are radial variables, $γ$ is an angle and $κ$ is a complex parameter. The sum depends, as far as the radial variables and the angle are concerned, on their combinations $r+r_{0}$ and $(r^{2}+r_{0}^{\,2}-rr_{0}\cos(γ))^{1/2}$. This addition formula generalizes in some respect Gegenbauer's Addition Theorem and follows rather straightforwardly from some already known results, particularly from Hostler's formula for Coulomb Green's function. In addition, several complementary summation formulas are derived. They suggest that a further extension of this addition formula may be possible.

math-ph↗

arXiv2021

On diagonalizable quantum weighted Hankel matrices

A semi-infinite weighted Hankel matrix with entries defined in terms of basic hypergeometric series is explicitly diagonalized as an operator on $\ell^{2}(\mathbb{N}_{0})$. The approach uses the fact that the operator commutes with a diagonalizable Jacobi operator corresponding to Al-Salam-Chihara orthogonal polynomials. Yet another weighted Hankel matrix, which commutes with a Jacobi operator associated with the continuous $q$-Laguerre polynomials, is diagonalized. As an application, several new integral formulas for selected quantum orthogonal polynomials are deduced. In addition, an open research problem concerning a quantum Hilbert matrix is also mentioned.

math.CA↗

arXiv2019

New explicitly diagonalizable Hankel matrices related to the Stieltjes-Carlitz polynomials

Four new examples of explicitly diagonalizable Hankel matrices depending on a parameter $k\in(0,1)$ are presented. The Hankel matrices are regarded as matrix operators on the Hilbert space $\ell^{2}(\mathbb{N}_{0})$ and the solution of the spectral problem is based on an application of the commutator method. Each of the Hankel matrices commutes with a Jacobi matrix which is related to a particular family of the Stieltjes-Carlitz polynomials. More examples of explicitly diagonalizable structured matrix operators are obtained when taking into account also weighted Hankel matrices.

math.SP↗

arXiv2019

On Hankel matrices commuting with Jacobi matrices from the Askey scheme

A complete characterization is provided of Hankel matrices commuting with Jacobi matrices which correspond to hypergeometric orthogonal polynomials from the Askey scheme. It follows, as the main result of the paper, that the generalized Hilbert matrix is the only prominent infinite-rank Hankel matrix which, if regarded as an operator on $\ell^{2}(\mathbb{N}_{0})$, is diagonalizable by application of the commutator method with Jacobi matrices from the mentioned families.

math.CA↗

arXiv2018

Spectral representation of some weighted Hankel matrices and orthogonal polynomials from the Askey scheme

We provide an explicit spectral representation for several weighted Hankel matrices by means of the so called commutator method. These weighted Hankel matrices are found in the commutant of Jacobi matrices associated with orthogonal polynomials from the Askey scheme whose Jacobi parameters are polynomial functions of the index. We also present two more results of general interest. First, we give a complete description of the commutant of a Jacobi matrix. Second, we deduce a necessary and sufficient condition for a weighted Hankel matrix commuting with a Jacobi matrix to determine a unique self-adjoint operator on $\ell^{2}(\mathbb{N}_{0})$.

math.SP↗