Related papers: On the Markov sequence problem for Jacobi polynomi…

We identify the Atkin polynomials in terms of associated Jacobi polynomials. Our identificationthen takes advantage of the theory of orthogonal polynomials and their asymptotics to establish many new properties of the Atkin polynomials.…

We develop direct and inverse scattering theory for Jacobi operators with steplike coefficients which are asymptotically close to different finite-gap quasi-periodic coefficients on different sides. We give a complete characterization of…

We recall criteria on the spectrum of Jacobi matrices such that the corresponding isospectral torus consists of periodic operators. Motivated by those known results for Jacobi matrices, we define a new class of operators called GMP…

We characterize the atomic probability measure on $\mathbb{R}^d$ which having a finite number of atoms. We further prove that the Jacobi sequences associated to the multiple Hermite (resp. Laguerre, resp. Jacobi) orthogonal polynomials are…

Jacobi polynomials are mapped onto the continuous Hahn polynomials by the Fourier transform and the orthogonality relations for the continuous Hahn polynomials then follow from the orthogonality relations for the Jacobi polynomials and the…

We present an unified framework to identify spectra of Jacobi matrices. We give applications to long-standing conjecture of Chihara concerning one-quarter class of orthogonal polynomials, to the conjecture posed by Roehner and Valent…

We present certain results on the direct and inverse spectral theory of the Jacobi operator with complex periodic coefficients. For instance, we show that any $N$-th degree polynomial whose leading coefficient is $(-1)^N$ is the Hill…

We consider the generic quantum superintegrable system on the $d$-sphere with potential $V(y)=\sum_{k=1}^{d+1}\frac{b_k}{y_k^2}$, where $b_k$ are parameters. Appropriately normalized, the symmetry operators for the Hamiltonian define a…

A method of generating differential operators is used to solve the spectral problem for a generalisation of the Sylvester-Kac matrix. As a by-product, we find a linear differential operator with polynomial coefficients of the first order…

We investigate Ambarzumian-type mixed inverse spectral problems for Jacobi matrices. Specifically, we examine whether the Jacobi matrix can be uniquely determined by knowing all but the first $m$ diagonal entries and a set of $m$ ordered…

We find all spectral type differential equations satisfied by the symmetric generalized ultraspherical polynomials which are orthogonal on the interval [-1,1] with respect to the classical symmetric weight function for the Jacobi…

Jacobi sums and cyclotomic numbers are the important objects in number theory. The determination of all the Jacobi sums and cyclotomic numbers of order $e$ are merely intricate to compute. This paper presents the lesser numbers of Jacobi…

I explore some consequences of a groundbreaking result of Breimesser and Pearson on the absolutely continuous spectrum of one-dimensional Schr"odinger operators. These include an Oracle Theorem that predicts the potential and rather general…

Generalized Jacobi polynomials are orthogonal polynomials related to a weight function which is smooth and positive on the whole interval of orthogonality up to a finite number of points, where algebraic singularities occur. The influence…

This article is devoted to the study of Jack connection coefficients, a generalization of the connection coefficients of the classical commutative subalgebras of the group algebra of the symmetric group closely related to the theory of Jack…

We propose a spectral collocation method, based on the generalized Jacobi wavelets along with the Gauss-Jacobi quadrature formula, for solving a class of third-kind Volterra integral equations. To do this, the interval of integration is…

We introduce a function of the density of states for periodic Jacobi matrices on trees and prove a useful formula for it. This allows new, streamlined proofs of the gap labeling and Aomoto index theorems. We prove a version of this new…

The Jacobian conjecture over a field of characteristic zero is considered directly in view of the nonlinear partial differential equations it is associated with. Exploring the integrals of such partial differential equations, this work…

Recently, general fractional calculus was introduced by Kochubei (2011) and Luchko (2021) as a further generalisation of fractional calculus, where the derivative and integral operator admits arbitrary kernel. Such a formalism will have…

The Jacobi set is a useful descriptor of mutual behavior of functions defined on a common domain. We introduce the piecewise linear Jacobi set for general vector fields on simplicial complexes. This definition generalizes the definition of…