Related papers: On the Weyl function for complex Jacobi matrices
We consider the cases of the self-adjoint and skew-self-adjoint discrete Dirac systems, obtain explicit expressions for reflection coefficients and show that rational reflection coefficients and Weyl functions coincide.
In a previous paper I have defined a new basis for the representation ring of a Weyl group. In this paper we show that the new basis is related to the standard basis by an upper triangular unipotent matrix. We also give a new…
In this paper we use the orthogonal system of the Jacobi polynomials as a tool to study the Riemann-Liouville fractional integral and derivative operators on a compact of the real axis.This approach has some advantages and allows us to…
The present paper deals with the representation theory of the reflection equation algebra, connected with a Hecke type R-matrix. Up to some reasonable additional conditions the R-matrix is arbitrary (not necessary originated from quantum…
We study the Jacobian conjecture for Keller maps $f:X_0:=\mathbf{A}^n\rightarrow Y_0:=\mathbf{A}^n$ in characteristic $0$ and attempt to prove it. We are quite aware of the fact that many people have tried to prove the Jacobian conjecture…
One may consider the generalization of Jacobi polynomials and the Jacobi function of the second kind to a general function where the index is allowed to be a complex number instead of a non-negative integer. These functions are referred to…
We develop the theory of Hermitian Jacobi forms of lattice index, for both definite and indefinite Hermitian lattices. We also prove a theta decomposition theorem for vector-valued Jacobi forms (both in the orthogonal and Hermitian…
A method for successive synthesis of the Weyl matrix on the square lattice is proposed. It allows one to compute the Weyl matrix of a large graph by adding new edges and solving elementary systems of linear algebraic equations at each step.…
We give a birational realization of affine Weyl group of type $A^{(1)}_{m-1} \times A^{(1)}_{n-1}$. We apply this representation to construct some discrete integrable systems and discrete Painlev\'e equations. Our construction has a…
We prove some differential equations for the Riemann theta function associated to the Jacobian of a Riemann surface. The proof is based on some variants of a formula by Fay for the theta function, which are motivated by their analogues in…
In this note, we briefly introduce the background and motivation of the collaborative work [arXiv:2508.20797], and provide an outline of the main results. The latter relates to matrix and higher order scalar differential equations satisfied…
We consider a class of Jacobi matrices with unbounded entries in the so called critical (double root, Jordan box) case. We prove a formula for the spectral density of the matrix which relates its spectral density to the asymptotics of…
We propose a gradient-based Jacobi algorithm for a class of maximization problems on the unitary group, with a focus on approximate diagonalization of complex matrices and tensors by unitary transformations. We provide weak convergence…
In this paper we obtain a set of five new transmutations of the mother formula. Further, we obtain the second set of ten exact metafunctional equations by crossbreeding on every two elements of the previous set. Elements of the last set…
In 1992 Wirthm\"{u}ller showed that for any irreducible root system not of type $E_8$ the ring of weak Jacobi forms invariant under Weyl group is a polynomial algebra. However, it has recently been proved that for $E_8$ the ring is not a…
In this paper we introduce the notion of infinite dimensional Jacobi structure to describe the geometrical structure of a class of nonlocal Hamiltonian systems which appear naturally when applying reciprocal transformations to Hamiltonian…
This paper builds on the notion of the so-called orthogonal derivative, where an n-th order derivative is approximated by an integral involving an orthogonal polynomial of degree n. This notion was reviewed in great detail in a paper in J.…
We prove that the ring of Weyl invariant $E_8$ weak Jacobi forms is isomorphic to that of joint covariants of a binary sextic and a binary quartic form. The ring is therefore finitely generated. A minimal basis of generators is obtained…
In this paper, the fractional differential matrices based on the Jacobi-Gauss points are derived with respect to the Caputo and Riemann-Liouville fractional derivative operators. The spectral radii of the fractional differential matrices…
A transfer matrix function representation of the fundamental solution of the general-type discrete Dirac system, corresponding to rectangular Schur coefficients and Weyl functions, is obtained. Connections with Szeg\"o recurrence, Schur…