arXiv subjects
Publications and source records attributed to V. -B. K. Rogov.
The connections between q-Bessel functions of three types and q-exponential of three types are established. The q-exponentials and the q-Bessel functions are represented as the Laurent series. The asymptotic behaviour of the q-exponentials and the q-Bessel functions is investigated.
The aim of this paper is to give a group theoretical interpretation of the three types of Bessel-Jackson functions. We consider a family of quantum Lorentz groups and a family of quantum Lobachevsky spaces. For three members of quantum Lobachevsky spaces the Casimir operators give rise to the two-body relativistic open Toda lattice Hamiltonians. Their eigen-functions are the modified Bessel-Jackson functions of three types. We construct the principal series of unitary irreducible representations of the quantum Lorentz groups. Special matrix elements in the irreducible spaces are the Bessel-Macdonald-Jackson functions. They are the wave functions of the two-body relativistic open Toda lattice. We obtain integral representations for these functions.
The q-Bessel-Macdonald functions of kinds 1, 2 and 3 are considered. Their representations by classical integral are constructed.
The modified q-Bessel functions and the q-Bessel-Macdonald functions of the first and second kind are introduced. Their definition is based on representations as power series. Recurrence relations, the q-Wronskians, asymptotic decompositions and q-integral representations are received. In addition, the q-Bessel-Macdonald function of kind 3 is determined by its q-integral representation.
The functions on a lattice generated by the integer degrees of $q^2$ are considered, 0<q<1. The $q^2$-translation operator is defined. The multiplicators and the $q^2$-convolutors are defined in the functional spaces which are dual with respect to the $q^2$-Fourier transform. The $q^2$-analog of convolution of two $q^2$-distributions is constructed. The $q^2$-analog of an arbitrary (non integer) order derivative is introduced
We define a q-analog of the modified Bessel and Bessel-Macdonald functions. As for the q-Bessel functions of Jackson there is a couple of functions of the both kind. They are arisen in the Harmonic analysis on quantum symmetric spaces similarly to their classical counterpart. Their definition is based on the power expansions. We derive the recurrence relations, difference equations, q-Wronskians, and an analog of asymptotic expansions which turns out is exact in some domain if $q\neq 1$. Some relations for the basic hypergeometric function which follow from this fact are discussed.
We consider the quantum Lobachevsky space ${\bf L}_q^3$, which is defined as subalgebra of the Hopf algebra ${\cal A}_q(SL_2({\bf C}))$. The Iwasawa decomposition of ${\cal A}_q(SL_2({\bf C}))$ introduced by Podles and Woronowicz allows to consider the quantum analog of the horospheric coordinates on ${\bf L}_q^3$. The action of the Casimir element, which belongs to the dual to ${\cal A}_q$ quantum group $U_q(SL_2({\bf C}))$, on some subspace in ${\bf L}_q^3$ in these coordinates leads to a second order difference operator on the infinite one-dimensional lattice. In the continuos limit $q\rightarrow 1$ it is transformed into the Schrödinger Hamiltonian, which describes zero modes into the Liouville field theory (the Liouville quantum mechanics). We calculate the spectrum (Brillouin zones) and the eigenfunctions of this operator. They are $q$-continuos Hermit polynomials, which are particular case of the Macdonald or Rogers-Askey-Ismail polynomials. The scattering in this problem corresponds to the scattering of first two level dressed excitations in the $Z_N$ Baxter model in the very peculiar limit when the anisotropy parameter $\ga$ and $N~\rightarrow\infty$, or, equivalently, $(\ga, N)\rightarrow 0$.