On the location of the discrete spectrum for complex Jacobi matrices

We study spectrum inclusion regions for complex Jacobi matrices that are compact perturbations of the discrete Laplacian. The condition sufficient for the lack of a discrete spectrum for such matrices is given.

     

Uncertainty principles for Jacobi expansions

In this paper an uncertainty principle for Jacobi expansions is derived, as a generalization of that for ultraspherical expansions by Rösler and Voit. Indeed a stronger inequality is proved, which is new even for Fourier cosine or ultraspherical expansions. A complex base of exponential type on the torus related to Jacobi polynomials is introduced, which are the eigenfunctions both of certain differential-difference operators of the first order and the second order. An uncertainty principle related to such exponential base is also proved.     

Some generating functions of Jacobi polynomials

In this paper, Weisner’s group-theoretic method of obtaining generating functions is utilized in the study of Jacobi polynomialsP> _n ^(a,ß)(x) by giving suitable interpretations to the index (n) and the parameter (β) to find out the elements for constructing a six-dimensional Lie algebra.     

Discrete spectrum for complex perturbations of periodic Jacobi matrices

We study spectrum inclusion regions for complex Jacobi matrices, which are compact perturbations of real periodic Jacobi matrix. The condition sufficient for the lack of the discrete spectrum for such matrices is given.     

On Ramanujan asymptotic expansions and inequalities for hypergeometric functions

In this paper we first discuss refinement of the Ramunujan asymptotic expansion for the classical hypergeometric functionsF(a,b;c;x), c ≤a + b, near the singularityx = 1. Further, we obtain monotonous properties of the quotient of two hypergeometric functions and inequalities for certain combinations of them. Finally, we also solve an open problem of finding conditions ona, b > 0 such that 2F(−a,b;a +b;r ²) < (2−r ²)F(a,b;a +b;r ²) holds for all r∈(0,1).     

Unclosed asymptotic expansions and mock theta functions

In his last letter to Hardy, Ramanujan defined 17 functions f(q), (|q|<1), which he called mock theta functions. Each f(q) has infinitely many exponential singularities at roots of unity, and under radial approach to every such singularity, f(q) has an asymptotic approximation consisting of a finite number of terms with closed exponential factors, plus an error term O(1). We give an example of a q-series in Eulerian form having an approximation with an unclosed exponential factor. Complete asymptotic expansions as q→1 of some shifted q-factorials are given in terms of polylogarithms and Bernoulli polynomials. Supported by the Natural Sciences and Engineering Research Council of Canada.     

Spectral properties of Jacobi matrices by asymptotic analysis

We consider two classes of Jacobi matrix operators in l² with zero diagonals and with weights of the form n^α+c_n for 0<α1 or of the form n^α+c_nn^α−1 for α>1, where {c_n} is periodic. We study spectral properties of these operators (especially for even periods), and we find asymptotics of some of their generalized eigensolutions. This analysis is based on some discrete versions of the Levinson theorem, which are also proved in the paper and may be of independent interest.     

A note on exact finite difference schemes for the differential equations satisfied by the Jacobi cosine and sine functions

We construct the exact finite difference equation discretizations for the nonlinear differential equations whose solutions are the Jacobi cosine and sine functions. Our derivations clarify and extend previous work done on this topic.     

Jacobi matrices generated by ratios of hypergeometric functions

A problem of determining zeroes of the Gauss hypergeometric function goes back to Klein, Hurwitz, and Van Vleck. In this very short note we show how ratios of hypergeometric functions arise as m-functions of Jacobi matrices and we then revisit the problem based on the recent developments of the spectral theory of non-Hermitian Jacobi matrices.     

Divergent Cesàro and Riesz means of Jacobi and Laguerre expansions

We show that for below certain critical indices there are functions whose Jacobi or Laguerre expansions have almost everywhere divergent Cesàro and Riesz means of order .

     

Sum rules for Jacobi matrices and divergent Lieb-Thirring sums

Let E_j be the eigenvalues outside [-2,2] of a Jacobi matrix with a_n-1∈?² and b_n→0, and μ^′ the density of the a.c. part of the spectral measure for the vector δ₁. We show that if b_n∉?⁴, b_n+1-b_n∈?², then

     

广义Jacobi矩阵的逆特征问题

本文主要讨论广义Jacobi阵及多个特征对的广义Jacobi阵逆特征问题.通过相似变换将广义Jacobi阵变换为三对角对称矩阵,其特征不变、特征向量只作线性变换,再应用前人理论求得广义Jacobi阵元素ai,|bi|,|ci|有唯一解的充要条件及其具体表达式.     

Jost functions and Jost solutions for Jacobi matrices, I. A necessary and sufficient condition for Szegő asymptotics

We provide necessary and sufficient conditions for a Jacobi matrix to produce orthogonal polynomials with Szegő asymptotics off the real axis. A key idea is to prove the equivalence of Szegő asymptotics and of Jost asymptotics for the Weyl solution. We also prove L² convergence of Szegő asymptotics on the spectrum.     

Two new asymptotic expansions of the ratio of two gamma functions

Formal expansions, giving as particular cases semiasymptotic expansions, of the ratio of two gamma functions are obtained.     

Mourre's theory for some unbounded Jacobi matrices

We show a Mourre estimate for a class of unbounded Jacobi matrices. In particular, we deduce the absolute continuity of the spectrum of such matrices. We further conclude some propagation theorems for them.     

完全次对称雅可比矩阵的某些性质

是在对完全对称雅可比矩阵及相应次对称矩阵对比研究的基础上,导出了完全次对称雅可比矩阵的特征值和相应特征向量之间的某些十分有趣的性质.     

H~1-Estimates of the Littlewood-Paley and Lusin Functions for Jacobi Analysis Ⅱ

Let $({\Bbb R}_+,*,\Delta)$ be the Jacobi hypergroup. We introduce analogues of the Littlewood-Paley $g$ function and the Lusin area function for the Jacobi hypergroup and consider their $(H^1, L^1)$ boundedness. Although the $g$ operator for $({\Bbb R}_+,*,\Delta)$ possesses better property than the classical $g$ operator, the Lusin area operator has an obstacle arisen from a second convolution. Hence, in order to obtain the $(H^1, L^1)$ estimate for the Lusin area operator, a slight modification in its form is required.