论文标题
与痕迹类反向雅各比矩阵相对应的正交多项式的家族
A family of orthogonal polynomials corresponding to Jacobi matrices with a trace class inverse
论文作者
论文摘要
假设$ \ {a_ {n}; \，n \ geq0 \} $是一个正数，$ \ sum a_ {n}^{\， - 1} <\ infty $。令$α_{n} = ka_ {n} $，$β_{n} = a_ {n}+k^{2} a_ {n-1} $，其中$ k \ in（0,1）$是一个参数，然后让$ \ \ \ {p_ {n}（x）（x）（x）（x）（x）（x）（x）（x） \ [α_{0} p_ {1}（x）+（β_{0} -x）p_ {0}（x）= 0，\α_{n} p_ {n+1}（x）+（x） 对于$ n \ geq1 $，带有$ p_ {0}（x）= 1 $。令$ j $为相应的jacobi（tridiagonal）矩阵，即$ j_ {n，n} =β_{n} $，$ j_ {n，n，n+1} = j_ {n+1，n+1，n} =α_{n} $ for $ n \ geQ0 $。然后存在$ j^{ - 1} $，属于跟踪类。我们为$ p_ {n}（x）$以及$ j $的特征函数得出一个明确的公式，并描述了多项式序列的正交性度量。特定情况下，介绍和研究了修改后的$ Q $ -Laguerre多项式。
Assume that $\{a_{n};\,n\geq0\}$ is a sequence of positive numbers and $\sum a_{n}^{\,-1}<\infty$. Let $α_{n}=ka_{n}$, $β_{n}=a_{n}+k^{2}a_{n-1}$ where $k\in(0,1)$ is a parameter, and let $\{P_{n}(x)\}$ be an orthonormal polynomial sequence defined by the three-term recurrence \[ α_{0}P_{1}(x)+(β_{0}-x)P_{0}(x)=0,\ α_{n}P_{n+1}(x)+(β_{n}-x)P_{n}(x)+α_{n-1}P_{n-1}(x)=0 \] for $n\geq1$, with $P_{0}(x)=1$. Let $J$ be the corresponding Jacobi (tridiagonal) matrix, i.e. $J_{n,n}=β_{n}$, $J_{n,n+1}=J_{n+1,n}=α_{n}$ for $n\geq0$. Then $J^{-1}$ exists and belongs to the trace class. We derive an explicit formula for $P_{n}(x)$ as well as for the characteristic function of $J$ and describe the orthogonality measure for the polynomial sequence. As a particular case, the modified $q$-Laguerre polynomials are introduced and studied.
