广义Hilbert矩阵的显可对角化加权Hankel矩阵族
Family of explicitly diagonalizable weighted Hankel matrices generalizing the Hilbert matrix
摘要：
引入了一个三参数加权Hankel矩阵族$B=B(a,b,c)$，其元素为\[ B_{j,k}=\frac{Γ(j+k+a)}{Γ(j+k+b+c)}\,\sqrt{\frac{Γ(j+b)Γ(j+c)Γ(k+b)Γ(k+c)}{Γ(j+a)\, j!\,Γ(k+a)\, k!}}\,, \] $j,k\in\mathbb{Z}_{+}$，假设$a$，$b$，$c$为正，$a<b+c$，$b<a+c$，$c\leq a+b$ .著名的希尔伯特矩阵是作为一个特殊的情况。的直接和$B(a,b,c)\oplus B(a+1,b+1,c)$示出与扩张算子的离散模拟交换。由此可知，存在一个与$B(a,b,c)$可交换的三参数实对称Jacobi矩阵族$T(a,b,c)$.与$T(a,b,c)$相关的正交多项式是连续的对偶哈恩多项式。因此，可以显式地构造对角化$T(a,b,c)$的酉映射$U$。同时，$U$对角化$B(a,b,c)$和这个矩阵算子的频谱被证明是纯绝对连续的，并填充区间$[0,M(a,b,c)]$，其中$M(a,b,c)$是明确已知的。如果放宽假设$c\leq a+b$，而关于$a$、$b$、$c$的其余不等式都被假定为有效，则频谱还包含位于阈值$M(a,b,c)$之上的有限离散部分。同样，所有特征值和特征向量都被明确描述。
英文摘要：
A three-parameter family $B=B(a,b,c)$ of weighted Hankel matrices is introduced with the entries \[ B_{j,k}=\frac{Γ(j+k+a)}{Γ(j+k+b+c)}\,\sqrt{\frac{Γ(j+b)Γ(j+c)Γ(k+b)Γ(k+c)}{Γ(j+a)\, j!\,Γ(k+a)\, k!}}\,, \] $j,k\in\mathbb{Z}_{+}$, supposing $a$, $b$, $c$ are positive and $a<b+c$, $b<a+c$, $c\leq a+b$. The famous Hilbert matrix is included as a particular case. The direct sum $B(a,b,c)\oplus B(a+1,b+1,c)$ is shown to commute with a discrete analog of the dilatation operator. It follows that there exists a three-parameter family of real symmetric Jacobi matrices, $T(a,b,c)$, commuting with $B(a,b,c)$. The orthogonal polynomials associated with $T(a,b,c)$ turn out to be the continuous dual Hahn polynomials. Consequently, a unitary mapping $U$ diagonalizing $T(a,b,c)$ can be constructed explicitly. At the same time, $U$ diagonalizes $B(a,b,c)$ and the spectrum of this matrix operator is shown to be purely absolutely continuous and filling the interval $[0,M(a,b,c)]$ where $M(a,b,c)$ is known explicitly. If the assumption $c\leq a+b$ is relaxed while the remaining inequalities on $a$, $b$, $c$ are all supposed to be valid, the spectrum contains also a finite discrete part lying above the threshold $M(a,b,c)$. Again, all eigenvalues and eigenvectors are described explicitly.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
