论文标题
等级和连续程度的Laguerre多项式的零隔离
Interlacing of zeros of Laguerre polynomials of equal and consecutive degree
论文作者
论文摘要
我们研究了拉瓜雷尔多项式的零零的交织属性$ l_ {n}^{（α）}（x）$和$ l_ {n+1}^{（α+k）}（x）}（x），$ $ $ $ $ $ a } \} $。我们证明，总的来说，这些多项式的零是部分而不是充分的。尖锐的$ t $间隔，其中两个相等程度的laguerre多项式的零在$ l_n^{（α）}（x）$和$ l_n^{（α+t）}（x+t）}（x）$中都在每个$ n \ in \ mathbb {n} $ in $ n} $ an \ cite {drmu2}，以及尖锐的$ t $间隔，其中两个连续两个度的laguerre多项式的零元素$ l_n^{（α）}（x）$和$ l_ {n-1}^{n-1}^{（α+t）}（α+t）}（x）$ n $ n $ n $ n $ n $ n \ n $ n \ n \ n \ n \ n \ n \ n \ n \ n \ n \ n \ n \ n \ n \ n \ n \ n \ n \ n \ n \ n \ n \ n \ n \ n \ n \ n是$ 0 \ leq t \ leq 2，$ \ cite {drmu1}。我们以$ n \ in \ mathbb {n} $和$α，$ $ $ $ $ a，确定$ l_n^{（α）}（x）$的部分或完全交织来得出条件。我们还证明，部分交织在$ l_n^{（α）}（x）$和$ l_ {n -1}^{（α + 2 + 2 + k）}（x）$时$ k \ in {\ in {\ k \ in {\ {1,2} \}，$ n \ in \ in \ n n \ n n \ n} $ n} $ n} $ n} $ n之间。提供了交错及其故障的数值插图。
We investigate interlacing properties of zeros of Laguerre polynomials $ L_{n}^{(α)}(x)$ and $ L_{n+1}^{(α+k)}(x),$ $ α> -1, $ where $ n \in \mathbb{N}$ and $ k \in {\{ 1,2 }\}$. We prove that, in general, the zeros of these polynomials interlace partially and not fully. The sharp $t-$interval within which the zeros of two equal degree Laguerre polynomials $ L_n^{(α)}(x)$ and $ L_n^{(α+t)}(x)$ are interlacing for every $n \in \mathbb{N}$ and each $ α> -1$ is $ 0 < t \leq 2,$ \cite{DrMu2}, and the sharp $t-$interval within which the zeros of two consecutive degree Laguerre polynomials $ L_n^{(α)}(x)$ and $ L_{n-1}^{(α+t)}(x)$ are interlacing for every $n \in \mathbb{N}$ and each $ α> -1$ is $ 0 \leq t \leq 2,$ \cite{DrMu1}. We derive conditions on $n \in \mathbb{N}$ and $α,$ $ α> -1$ that determine the partial or full interlacing of the zeros of $ L_n^{(α)}(x)$ and the zeros of $ L_n^{(α+ 2 + k)}(x),$ $ k \in {\{ 1,2 }\}$. We also prove that partial interlacing holds between the zeros of $ L_n^{(α)}(x)$ and $ L_{n-1}^{(α+ 2 +k )}(x)$ when $ k \in {\{ 1,2 }\},$ $n \in \mathbb{N}$ and $ α> -1$. Numerical illustrations of interlacing and its breakdown are provided.
