We investigate interlacing properties of zeros of Laguerre polynomials $L_n^{\left(\alpha \right)}\left(x\right)$ and $L_{n+1}^{(\alpha +k)}\left(x\right),$ $\alpha >-1,$ 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 $L_n^{\left(\alpha \right)}\left(x\right)$ and $L_n^{(\alpha +t)}\left(x\right)$ are interlacing for every $n\in \mathbb{N}$ and each $\alpha >-1$ is $0<t\le 2,$ [Driver K, Muldoon ME. Sharp interval for interlacing of zeros of equal degree Laguerre polynomials. J Approx Theory, to appear.], and the sharp t-interval within which the zeros of two consecutive degree Laguerre polynomials $L_n^{\left(\alpha \right)}\left(x\right)$ and $L_{n-1}^{(\alpha +t)}\left(x\right)$ are interlacing for every $n\in \mathbb{N}$ and each $\alpha >-1$ is $0\le t\le 2,$ [Driver K, Muldoon ME. Common and interlacing zeros of families of Laguerre polynomials. J Approx Theory. 2015;193:89–98]. We derive conditions on $n\in \mathbb{N}$ and α, $\alpha >-1$ that determine the partial or full interlacing of the zeros of $L_n^{\left(\alpha \right)}\left(x\right)$ and the zeros of $L_n^{(\alpha +2+k)}\left(x\right),$ $k\in \{1,2\}$. We also prove that partial interlacing holds between the zeros of $L_n^{\left(\alpha \right)}\left(x\right)$ and $L_{n-1}^{(\alpha +2+k)}\left(x\right)$ when $k\in \{1,2\},$ $n\in \mathbb{N}$ and $\alpha >-1$. Numerical illustrations of interlacing and its breakdown are provided.

我们研究了 Laguerre 多项式的零点的交错性质 ${升}_n^{\left(\alpha \right)}\left(X\right)$ 和 ${升}_{n+1}^{(\alpha +克)}\left(X\right),$ $\alpha >-1,$ 在哪里 $n\in \mathbb{N}$ 和 $克\in \{1,2\}$. 我们证明，一般而言，这些多项式的零部分交错而不是完全交错。两个等次拉盖尔多项式的零点的尖锐t区间${升}_n^{\left(\alpha \right)}\left(X\right)$ 和 ${升}_n^{(\alpha +吨)}\left(X\right)$ 为每一个交错 $n\in \mathbb{N}$ 和每个 $\alpha >-1$ 是 $0<吨\le 2,$[司机K，Muldoon ME。等次拉盖尔多项式零点交错的锐间隔。J Approx Theory，出现。]，以及两个连续阶 Laguerre 多项式的零点的尖锐t区间${升}_n^{\left(\alpha \right)}\left(X\right)$ 和 ${升}_{n-1}^{(\alpha +吨)}\left(X\right)$ 为每一个交错 $n\in \mathbb{N}$ 和每个 $\alpha >-1$ 是 $0\le 吨\le 2,$[司机K，Muldoon ME。拉盖尔多项式族的公共和交错零点。J 近似理论。2015;193:89-98]。我们推导出条件$n\in \mathbb{N}$和α ,$\alpha >-1$ 确定零点的部分或全部交错 ${升}_n^{\left(\alpha \right)}\left(X\right)$ 和零点 ${升}_n^{(\alpha +2+克)}\left(X\right),$ $克\in \{1,2\}$. 我们还证明了部分交错在零点之间成立${升}_n^{\left(\alpha \right)}\left(X\right)$ 和 ${升}_{n-1}^{(\alpha +2+克)}\left(X\right)$ 什么时候 $克\in \{1,2\},$ $n\in \mathbb{N}$ 和 $\alpha >-1$. 提供了隔行扫描及其分解的数字说明。