论文标题
雅各比和超球多项式的零
Zeros of Jacobi and Ultraspherical polynomials
论文作者
论文摘要
假设$ \ {p_ {n}^{（α，β）}（x）\} _ {n = 0}^\ infty $是一系列jacobi多项式，$ $α，β> -1。 p_ {n}^{（（α，β）}（x）$和$ p_ {n+ k}^{（α+ t，β+ s）}（x）$如果$ s，t> 0 $和$ k \ in \ mathbb {n} n}。 P_{n}^{(α,β)}(x)$ and $ P_{n+1}^{(α, β+ 1 )}(x),$ $ α> -1, β> 0, $ $ n \in \mathbb{N},$ are partially, but in general not fully, interlacing depending on the values of $α, β$ and $n.$ A similar result在$ p_ {n}^{（α，β）}（x）$和$ p_ {n+ 1}^{（α+ 1，β+ 1，β+ 1）}（x）（x）（x）（x）（$ a> $α> -1，β> -1。 p_ {n}^{（α，β）}（x）$和$ p_ {n}^{（α-t，β+ s）}（x）}（x）$以$α -t> -1，β> -1，β> -1，$ 0 \ $ 0 \ leq t，s \ leq 2的相互交织，但s s \ leq 2 note n of z note n of z note n of z in n of z note in in n in n in n in n in n in n in n in n in n in n of z in n of z in conters in n in in conterer，一般而言，一般而言。 p_ {n}^{（α，β）}（x）$和$ p_ {n}^{（α+ 1，β+ 1）}（x），$当$α> -1，β> -1。$。$。还考虑了雅各比多项式的对称情况$α=β=λ-1/2 $。我们证明超球多项式的零零$ c_ {n}^{（λ）}（x）$和$ c_ {n + 1}^{（λ + 1）}（λ + 1）}（x），$ $ $ucλ> -1/2 $部分是部分，但总体而言并非完全，插入。相等程度超球的零元素的交错$ c_ {n}^{（λ）}（x）$和$ c_ {n}^{（λ+3）}（x）（x），$ $ $ucλ> -1/2，还讨论了$。
Suppose $\{P_{n}^{(α, β)}(x)\}_{n=0}^\infty $ is a sequence of Jacobi polynomials with $ α, β>-1.$ We discuss special cases of a question raised by Alan Sokal at OPSFA in 2019, namely, whether the zeros of $ P_{n}^{(α,β)}(x)$ and $ P_{n+k}^{(α+ t, β+ s )}(x)$ are interlacing if $s,t >0$ and $ k \in \mathbb{N}.$ We consider two cases of this question for Jacobi polynomials of consecutive degree and prove that the zeros of $ P_{n}^{(α,β)}(x)$ and $ P_{n+1}^{(α, β+ 1 )}(x),$ $ α> -1, β> 0, $ $ n \in \mathbb{N},$ are partially, but in general not fully, interlacing depending on the values of $α, β$ and $n.$ A similar result holds for the extent to which interlacing holds between the zeros of $ P_{n}^{(α,β)}(x)$ and $ P_{n+1}^{(α+ 1, β+ 1 )}(x),$ $ α>-1, β> -1.$ It is known that the zeros of the equal degree Jacobi polynomials $ P_{n}^{(α,β)}(x)$ and $ P_{n}^{(α- t, β+ s )}(x)$ are interlacing for $ α-t > -1, β> -1, $ $0 \leq t,s \leq 2.$ We prove that partial, but in general not full, interlacing of zeros holds between the zeros of $ P_{n}^{(α,β)}(x)$ and $ P_{n}^{(α+ 1, β+ 1 )}(x),$ when $ α> -1, β> -1.$ We provide numerical examples that confirm that the results we prove cannot be strengthened in general. The symmetric case $α= β= λ-1/2$ of the Jacobi polynomials is also considered. We prove that the zeros of the ultraspherical polynomials $ C_{n}^{(λ)}(x)$ and $ C_{n + 1}^{(λ+1)}(x),$ $ λ> -1/2$ are partially, but in general not fully, interlacing. The interlacing of the zeros of the equal degree ultraspherical polynomials $ C_{n}^{(λ)}(x)$ and $ C_{n}^{(λ+3)}(x),$ $ λ> -1/2,$ is also discussed.
