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arXiv 0901.0817math.CA
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Meixner多项式和Krawtchouk多项式的零点

Zeros of Meixner and Krawtchouk polynomials

A Jooste, K Jordaan, F Tookos

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摘要：

我们研究了一族超几何多项式$_2F_1(-n,-x;a;t)$、$n\in\nn$的零点，对于参数$a$和$t$的某些值，这些多项式被称为Meixner多项式。当$a=-N$、$N\in\nn$和$t=\frac1{p}$时，多项式$K_n(x;p,N)=(-N)_n\phantom{}_2F_1(-n,-x;-N;\frac1{p})$、$n=0,1,...N$、$0<p<1$被称为Krawtchouk多项式。本文证明了正交多项式$K_{n}(x;p,a)$，$0<p<1$和$a>n-1$，准正交多项式$K_{n}(x;p,a)$，$k-1<a<k$，$k=1,...,n-1$和$p>1$或$p<0$以及非正交多项式$K_{n}(x;p,N)$，$0<p<1$和$n=N+1,N+2,...$的零点位置的结果.我们还表明，多项式$K_{n}(x;p,a)$，$a\in \rr$是真正的根时，$p\rightarrow 0$我们使用一个广义Sturmian序列参数和离散正交的Krawtchouk多项式的某些参数值，以证明所有的零迈斯纳多项式是真正的和积极的参数范围内，他们不再是正交的。

英文摘要：

We investigate the zeros of a family of hypergeometric polynomials $_2F_1(-n,-x;a;t)$, $n\in\nn$ that are known as the Meixner polynomials for certain values of the parameters $a$ and $t$. When $a=-N$, $N\in\nn$ and $t=\frac1{p}$, the polynomials $K_n(x;p,N)=(-N)_n\phantom{}_2F_1(-n,-x;-N;\frac1{p})$, $n=0,1,...N$, $0<p<1$ are referred to as Krawtchouk polynomials. We prove results for the zero location of the orthogonal polynomials $K_{n}(x;p,a)$, $0<p<1$ and $a>n-1$, the quasi-orthogonal polynomials $K_{n}(x;p,a)$, $k-1<a<k$, $k=1,...,n-1$ and $p>1$ or $p<0$ as well as the non-orthogonal polynomials $K_{n}(x;p,N)$, $0<p<1$ and $n=N+1,N+2,...$. We also show that the polynomials $K_{n}(x;p,a)$, $a\in \rr$ are real-rooted when $p\rightarrow 0$ We use a generalised Sturmian sequence argument and the discrete orthogonality of the Krawtchouk polynomials for certain parameter values to prove that all the zeros of Meixner polynomials are real and positive for parameter ranges where they are no longer orthogonal.

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