论文标题
Turán的不平等现象从Chebyshev到Laguerre多项式
Turán inequalities from Chebyshev to Laguerre polynomials
论文作者
论文摘要
令$ g $和$ h $是实值算术功能，正常式和标准化。在以下一般方案中的具体选择，递归定义的多项式\ begin {equation*} p_n^{g，h}（x）（x）：= \ frac {x} {x} {h（n）} \ sum_ {k = 1}初始值$ p_ {0}^{g，h}（x）= 1 $编码有关几个经典的，广泛研究的多项式的信息。这包括第二类的Chebyshev多项式，相关的Laguerre多项式和Nekrasov-Okounkov多项式。在本文中，我们证明，对于$ g（n）= n $和固定$ h $，我们获得了正交多项式序列，用于积极的确定功能。令$ h（n）= n^s $，$ 0 \ leq s \ leq 1 $。然后，该序列满足了Turán的不平等，以$ x \ geq 0 $。
Let $g$ and $h$ be real-valued arithmetic functions, positive and normalized. Specific choices within the following general scheme of recursively defined polynomials \begin{equation*} P_n^{g,h}(x):= \frac{x}{h(n)} \sum_{k=1}^{n} g(k) \, P_{n-k}^{g,h}(x), \end{equation*} with initial value $P_{0}^{g,h}(x)=1$ encode information about several classical, widely studied polynomials. This includes Chebyshev polynomials of the second kind, associated Laguerre polynomials, and the Nekrasov--Okounkov polynomials. In this paper we prove that for $g(n)=n$ and fixed $h$ we obtain orthogonal polynomial sequences for positive definite functionals. Let $h(n)=n^s$ with $0 \leq s \leq 1 $. Then the sequence satisfies Turán inequalities for $x \geq 0$.
