论文标题
KAC-Geronimus多项式的预期实际零数量的渐近扩展
An asymptotic expansion for the expected number of real zeros of Kac-Geronimus polynomials
论文作者
论文摘要
令$ \ {φ_i（z;α）\} _ {i = 0}^\ infty $，对应于$α\ in（-1,1）$，是正式的geronimus polynomials。我们研究了预期的真实零数量的渐近行为，例如$ \ mathbb e_n（α）$，是随机多项式的\ [p_n（z）：= \ sum_ {i = 0}^nη_i或标准高斯随机变量。当$α= 0 $时，$φ_i（z; 0）= z^i $和$ p_n（z）$称为kac多项式。在这种情况下，威尔金斯表明$ \ mathbb e_n（0）$承认表格的渐近扩展\ [\ mathbb e_n（0）\ sim \ sim \frac2π\ log（n+1）+sum_+sum_+sum_ {p = 0}在这项工作中，我们获得了$ \ Mathbb e（α）$的类似扩展，以$α\ neq 0 $。事实证明，在这种情况下，渐近学的领先术语是$（1/π）\ log（n+1）$。
Let $ \{φ_i(z;α)\}_{i=0}^\infty $, corresponding to $ α\in(-1,1) $, be orthonormal Geronimus polynomials. We study asymptotic behavior of the expected number of real zeros, say $ \mathbb E_n(α) $, of random polynomials \[ P_n(z) := \sum_{i=0}^nη_iφ_i(z;α), \] where $ η_0,\dots,η_n $ are i.i.d. standard Gaussian random variables. When $ α=0 $, $ φ_i(z;0)=z^i $ and $ P_n(z)$ are called Kac polynomials. In this case it was shown by Wilkins that $ \mathbb E_n(0)$ admits an asymptotic expansion of the form \[ \mathbb E_n(0) \sim \frac2π\log(n+1) + \sum_{p=0}^\infty A_p(n+1)^{-p} \] (Kac himself obtained the leading term of this expansion). In this work we obtain a similar expansion of $ \mathbb E(α) $ for $ α\neq 0 $. As it turns out, the leading term of the asymptotics in this case is $ (1/π)\log(n+1) $.
