Properties of powers of functions satisfying second-order linear differential equations with applications to statistics
Properties of powers of functions satisfying second-order linear differential equations with applications to statistics
复制标题
满足二阶线性微分方程的函数幂的性质及其在统计中的应用
DOI：
10.1007/s13160-015-0179-3
复制
发表时间：
2015
影响因子：
0.9
通讯作者：
Toshinori Oaku
中科院分区：
文献类型：
--
作者：
Naoki;Marumo;Toshinori Oaku
我们推导出满足二阶线性微分方程的函数的幂的性质。特别证明了函数的次方满足一阶微分方程，并给出了求微分方程的简单方法。我们还确定微分方程的指数并导出多项式次数的界限，多项式是微分方程中的系数。该界限对应于函数傅里叶变换的折卷积所满足的微分方程的阶。这些结果应用于统计学中使用的一些概率密度函数。
We derive properties of powers of a function satisfying a second-order linear differential equation. In particular we prove that then-th power of the function satisfies an-th order differential equation and give a simple method for obtaining the differential equation. Also we determine the exponents of the differential equation and derive a bound for the degree of the polynomials, which are coefficients in the differential equation. The bound corresponds to the order of differential equation satisfied by then-fold convolution of the Fourier transform of the function. These results are applied to some probability density functions used in statistics.