Properties of powers of functions satisfying second-order linear differential equations with applications to statistics
Classical Analysis and ODEs
2015-07-29 v1 Statistics Theory
Statistics Theory
Abstract
We derive properties of powers of a function satisfying a second-order linear differential equation. In particular we prove that the n-th power of the function satisfies an (n+1)-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 the n-fold convolution of the Fourier transform of the function. These results are applied to some probability density functions used in statistics.
Keywords
Cite
@article{arxiv.1405.4451,
title = {Properties of powers of functions satisfying second-order linear differential equations with applications to statistics},
author = {Naoki Marumo and Toshinori Oaku and Akimichi Takemura},
journal= {arXiv preprint arXiv:1405.4451},
year = {2015}
}