Properties of powers of functions satisfying second-order linear differential equations with applications to statistics

TL;DR

This paper investigates the properties of powers of functions that satisfy second-order linear differential equations, deriving higher-order equations for these powers and applying findings to statistical probability density functions.

Contribution

It introduces a method to derive higher-order differential equations for powers of functions satisfying second-order equations, with applications to statistical distributions.

Findings

The n-th power of such functions satisfies an (n+1)-th order differential equation.

A simple method for deriving these differential equations is provided.

Bounds for polynomial degrees in the equations are established and related to convolutions in Fourier space.

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.