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arXiv · 1405.4451

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

Naoki Marumo·Toshinori Oaku·Akimichi Takemura
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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.

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Naoki Marumo, Toshinori Oaku, Akimichi Takemura. 2014-05-18. Properties of powers of functions satisfying second-order linear differential equations with applications to statistics. https://arxiv.org/abs/1405.4451

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