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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Acknowledgments
We are grateful to C. Koutschan for computation of (16). The third author is supported by JSPS KAKENHI Grant Number 25220001.
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Marumo, N., Oaku, T. & Takemura, A. Properties of powers of functions satisfying second-order linear differential equations with applications to statistics. Japan J. Indust. Appl. Math. 32, 553–572 (2015). https://doi.org/10.1007/s13160-015-0179-3
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DOI: https://doi.org/10.1007/s13160-015-0179-3