Properties of powers of functions satisfying second-order linear   differential equations with applications to statistics
  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. 
Naoki Marumo and Toshinori Oaku and Akimichi Takemura
math.CA math.ST stat.TH