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

  • Naoki Marumo
  • Toshinori Oaku
  • Akimichi TakemuraEmail author
Original Paper Area 2


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.


Characteristic function Exponents Holonomic function Indicial equation Skewness 

Mathematics Subject Classification

16S32 62E15 



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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Copyright information

© The JJIAM Publishing Committee and Springer Japan 2015

Authors and Affiliations

  • Naoki Marumo
    • 1
  • Toshinori Oaku
    • 2
  • Akimichi Takemura
    • 1
    Email author
  1. 1.Graduate School of Information Science and TechnologyUniversity of TokyoTokyoJapan
  2. 2.Department of MathematicsTokyo Woman’s Christian UniversityTokyoJapan

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