Remarks on an 1828 theorem of Clausen

TL;DR

This paper explores explicit calculations related to Clausen's 1828 theorem, revealing that powers of solutions to second order linear differential equations satisfy higher-order linear differential equations, extending known results to fourth and fifth powers.

Contribution

It demonstrates that powers of solutions to second order linear differential equations satisfy higher-order linear differential equations, including new results for fourth and fifth powers.

Findings

Fourth powers satisfy a linear differential equation of order five.

Fifth powers satisfy a linear differential equation of order six.

Any m-th power of a solution satisfies an effectively computable linear differential equation of order m+1.

Abstract

We list some explicit calculations related to a theorem of Clausen originally published in 1828, more commonly known as the result that describes the linear third order differential equation satisfied by the squares and the product of any two solutions of a linear second order differential equation in the real domain. The case of the cube of a solution dates to Appell, 1880. Although not commonly known and perhaps even new to some extent we show that the fourth and fifth powers of such solutions actually satisfy a linear differential equation of order five and six respectively provided the coefficients are sufficiently smooth. Indeed, it is the case that given any solution of a linear second order equation its m-th power satisfies an effectively computable linear differential equation of order m+1.