Remarks on an 1828 theorem of Clausen

Details Remarks on an 1828 theorem of Clausen Remarks on an 1828 theorem of Clausen

2008-06-18

arxiv.org/abs/0806.2961v1

Mathematics

Classical Analysis and ODEs

Preprint, unpublished

Scientific paper

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.

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