1School of Mathematics and Statistics, The University of Birmingham, Edgbaston, Birmingham, B152TT, UK.
2Department of Mathematics, Baylor University, One Bear Place #97328, Waco, Texas, 76798-7328, USA.
*E-mail: W.N.Everitt@bham.ac.uk
**E-mail: Lance_Littlejohn@baylor.edu
***E-mail: Davut_Tuncer@baylor.edu
Dedicated to Dave Race, our long-time colleague and friend.
In this paper, we prove some general results about Lagrangian symmetric ordinary differential expressions ℓ[·] of order n when the coefficients of ℓ[·] are sufficiently smooth. In particular, for a natural number j, we show that under increased smoothness conditions on the coefficients, the jth composite power ℓj[·] of ℓ[·] is also Lagrangian symmetric. More generally, we show that if w is a symmetry factor for ℓ[·], then w is also a symmetry factor for ℓj[·]. Several classical second-order examples are given to illustrate these results and their jth composite powers, for each j ∈ ℕ, are given explicitly in Lagrangian symmetric form.
Lagrange symmetric ordinary differential expression, Lagrange symmetrizable differential expression, symmetry factor, composite powers of differential expressions, locally absolutely continuous function, compact support, Stirling numbers of the second kind, Jacobi-Stirling numbers
