Symmetric and Symmetrisable Differential Expressions

In 1960, H. L. Krall showed that every formally symmetric differential expression with sufficiently smooth real‐valued coefficients can be written in the form:∑ k = 0 nα 2 ky ( 2 k ) + ∑ k = 1 n∑ s = k n( 2 k - 1 2 s )2 2 s - 2 k + 2 + 1 s - k + 1B 2 s - 2 k + 2α 2 s ( 2 s - 2 k + 1 )y ( 2 k - 1 )where the B 2 i are the Bernoulli numbers. Based on this formula, Littlejohn found necessary and sufficient conditions on when an even‐order real differential expression L[·] possesses a symmetry factor, i.e. a function f ( x ) such that fL [·] is formally symmetric. In this paper, the authors generalize the work of both Krall and Littlejohn to differential expressions of arbitrary order with complexvalued coefficients.