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Symmetric and Symmetrisable Differential Expressions
Author(s) -
Littlejohn Lance L.,
Race David
Publication year - 1990
Publication title -
proceedings of the london mathematical society
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 1.899
H-Index - 65
eISSN - 1460-244X
pISSN - 0024-6115
DOI - 10.1112/plms/s3-60.2.344
Subject(s) - mathematics , order (exchange) , differential (mechanical device) , function (biology) , symmetry (geometry) , combinatorics , bernoulli's principle , expression (computer science) , pure mathematics , mathematical analysis , geometry , physics , computer science , thermodynamics , finance , evolutionary biology , biology , economics , programming language
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.