Algebraic dependence of commuting differential operators

AbstractIt was proved by Burchnall and Chaundy (Proc. London. Math. Soc. 21(2) (1922) 420–440) that commuting elements in a certain algebra of differential operators are algebraically dependent, a result which has since found a use in a method for solving some non-linear PDEs using, amongst other things, algebraic geometry. The original proof used methods of functional analysis. This paper presents a new proof, which merely uses simple combinatorics and elementary linear algebra, but which still yields a generalized form of the result