Commutative ordinary differential operators. II.—The identity P n = Q m

Abstract

In a paper previously communicated to the Society we developed the general theory of ordinary differential operators P, Q of interprime orders m, n , which satisfy the commutative identity PQ = QP. Such a pair of operators also satisfy a second identity f (P, Q) ≡ P n — Q m + ... = 0, where, if we replace P, Q by two-way co-ordinates p, q , the equation f ( p , q ) = 0 defines a curve which is integral though not necessarily rational. The expression of P, Q depends upon parameters defined by equations of Abelian form involving transcendents of genus equal to that of the curve, which, if finite double points are absent, is g = ½ ( m — 1) ( n — 1). If the curve possesses finite double points, operators satisfying f (P, Q) = 0 can still be constructed by the method of C. O. 2, section IV, but the genus of the transcendents appearing will be less than ½ ( m — 1) ( n — 1). This number, however, in virtue of its position in the theory of lattice integers, is still of considerable importance.