The sign of Lommel’s function

Lommel's function s,,v(x) is a particular solution of the differential equation X2y" + Xy' + (x2 _ v2)y = X# +1. It is shown here that sg,v(x)> 0 for x >0, if IL=2 and IvI ' and IvI <,u. This includes earlier results of R. G. Cooke's. The sign of sg,v(x) for other values of ,u and v is also discussed.