Irregular Differential Systems of Order Two and the Related Expansion Problems

We have prevously discussed the similarities between the series of Fourier and of Birkhoff.tSince a series of Birkhoff is defined by a linear homogeneous differential system of the wth order in which the boundary conditions are of regular type,î it is natural to attempt an extension of the methods there employed to some systems with irregular boundary conditions.We shall discuss here the case n = 2, with the hope of giving a comparatively exhaustive treatment of the narrowed topic.From our point of view, it is not essential in this discussion that a series be treated with regard to its convergence: a sum by appropriate means we consider equally valuable.A treatment of the convergence of the formal expansions for a function restricted to have a certain number of derivatives and to satisfy certain boundary conditions has come to our attention since the completion of this paper.§ As Professor Jackson has suggested to the writer, the methods of Wilder in a similar problem could be applied to this end, as is obvious from a comparison of the formulas of this paper with his.||It should be noted, however, that under our discussion of systems of type 1, Case I, the series for the function 1 can be seen to be divergent, so that such results are not so useful as it might appear.We note that the series discussed in this paper are entirely different from those discussed by Jackson and Hopkins in the case n = 3.%