Expansion theory associated with linear differential equations and their regular singular points

Abstract

Introduction. Birkhofft has developed an expansion theory associated with linear differential equations on an interval of the real axis containing no singular points of the equation. The purpose of this paper is to develop a similar theory for the domain of complex variables in a region containing a regular singular point of the equation. To take the place of the two-point boundary conditions of the real variable case, we have corresponding relations between the values of the solution and its derivatives at a given point, and the values obtained by analytic continuation about the regular singular point back to the given point. The boundary conditions, n in number, are assumed to be linear, homogeneous and linearly independent, the order of the equation being n. An expansion theory associated with the linear equation of the second order with polynomial coefficients has been developed by 0. Volk.$ The method used in this paper is different from that employed by Volk and the coefficients are not assumed to be polynomials. The results of this paper and those of Volk overlap each other for the second-order equation, but neither includes the whole of the other. The NeumannGegenbauer expansions in Bessel's functions are obtained as special cases. The Legendre and hypergeometric differential equations give rise to expansion theories if infinity is considered as the regular singular point and the parameter is suitably chosen as a function of the arbitrary constants of the respective equations. The expansion theory for the hypergeometric equation as developed by Reinsch? is not included as a special case, since the parameter does not enter in the same way. In another paper it is hoped to extend the present theory so as to include the latter as a special case of a more general theory. The Laurent expansion is included as a special case arising from differential equations of every order. A solution of a differential equation with a regular singular point is, in