An integral representation of solutions and the problem of coupling factors for a linear differential equation

The equation $x^2\varphi''-(x^3+a_2x^2+a_1x+a_0)\varphi=0$, which is encountered in problems of mechanics, is considered in the work. It has two singular points: a regular singular point at zero and an irregular singular point at infinity. A fundamental family of solutions (f.f.s.) can be constructed in the form of integrals of Mellin-Barns type where the integrand satisfies a linear difference equation with polynomial coefficients. On the basis of this representation a f.f.s. is constructed in a neighborhood of zero, and its asymptotics at infinity is found. The coefficients of this asymptotics (the coupling factors) can be represented in the form of analytic expressions containing certain solutions of the difference equation adjoint to the difference equation previously mentioned. In contrast to previous works of the author, the general case of the initial equation is investigated.