Ordinary differential operators and the integral representation of sums of certain power series

The explicit form of the eigenvalues and eigenfunctions is known for certain operators generated by symmetric differential expressions with constant coefficients and self-adjoint boundary conditions in the space of Lebesgue square-integrable functions on an interval, and their resolvents are known to be integral operators. According to the spectral theorem, the kernels of these resolvents satisfy a certain bilinear relation. Moreover, each such kernel is the Green's function of some self-adjoint boundary value problem and the method of constructing it is well known. Consequently, the Green's functions of these problems can be expanded in a series of eigenfunctions. In this paper, the identities obtained in this way are applied to construct an integral representation of sums of certain power series and special functions, and in particular, to evaluate sums of some converging number series.