Copyright © 1998 S. A. Saleh. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
Consider the eigenvalue problem which is given in the interval by the
and multi-point conditions
where is sufficiently smooth function defined in the interval . We assume that the points
divide the interval to commensurable parts and
be the eigenvalues of the problem (0.1)-(0.2) for which we shall assume that they are
simple, where , are positive integers and suppose that are the residue of Green's
for the problem (0.1)-(0.2) at the points . The aim of this work is to calculate the regularized sum which is given by the form:
The above summation can be represented by the coefficients of the asymptotic expansion of the function in negative powers of
. In series (0.3) is an integer, while
is a function of variables
, and defined in the square which ensure the convergence
of the series (0.3).