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.

#### Abstract

Consider the eigenvalue problem which is given in the interval [0,π] by the
differential equation
−y″(x)+q(x)y(x)=λy(x); 0≤x≤π (0,1)
and multi-point conditions
U1(y)=α1y(0)+α2y(π)+∑K=3nαKy(xKπ)=0,U2(y)=β1y(0)+β2y(π)+∑K=3nβKy(xKπ)=0, (0,2)
where q(x) is sufficiently smooth function defined in the interval [0,π]. We assume that the points
X3,X4,…,Xn divide the interval [0,1] to commensurable parts and
α1β2−α2β1≠0. Let
λk,s=ρk,s2 be the eigenvalues of the problem (0.1)-(0.2) for which we shall assume that they are
simple, where k,s, are positive integers and suppose that Hk,s(x,ξ) are the residue of Green's
function
G(x,ξ,ρ) for the problem (0.1)-(0.2) at the points ρk,s. The aim of this work is to calculate the regularized sum which is given by the form:
∑(k)∑(s)[ρk,sσHk,s(x,ξ)−Rk,s(σ,x,ξ,ρ)]=Sσ(x,ξ) (0,3)
The above summation can be represented by the coefficients of the asymptotic expansion of the function G(x,ξ,ρ) in negative powers of
k. In series (0.3) σ is an integer, while Rk,s(σ,x,ξ,ρ)
is a function of variables
x,ξ, and defined in the square [0,π]x[0,π] which ensure the convergence
of the series (0.3).