Regularized sum for eigenfunctions of multi-point problem in the commensurable case

Consider the eigenvalue problem which is given in the interval [ 0 , π ] by the differential equation − y ″ ( x ) + q ( x ) y ( x ) = λ y ( x ) ; 0 ≤ x ≤ π ( 0 , 1 ) and multi-point conditions U 1 ( y ) = α 1 y ( 0 ) + α 2 y ( π ) + ∑ K = 3 n α K y ( x K π ) = 0 , U 2 ( y ) = β 1 y ( 0 ) + β 2 y ( π ) + ∑ K = 3 n β K y ( x K π ) = 0 , ( 0 , 2 ) where q ( x ) is sufficiently smooth function defined in the interval [ 0 , π ] . We assume that the points X 3 , X 4 , … , X n divide the interval [ 0 , 1 ] to commensurable parts and α 1 β 2 − α 2 β 1 ≠ 0 . Let λ k , s = ρ k , s 2 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 H k , s ( x , ξ ) are the residue of Green's function G ( x , ξ , ρ ) for the problem (0.1)-(0.2) at the points ρ k , s . The aim of this work is to calculate the regularized sum which is given by the form: ∑ ( k ) ∑ ( s ) [ ρ k , s σ H k , s ( x , ξ ) − R k , s ( σ , x , ξ , ρ ) ] = S σ ( 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 R k , 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).