S. A. Saleh, "Regularized sum for eigenfunctions of multi-point problem in the commensurable case", International Journal of Mathematics and Mathematical Sciences, vol. 21, Article ID 479193, 14 pages, 1998. https://doi.org/10.1155/S0161171298000738
Regularized sum for eigenfunctions of multi-point problem in the commensurable case
Consider the eigenvalue problem which is given in the interval by the differential equation 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 . Let 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 function 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).
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