Results. Math. Online First c  2015 Springer Basel DOI 10.1007/s00025-015-0450-6 Results in Mathematics Multiple Completeness of the Root Functions for a Certain Class of Pencils of Ordinary Differential Operators with Constant Coefficients Victor S. Rykhlov Abstract. A class of polynomial pencils of ordinary differential operators with constant coefficients is considered in the article. The pencils from this class are generated by the n-th order ordinary differential expression and two-point boundary conditions. Coefficients of the differential ex- pression are supposed to be polynomials of the spectral parameter with constant coefficients. The boundary conditions are supposed to depend on the spectral parameter polynomially. It is assumed that the roots of the characteristic equation of the pencils from this class are simple, non-zero and lie on two rays emanating from the origin. The author investigates n-fold completeness of the root functions of the pencils from this class in the space of summable with square functions on the main segment. Sufficient conditions of the n-fold completeness of the root functions are obtained. The main idea of the method of the proof of the theorem is a new asymptotics of the characteristic determinant of the pencil. The presented results supplement previous results of the author. Mathematics Subject Classification. Primary 34L10; Secondary 34B07, 47E05. Keywords. Pencil of ordinary differential operators, root functions, multi- ple completeness, polynomial pencil of operators, system of root functions, eigenfunctions, associated functions. The results were obtained within the framework of the state task of Russian Ministry of Education and Science (Project 1.1520.2014K).