Differential operators on the projective plane

A formula for the addition of even spherical harmonics was obtained in [1]. This formula became the starting point for the derivation of the first regularized trace for the Laplace – Beltrami operator with potential on the projective plane. Due to this formula, it is not need to find asymptotic formulas for the associated Legendre polynomials by the three parameters, which was an impossible problem for a long time. The obtained results are the basis for the calculation of the perturbation theory corrections and allow the following application for the formulas for regularized traces of elliptic differential operators. In this paper we consider the problem of summation of divergent series class. We propose a method for calculating of the corrections of perturbation theory for a differential operator with potential on the real projective plane. The method is applicable, in particular, to take sum of series with factorial growth of its members.