Implementation of Koshliakov's zeta function in “Kryloff for Sage” package

The acceleration of convergence of Fourier series by A.N. Krylov's method is considered for the case of Robin boundary conditions. In this case, it is necessary to sum trigonometric series containing roots of transcendental equations instead of natural values of the summation index. Summation of series of this class has attracted great interest of researchers in recent years. For the series of the class under consideration, many important and beautiful results have been obtained on their summation in the nal form, revealing connections with classical special functions. At the same time, it is possible to propose the possibility of implementing Krylov's method of accelerating convergence for them. The basic constructions of Krylov's method can be implemented in this case as well. In a situation where the right endpoint contains a condition of the third kind, and the left endpoint satis es the homogeneous Dirichlet condition, the generalization of the Hurwitz ζ-function given by N.S. Koshliakov is very useful. This generalization is of interest beyond mathematical physics as well. We present an implementation of Koshliakov's approaches in the “Krylo for Sage” package and try to answer the question of which results on symbolic summation of ordinary Fourier series can be extended to the case of Robin boundary conditions.