On the Summation of Fourier Series over the Roots of Transcendental Equations Using Annihilation

Trigonometric series in the roots of transcendental equations are considered. Similar series arise in problems of mathematical physics in the case of boundary conditions of the third kind. For this class of series, sufficient conditions for their representability in the finite form of elementary functions are indicated. We generalize to this case our previous works based on the use of finite expressions for divergent Fourier series. In this case, we obtain a simple method for determining the elementarity of the ζ-function of N.S. Koshlyakov and its generalizations. These considerations are used to create new functions of symbolic summation in computer algebra systems.