On the Summation of Fourier and Bessel Series

Using Fourier and Hankel transform techniques and a complex contour integration method that arises in interpolating electrical signals [H. D. Helms and J. B. Thomas, Proc. IRE50 (1962), 179-184; M. D. Rawn, SIAM J. Appl. Math., April 1989], we are able to derive many classical Fourier summation formulae and many interesting new Bessel summations that involve the zeros of the Bessel functions of the first kind. Our method is similar in form to the one employed by P. J. Forrester [Rocky Mountain J. Math. 13, (1983), 557-572], and M. L. Glasser [Math. Comp.37, (1981), 499-501]. Finally, we extend several of the classical Fourier summations to character series, in the spirit of [B. C. Berndt, Publ. Elektrotehnickog Fak. Univ. Beograder, No. 381-409 (1972), 25-29].