Abstract:
We consider sums over the square lattice which depend only on radial distance, and
provide formulas which enable sums of functions with Neumann series to be reexpressed
as combinations of hypergeometric series. We illustrate the procedure using
trigonometric sums previously studied by Borwein and Borwein, sums combining
logarithms, Bessel functions Jλ, and powers of distance, and sums of Neumann
functions. We also exhibit sums which may be evaluated analytically and recurrence
formulas linking sums.
