R. B. Paris
We examine convergent representations for the sum of Bessel functions X∞ n=1 Jµ(na)Jν(nb) nα for µ, ν ≥ 0 and positive values of a and b. Such representations enable easy computation of the series in the limit a, b → 0+. Particular attention is given to logarithmic cases that occur both when a = b and a 6= b for certain values of α, µ and ν. The series when the first Bessel function is replaced by the modified Bessel function Kµ(na) is also investigated, as well as the series with two modified Bessel functions.