Abstract
This paper presents the proof of identities on finite series related to Bessel function. By the different modification of the product of J_-v-k(x) and a series Σ^∞_i=0 J_v+2i(x), we derive two finite series that look different but have the same summation value. The following is an outline of derivation. One finite series is derived by multiplying the power expansion of J_-v-k(x) and that of Σ^∞_i=0 J_v+2i(x), rearranging with the same power, rewriting with Pochhammer's symbol and applying the summation theorem for generalized hypergeometric series. The other finite series is derived by using the power expansion of the product of two Bessel functions in Σ^∞_j=0 J_-v-k(x) J_v+2j(x), rearranging with the same power, rewriting with Pochhammer's symbol and applying the summation theorem. Also for J_-v-k(x) Σ^∞_i=0 (v+2i) J_v+2i(x), we can derive another pair of two finite series that look different but have the same summation value in a similar way.
