Probability distribution and a self-reciprocal relation involving confluent hypergeometric functions

The mixture model formulation of the non-central negative binomial distribution eλqv+1L(v)k(−λq) where λ >0, v > −1, 0 < p = 1 − q < 1, and k = 0, 1, 2, … is used to provide a probabilistic proof of the following self-reciprocal relation involving confluent hypergeometric functions under Hankel transform of order v: y)12Jv(xy)xv+12 e−x22|F|(−n; 1+v2−n2; x22)dx = (−1)nyv+12e−y22|F|(−n; 1+v2−n2;y22).