A Darling–Siegert formula relating some Bessel integrals and random walks

Abstract

The combinatorial identity ∑ k = 0 r 2 k + m k + m 2 r - 2 k + m r - k + m 1 k + m + 1 = 1 m + 1 2 r + 2 m + 1 r + 2 m + 1 , for m = 0 , 1 , … , emerging in the study of random flights in the space R 4 is examined. A probabilistic interpretation of this formula based on the first-passage time and the time of first return to zero of symmetric random walks is given. A combinatorial proof of this result is also provided. A detailed analysis of the first-passage time distribution is presented together with its fractional counterpart.