On the moment distance of Poisson processes

Consider two identical and independent Poisson processes with arrival rate λ > 0 and respective arrival times X1, X2, … and Y1, Y2, … on a line. We give a closed analytical formula for the E[|Xk + r − Yk|a], for any integer k ⩾ 1, r ⩾ 0 and a ⩾ 1. The expected absolute difference of the arrival times to the power a between two identical and independent Poisson processes we represent as the combination of the Pochhammer polynomials.Especially, for r = 0 and any positive integer a, the following identity is valid E|Xk-Yk|a=a!λaΓa2+kΓ(k)Γa2+1,\begin{equation*} \mathbf {E}\left[|X_k-Y_k|^a\right]=\frac{a!}{\lambda ^a}\frac{\Gamma \left(\frac{a}{2}+k\right)}{\Gamma (k)\Gamma \left(\frac{a}{2}+1\right)}, \end{equation*}where Γ(z) is Gamma function.