On the moment distance of Poisson processes

Consider the distance between two i.i.d. and independent Poisson processes with arrival rate $\lambda >0$ and respective arrival times $X_1,X_2,\dots$ and $Y_1,Y_2,\dots$ on a line. We give a closed analytical formula for the %expected distance to the power $a$ $\text{E}\mathrm{\mid }{X}_{k+r}-Y_k{\mathrm{\mid }}^a,$ for any integer $k\ge 1,r\ge 0$ and $a\ge 1.$ The expected difference of the arrival times to the power $a$ between two i.i.d. 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 $$\text{E}\mathrm{\mid }{X}_k-Y_k{\mathrm{\mid }}^a=\frac{a!}{{\lambda }^a}\frac{\mathrm{\Gamma }(\frac{a}{2}+k)}{\mathrm{\Gamma }(k)\mathrm{\Gamma }(\frac{a}{2}+1)},$$ where $\mathrm{\Gamma }(z)$ is Gamma function.