Computation of Al-Salam Carlitz and Askey-Wilson Moments using Motzkin Paths

In this paper we study the moments of polynomials from Askey scheme, and focus on Askey-Wilson polynomials. More precisely, give a combinatorial proof for case where $d=0$. Their values have already been computed by Kim Stanton in 2015, however, is not completely combinatorial, which means that an explicit bijection has exhibited yet. work, use new approach simpler Al-Salam-Carlitz, using sign reversing involution directly operates Motzkin path. We then generalize method to with $d=0$ only, providing first fully case.