Moments of Askey–Wilson polynomials

New formulas for the n  th moment μn(a,b,c,d;q)μn(a,b,c,d;q) of the Askey–Wilson polynomials are given. These are derived using analytic techniques, and by considering three combinatorial models for the moments: Motzkin paths, matchings, and staircase tableaux. A related positivity theorem is given and another one is conjectured.