A Catalan-Hankel Determinant Evaluation

Let Ck = ( 2k k ) /(k+1) denote the k-th Catalan number and put ak(x) = Ck+Ck−1x+· · ·+C0x . Define the (n+1)×(n+1) Hankel determinant by setting Hn(x) = det[ai+j(x)]0≤i,j≤n. Even though Hn(x) does not admit a product form evaluation for arbitrary x, the recently introduced technique of γ-operators is applicable. We illustrate this technique by evaluating this Hankel determinant as Hn(x) = n ∑