Orthogonal polynomials, Catalan numbers, and a general Hankel determinant evaluation

In this paper we deal with Hankel determinants of the form where r is a non-negative integer, an+r(x)=an+r+an+r-1x+⋯+a0xn+r and (an)n⩾0 is a sequence complex numbers. When a0≠0 and the Hankel determinants associated with the sequence (an+r+1)n⩾0 are not identically zero, we show that is a sequence of polynomials satisfying a three term recurrence relation. We illustrate our result by evaluating the Hankel determinant associated with the sequence for r=0 and r=1.