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

Abstract

In this paper we deal with Hankel determinants of the form det[ai+j+r(x)]i,j=0n, 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 (det[ai+j+r(x)]i,j=0n)n⩾0 is a sequence of polynomials satisfying a three term recurrence relation. We illustrate our result by evaluating the Hankel determinant associated with the sequence det∑ν=0l+k+r1l+k+r+1-ν2(l+k+r-ν)l+k+r-νxνl,k=0n, for r=0 and r=1.