# Hankel Determinants of Some Sequences of Polynomials

Abstract. Ehrenborg gave a combinatorial proof of Radoux's theorem which states that the determinant of the (n+1)x(n+1) dimensional Hankel matrix of exponential polynomials is xn(n+1)/2 \prod_{i=0}^n i!. This proof also shows the result that the (n+1)x(n+1) Hankel matrix of factorial numbers is \prod_{k=1}^n (k!)2. We observe that two polynomial generalizations of factorial numbers also have interesting determinant values for Hankel matrices.

A polynomial generalization of the determinant of the Hankel matrix with entries being fixed-point free involutions on the set [2n] is given next. We also give a bivariate non-crossing analogue of a theorem of Cigler about the determinant of a similar Hankel matrix.

Received: October 4, 2009. Revised: April 4, 2010. Accepted: April 17, 2010. Final Version: April 20, 2010.

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