Hankel Determinants of Some Sequences of Polynomials

Sivaramakrishnan Sivasubramanian

Séminaire lotharingien de combinatoire, Tome 63 (2010)

Résumé

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 x^n(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.