Séminaire Lotharingien de Combinatoire, B63d (2010), 8 pp.
Sivaramakrishnan Sivasubramanian
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.
The following versions are available: