Addition formulas for polynomials built on classical combinatorial sequences
This short article contains quite a number of interesting results concerning Catalan numbers, Catalan polynomials, Stirling numbers of the second kind, Bell polynomials, Hermite polynomials and derangement polynomials in relation to explicit computation of several Hankel determinants as a continuation of the author's earlier results (the references list five papers, all by the present author). The first result states that the Hankel matrix of any order built on Catalan numbers has determinant 1. Then he obtains addition formulae and computes the Hankel determinant for Catalan polynomials and related ones defined by \(Q_n(x)= \sum^n_{k=0} {n\choose k}^2 x^k\), as well as for Bell polynomials related to Stirling numbers of the second kind \[ \det\bigl( B_{i+j}(x) \bigr)_{i,j=0, \dots,n}= x^{n(n+1)/2} \prod^n_{k=0} k!. \] A purely combinatorial enumerative proof for this formula was recently given by \textit{R. Ehrenborg} [Am. Math. Mon. 107, 557-560 (2000; Zbl 0985.15006)]. The author's earlier proof [Bull. Soc. Math. Belg., Sér. B 31, 49-55 (1979; Zbl 0439.10006)] used a Silvester's identity.
scientific article; zbMATH DE number 1423337