Hankel determinants of some polynomials arising in combinatorial analysis
Evaluations of special Hankel determinants (this is a determinant of the form \(\det(c_{i+j})_{1\leq i,j\leq n}\)) have always fascinated many people, as there are a lot of beautiful such results. It is classical, but unfortunately not wide-spread enough, that the deep reason for this phenomenon is that evaluations of Hankel determinants are intimately tied to orthogonal polynomials. More precisely, given a sequence of orthogonal polynomials with an associated measure, the \(i\)th moment of which is \(c_i\), the Hankel determinant \(\det(c_{i+j})_{1\leq i,j\leq n}\) is simply a product of the coefficients which appear in the three-term recurrence that the orthogonal polynomials must satisfy, thanks to Favard's theorem. In particular, most of the found evaluations of Hankel determinants do in fact come from known facts about hypergeometric orthogonal polynomials in the Askey scheme. The purpose of the article under review is to put emphasis on this point, and to discuss this matter by considering several examples. (For a similar treatment, more examples and references, see Section~2.7 and Theorem~52, plus the subsequent discussion, in the reviewer's survey [Sémin. Lothar. Comb. 42, B42q (1999; Zbl 0923.05007)].) In particular, the author derives an evaluation of a Hankel determinant with entries being certain hypergeometric \(_2F_1\)-series, and shows that several previously found Hankel determinant evaluations in the literature follow on specializing his result. Moreover he derives several evaluations which apparently have not been recorded earlier. The author also discusses three Hankel determinant evaluations, which he is not able to prove. However, two of these do in fact appear in the literature. Namely, the first of these, the evaluation of \(\det(B_{i+j})_{0\leq i,j\leq n}\), where \(B_k\) denotes the \(k\)th Bernoulli number, appears in \textit{W. A. Al-Salam} and \textit{L. Carlitz} [Portugaliae Math. 18, 91-99 (1959; 093.01504)]. The second, the evaluation of \(\det(B_{i+j+1})_{0\leq i,j\leq n}\), appears in \textit{M. Fulmek} and the reviewer [Eur. J. Comb. 21, 601-640 (2000); Theorem~23] as one evaluation in a whole family of evaluations of Hankel determinants with entries being certain linear combinations of Bernoulli numbers. (Also the first evaluation and the evaluation of \(\det(B_{i+j+2})_{0\leq i,j\leq n}\), which appears as (73) in the paper under review, are members of this family.) Finally, the third unproven evaluation, the evaluation of \(\det(P_{i+j}(t))_{0\leq i,j\leq n}\), where \(P_n(t)=\sum _{k=0} ^{n}{(-n)_k} t^k/{(b-n)_k}\), can be accomplished as follows: we apply the well-known identity (see Theorem~10 in the paper under review or Lemma 15 in the reviewer's survey [loc. cit.]) \[ \det_{0\leq i,j\leq n}(A_{i+j})= \det_{0\leq i,j\leq n}\left(\sum _{\ell=0} ^{i+j}\binom {i+j}\ell A_{\ell}x^{i+j-\ell}\right) \] with \(A_m=P_m(t)\) and \(x=-1\). Then, the resulting double sum on the right-hand side can be simplified by interchanging summations over \(\ell\) and \(k\), evaluating the (now) inner sum over \(\ell\) by means of Chu--Vandermonde summation, and evaluating the resulting (single) sum over \(k\) by means of the binomial theorem. As a result we obtain, up to sign, the determinant \(\det(\mu_{i+j})_{0\leq i,j\leq n}\), where \(\mu_0=1\) and \(\mu_m=-t(1-t)^{m-1}{m!} /{(1-b)_m}\) for \(m\geq 1\). Next, in the determinant, we split the \((0,0)\)-entry \(1\) in the form \(1=-\frac {t} {1-t}+\frac {1} {1-t}\), and then use linearity in the first row to infer that \[ \det_{0\leq i,j\leq n}(\mu_{i+j})= \det_{0\leq i,j\leq n}(\mu'_{i+j})+ \frac {1} {1-t} \det_{1\leq i,j\leq n}(\mu_{i+j}), \] where \(\mu'_m=-t(1-t)^{m-1}{m!} /{(1-b)_m}\) for all \(m\geq 0\). Both of the two determinants on the right-hand side are not difficult to evaluate, for example by using Theorem~26, (3.11), with \(q=1\), in the reviewer's survey [loc. cit.].
scientific article; zbMATH DE number 1534770