On a Remarkable Sequence of Polynomials
Alexander A. Kirillov, Anna Melnikov · 2001
A remarkable sequence of polynomials is considered. These polynomials in q describe in particular the number of solution to the equation X = 0 in triangular n × n matrices over a field q with q elements. They have at least three other important interpretations and a conjectural explicit expression in terms of the entries of the Catalan triangle. Resume Nous considerons une suite remarquable de polynomes. Ces polynomes en q decrivent en particulier le nombre de solutions de l’equation X = 0 dans les matrices n× n sur un corps q ayant q elements. Ils ont au moins trois autres interpretations importantes et une forme explicite conjecturale en termes des entrees du triangle de Catalan. Recently the first author has discovered a remarkable sequence of polynomials in one variable. We give below several different definitions which apparently lead to the same sequence of polynomials. 1. We start with the set An( q) of solutions to the equation X = 0 (1) in n × n upper-triangular matrices with elements from q. The cardinality of this set is, as we show below, a polynomial in q which will be denoted by An(q). Unfortunately, we do not know any direct recurrence relation between these polynomials. So, we will split the set An( q) into subsets consisting of matrices of a given rank r. The corresponding quantity is denoted by An(q) so that we have AMS 1980 Mathematics Subject Classification (1985 Revision): 15A57, 22E25, 05A15 ∗University of Pensylvania, Math. Dept., Philadelphia, PA 19104-6395, USA, and Institute for Problems of Information Transmission of RAS, B. Karetnyi, 19, Moscow 101 477, GSP-4, Russia †Weizmann Institute, Dept. of Pure Math., Rehovot, Israel We are grateful to Jacques Alev for the invitation to the Rencontre Franco-Belge which was very interesting and useful for all the participants. During this work we use the package “Mathematica” intensively. In this matter the first author has profited from the contacts with Herb Wilf and the second one – with Michael Shapiro both of whom we would like to thank. Societe Mathematique de France 36 A. A. KIRILLOV & A. MELNIKOV An(q) = ∑ r≥0 A r n(q). The new quantities satisfy the simple recurrence relations (see [1]) A n+1(q) = q r+1 ·A n (q) + (q n−r − q) · An(q); A 0 n+1(q) = 1 (2) which imply in particular that they are polynomials in q . One can express An(q) in terms of q-Hermite polynomials. Namely, in [1] the following equality is proved (2z) = ∑ r An(q) · q r(r−n) ·Hn−2r(z; q ), (3) where Hn(x; q) is the q-Hermite polynomial defined by