The Limiting Distribution of the Coefficients of the q-Catalan Numbers

We show that the limiting distributions of the coefficients of the qCatalan numbers and the generalized q-Catalan numbers are normal. Despite the fact that these coefficients are not unimodal for small n, we conjecture that for sufficiently large n, the coefficients are unimodal and even log-concave except for a few terms of the head and tail. Introduction The main objective of this paper is to show that the limiting distribution of the coefficients of the q-Catalan numbers is normal. The Catalan numbers Cn = 1 n+ 1 ( 2n n ) have many combinatorial interpretations; see Stanley [10]. The usual q-analog of the Catalan numbers is given by (1.1) Cn(q) = 1 [n+ 1] [ 2n n ] , where [n] = 1 + q + q + · · ·+ qn−1, and [ n k ] = [n]! [k]![n− k]! . There are also other types of q-analogs of the Catalan numbers; see, for example, Andrews [2], Gessel and Stanton [4], Krattenthaler [5]. We further consider the limiting distribution of the coefficients of the quotient of two products, which includes the result for the q-Catalan numbers as a special case. We conclude this paper with two conjectures on the unimodality and log-concavity for almost all the coefficients of the q-Catalan numbers and the generalized qCatalan numbers provided that n is sufficiently large. Received by the editors August 20, 2007. 2000 Mathematics Subject Classification. Primary 05A16, 60C05.