Catalan numbers, \(q\)-Catalan numbers and hypergeometric series
type: Item
instance of: scholarly article
title: Catalan numbers, \(q\)-Catalan numbers and hypergeometric series
zbMATH Open document ID: 0607.05006
DOI: 10.1016/0097-3165(87)90033-1
published in: Journal of Combinatorial Theory. Series A
publication date: 1987-01-01T00:00:00Z
review text: The author derives a new \(q\)-analog of the Catalan numbers: \[ {\mathcal C}_ n(\lambda,q)=q^{2n}\prod \frac{1+\lambda q^{2i-3}}{1-q^{2i}},\quad 1\leq i\leq n. \] The usual Catalan numbers are given by \({\mathcal C}_ n={\mathcal C}_ n(-1,1)\). In addition to this representation as a rational function in \(q\), Andrews' \(q\)-Catalan numbers have a natural combinatorial interpretation and satisfy \[ 0=\sum {\mathcal C}_ j(\lambda^{-1},q) {\mathcal C}_{n-j}(\lambda,q) (-\lambda q)^ j,\quad 0\leq j\leq n. \]
Mathematics Subject Classification ID: 05A30
Mathematics Subject Classification ID: 11B73
Mathematics Subject Classification ID: 05A15
Mathematics Subject Classification ID: 33D15
zbMATH DE Number: 3981173
zbMATH Keywords: q-analog
zbMATH Keywords: Catalan numbers
author: George E. Andrews
reviewed by: David M. Bressoud
MaRDI profile type: Publication
full work available at URL: 0097-3165(87)90033-1
OpenAlex ID: W2053642036
cites work: Applications of Basic Hypergeometric Functions
cites work: Q4146776
cites work: Q4760447
cites work: q-Catalan numbers
cites work: A Rogers-Ramanujan bijection
cites work: Q5508372
description: scientific article; zbMATH DE number 3981173