Catalan numbers, \(q\)-Catalan numbers and hypergeometric series
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 \[ 0=\sum {\mathcal C}_ j(\lambda^{-1},q) {\mathcal C}_{n-j}(\lambda,q) (-\lambda q)^ j,\quad 0\leq j\leq n. \]
scientific article; zbMATH DE number 3981173