Q-Pochhammer symbol
In the mathematical field of combinatorics, the q-Pochhammer symbol, also called the q-shifted factorial, is the product ( a ; q ) n = ∏ k = 0 n − 1 ( 1 − a q k ) = ( 1 − a ) ( 1 − a q ) ( 1 − a q 2 ) ⋯ ( 1 − a q n − 1 ) , {\displaystyle (a;q)_{n}=\prod _{k=0}^{n-1}(1-aq^{k})=(1-a)(1-aq)(1-aq^{2})\cdots (1-aq^{n-1}),} with ( a ; q ) 0 = 1. {\displaystyle (a;q)_{0}=1.} It is a q-analog of the Pochhammer symbol x ( n ) = x ( x + 1 ) … ( x + n − 1 ) {\displaystyle x^{(n)}=x(x+1)\dots (x+n-1)} , in the sense that lim q → 1 ( q x ; q ) n ( 1 − q ) n = x ( n ) .