Q-Pochhammer Symbol

The q-Pochhammer symbol, also called the q-shifted factorial, is a product defined in the mathematical field of combinatorics as (a;q) sub n equal to the product for k from 0 to n minus 1 of (1 minus a times q to the k), which expands to (1-a)(1-aq)(1-aq squared) and so on up to (1-a q to the n-1), with the convention that (a;q) sub 0 equals 1. It is the q-analog of the ordinary Pochhammer symbol used in classical analysis, and it converges back to that classical symbol as q approaches 1. Unlike the ordinary Pochhammer symbol, it extends naturally to an infinite product, which is an analytic function inside the unit disk and also makes sense as a formal power series in q; the special case of that infinite product taken at a equals q is Euler's function, important throughout combinatorics, number theory, and the theory of modular forms. The q-Pochhammer symbol is a basic building block for constructing q-analogs generally, especially within the theory of basic hypergeometric series. This description is adapted from Wikipedia contributors under CC BY-SA 4.0; changes were made. https://creativecommons.org/licenses/by-sa/4.0/