qfactorials

q-factorials are a q-analog of the ordinary factorial function. The ordinary factorial of a non-negative integer n, denoted by n!, is the product of all positive integers up to n. The q-factorial, denoted by n!_q, is defined for a parameter q.

The q-factorial is defined as the product of terms involving q: n!_q = \prod_{k=1}^{n} \frac{1-q^k}{1-q}. This can be rewritten as \frac{(1-q)(1-q^2)...(1-q^n)}{(1-q)^n}.

As the parameter q approaches 1, the q-factorial converges to the ordinary factorial. This can be seen by using L'Hôpital's rule on the terms of the product or by considering the definition in terms of q-numbers. The q-number [k]_q is defined as \frac{1-q^k}{1-q}, so the q-factorial can be expressed as n!_q = [1]_q [2]_q ... [n]_q.

q-factorials appear in various areas of combinatorics and number theory, particularly in the study of partitions and hypergeometric series. They are fundamental to the development of q-analogs of many combinatorial objects and identities, such as q-binomial coefficients. The concept of q-factorials is an extension of the familiar factorial function, introducing a parameter that allows for a richer mathematical structure.