Q-Pochhammer Symbol - Relationship To Other Q-functions

Relationship To Other Q-functions

Noticing that

we define the q-analog of n, also known as the q-bracket or q-number of n to be

From this one can define the q-analog of the factorial, the q-factorial, as

Again, one recovers the usual factorial by taking the limit as q approaches 1. This can be interpreted as the number of flags in an n-dimensional vector space over the field with q elements, and taking the limit as q goes to 1 yields the interpretation of an ordering on a set as a flag in a vector space over the field with one element.

A product of negative integer q-brackets can be expressed in terms of the q-factorial as:

From the q-factorials, one can move on to define the q-binomial coefficients, also known as Gaussian coefficients, Gaussian polynomials, or Gaussian binomial coefficients:

One can check that

One also obtains a q-analog of the Gamma function, called the q-gamma function, and defined as

This converges to the usual Gamma function as q approaches 1 from inside the unit disc.. Note that

for any x and

for non-negative integer values of n. Alternatively, this may be taken as an extension of the q-factorial function to the real number system.