While looking around for new mathematics to learn, I encountered the notion of the \(q\)-analogue. The \(q\)-analogue of some statement or concept is a parameterization of that statement in terms of some new parameter \(q\), which, when equal to \(1\), yields the original statement. If when \(q=1\) the statement is undefined or the statement involves some discontinuities, then we can determine the \(q\)-analogue when \(q\) approaches \(1\) from either the left or right. Since this is a very new concept to me, I’m just going to cover the absolute basics with the introduction of notions such as inversions in terms of the so-called “\(q\)-factorial”, the \(q\)-analogue of the derivative (as a brief introduction to \(q\)-calculus), and the \(q\)-analogue of the binomial theorem. We may also introduce the concept of a hypergeometric series, and will be using the Pochhammer notation for factorials.

Abstract

In this expository paper, we will outline several interesting properties of the \(q\)-factorial and its relationship with inversions, the \(q\)-derivative and the \(q\)-analogue of Taylor’s theorem, a theorem about the uniqueness of \(q\)-antiderivatives up to a constant alongside evaluation of definite \(q\)-integrals, as well as several perspectives and proofs of the \(q\)-binomial theorem. We will conclude with an introduction to the \(q\)-Pochhammer symbol, and use it to define the \(q\)-hypergeometric function and prove an interesting fact about the special \(q\)-hypergeometric series \({}_{1}\Phi_{0}\).

As of now, this paper is complete and can be found here.