Importance of $q$-analog

Question

I am currently studying q-analog, but I was actually confused on what its actual purpose is. Like I see all these manipulations using $q$, but I have little idea on what they represent. Sure the series and manipulations they look nice, and work out nicely, but again quite lost on what its actual purpose is.

I would appreciate some little background information and the actual use and intuitive interpretation of what they represent in mathematics.

Answer 1

The nice and gentle surveys

• The q-binomial Theorem by Shaun Cooper

• On the q-binomial coefficients and binomial congruences by Armin Straub

  might be helpful. The first presents a connection with enumeration of lattice paths.

The classic

• Integer Partitions by George E. Andrews and Kimmo Eriksson is the introductory text to integer partitions for beginners. $q$-related material starts with Chapter 7 Gaussian polynomials.

A book about the development of $q$-calculus from a historical perspective and providing many examples of applications is

• The History of q-Calculus and a new method by Thomas Ernst

Answer 2

Recently, some researchers are considering to apply $q$-analogs to numerical methods. For example, there is a $q$-Newton method (with local convergence):

Rajkovic, P. M., Stankovic, M. S., & Marinkovic, S. D. (2003). On $q$-iterative methods for solving equations and systems. Novi Sad J. Math, 33(2), 127-137.

A $q$-epsilon algorithm is also developed:

He, Y., Hu, X. B., & Tam, H. W. (2009). A $q$-difference version of the ϵ-algorithm. Journal of Physics A: Mathematical and Theoretical, 42(9), 095202.

$q$-analogs of numerical methods use $q$-derivative instead of the ordinary derivative. In other words, it provides derivative-free algorithms. This is the importance of $q$-analogs in numerical analysis.