# Importance of $q$-analog

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.

• Perhaps, instead of counting all elements in a set, you would like to find the distribution of some combinatorial statistic on that set. Use that statistic as the exponent of $q$, then. For example, the $q$-binomial coefficient gives the distribution of the inversion statistic on binary strings with a given number of $0$'s and $1$'s. – Alexander Burstein Apr 11 '18 at 5:35
• I am sorry, but maybe are there some sources where it explains these things in more detail. Im never satisfied with what they still mean. – Aurora Borealis Apr 11 '18 at 6:18
• Try Wikipedia or MathWorld pages on $q$-binomials, for example. Here is another source, but there are plenty more, just google some key words. – Alexander Burstein Apr 11 '18 at 6:25
• It is just that all these sources seem to be alittle more technical but ok. – Aurora Borealis Apr 11 '18 at 6:34
• “It’s not a bug, it’s a feature.” – Alexander Burstein Apr 11 '18 at 6:36

## 2 Answers

The nice and gentle surveys

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

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.