q-DERIVATIVE

Here is a nice diversion for anyone who knows what is the derivative of a simple function . The modern theory of differential and integral calculus began in the XVIIth century with the works of Newton and Leibniz. As it is well known, the derivative of a function with respect to the variable x is by definition :

Now, let us consider the following expression :

Of course, this is not valid when or but otherwise this alternative formula is equivalent to the usual derivative. You can convince yourself by writing , the term playing the role of .

At the beginning of the XXth century, F.H. Jackson studied this modified derivative and many of its consequences. The key concept is the q-derivative operator defined as follows when :

This q-derivative can be applied to functions not containing in their domain of definition. Then it reduces to the ordinary derivative when goes to :

Example : Compute the q-derivative of

One can easily check that the q-derivative operator is linear :

The product rule is slighlty modified but it approaches the usual product rule when q goes to 1 :

q-INTEGERS

There is an intriguing relation between arithmetics and the q-derivative. Indeed, if we compute then we notice an analogy with the usual derivation formula . It can be made explicit by defining q-integers :

As you may remember from the courses on geometric series, we have . Our previous computation thus becomes

We notice that q-integers coincide with usual integers when , that is to say :

Example : Recompute the q-derivative of by using q-integers

Here is an application of q-integers to the general Leibniz rule :

where we have defined the q-binomial coefficient :

and the q-factorial :

, .

q-INTEGRAL

As we can expect, the q-derivative has a reciprocal operation which is the indefinite q-integral. It is given by the Jackson integral :

If denotes the q-integral of then we can check that :

Example : Compute

since when

which is the q-analog of the usual integral

COMPLEMENTS

This is one of the most famous books for undergraduates :

Victor Kac and Pokman Cheung, Quantum Calculus, Springer, 2001