On the n th Quantum Derivative

The n th quantum derivative D n f ( x ) of the real‐valued function f is defined for each real non‐zero x aslim q → 1∑ k = 0 n( − 1 ) k[ ] q q ( k − 1 ) k / 2 f ( q n − k x )q ( n − 1 ) n / 2( q − 1 ) n x n,where[nk] qis the q ‐binomial coefficient. If the n th Peano derivative exists at x , which is to say that if f can be approximated by an n th degree polynomial at the point x , then it is not hard to see that D n f ( x ) must also exist at that point. Consideration of the function |1− x | at x = 1 shows that the second quantum derivative is more general than the second Peano derivative. However, it can be shown that the existence of the n th quantum derivative at each point of a set necessarily implies the existence of the n th Peano derivative at almost every point of that set.