Generalizations of The Derivative - Difference Operator, Q-analogues and Time Scales

Difference Operator, Q-analogues and Time Scales

• The q-derivative of a function is defined by the formula

For x nonzero, if f is a differentiable function of x then in the limit as q → 1 we obtain the ordinary derivative, thus the q-derivative may be viewed as its q-deformation. A large body of results from ordinary differential calculus, such as binomial formula and Taylor expansion, have natural q-analogues that were discovered in the 19th century, but remained relatively obscure for a big part of the 20th century, outside of the theory of special functions. The progress of combinatorics and the discovery of quantum groups have changed the situation dramatically, and the popularity of q-analogues is on the rise.

• The difference operator of difference equations is another discrete analog of the standard derivative.

• The q-derivative, the difference operator and the standard derivative can all be viewed as the same thing on different time scales.