Fractional Difference Graph of a Discrete Quadratic Function

简介

This is a three dimensional graph of the discrete function (x+1)(x) defined on the natural numbers. This is an analogue to the quadratic function in the nabla discrete calculus; in particular, this function is known as the rising function. One axis represents the continuous order of the Riemann-Liouville fractional difference ranging from 0 to 2. The x-axis ranges in discrete steps from 1 to 7. The zeroth order derivative (x+1)(x), the first derivative (2x+1), and the second derivative (2) are highlighted with a ridge. The Riemann-Liouville fractional difference is continuous with the respect to the order of the difference, so the transition between the zeroth, first, and second difference to the fractional differences in between results in a smooth graph in each discrete x-value.