Fractional Difference Graph of a Discrete Quadratic Function

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This is an illustration of a three-dimensional diagram representing the discrete function (x+1)(x), which has been defined across all natural numbers. This particular function serves as a direct analogue to the quadratic function within the realm of nabla discrete calculus, earning it the title of rising function. One axis displays a continuous sequence of Riemann-Liouville fractional differences, spanning from 0 to 2. The x-axis progresses in discrete increments from 1 to 7. The zeroth order derivative (x+1)(x), the first derivative (2x+1), and the second derivative (2) are clearly highlighted with a distinct ridge. Due to its continuous nature with respect to the order of differences, the Riemann-Liouville fractional difference exhibits a smooth transition between the zeroth, first, and second differences, as well as the fractional differences that fall in between, resulting in a seamless graph at each discrete x-value.