Grünwald–Letnikov derivative

Summary

The Grünwald–Letnikov derivative of real order $\alpha$ is a limit of weighted sums of values of the function. It extends the difference quotient that defines ordinary derivatives. The weights are generalised binomial coefficients $\binom{\alpha}{k}$ with alternating signs. For whole orders it gives the usual derivatives, and for smooth enough functions it agrees with the Riemann–Liouville derivative of fractional calculus. Its discrete form is the basis of many numerical methods for fractional differential equations.