Interpolating Discrete Functions through Riemann-Liouville Fractional Integrals

A method to find an analytic function passing through any countable set of points is provided, along with a proof of analyticity. This method is based on the fact that the formula f(n) = [ dⁿ/dtⁿ ∑ₖ₌₀∞ f(k) tᵏ/k! ]ₜ₌₀ can be extended using the Riemann-Liouville fractional derivative ξDₜᶻ (defined in the introduction), enabling evaluation of the derivative of any order z ∈ C: f(z) = [ ξDₜᶻ ∑ₖ₌₀∞ f(k) tᵏ/k! ]ₜ₌₀. This provides a simple formula to interpolate an arbitrary set of points of the form $(n, f(n))$, n ∈ N. Generalizations are provided to interpolate any countable set of points, given appropriate conditions.