A  Pseudogamma Function

Does there exist a psuedogamma function that (when substituted into the Cauchy formula for repeated integration) yields a fractional Differintegral that equals (or approximates) the Gr\”unwald-Letnikov derivative?

The Riemann–Liouville differintegral and the Caputo differintegral are different, so you could use either process (or maybe both) to check for agreement.

A Follow-up Question:

Do these integrals yield the expected values in the trivial cases? (Eg: trig functions, polynomials, exp() or Mittag-Leffler, Guassian Distribution, etc.)