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Interpolation of the generalized binomial coefficients underlie the representation of a particular class of fractional differintegro operators by convolution integrals and Cauchy-like complex contour integrals.

The basic differintegro operators can be characterized by their action on divided powers:

with and denoting the Hadamard finite part,

An can be factored out to reduce the relations to

which can also be expressed in terms of the modified Mellin transform

as

By using the binomial expansion and Euler’s reflection formula for the gamma function, these integrals can be related to sinc function interpolation of the generalized binomial coefficient:

and, with

so underlying the integral reps is a basic sinc function interpolation of the generalized binomial coefficient:

With umbral substitutions and Euler transformations, this can be related to a Newton series interpolation of the generalized binomial coefficients. The Bell / Touchard polynomials, , and the falling factorials , are umbral compositional inverses, so formally umbrally substituting the Bell polynomials for in the sinc interpolation formula gives

The last equality follows from the generalized Dobinski formula. Using the Euler transformation on this last expression gives

where

and we can identify

Now umbrally substituting the falling factorials for gives the Newton series interpolation

These maneuvers can be intimately related to the Mellin transform rep. using an operator rep for the Bell polynomials and the definition . For and ,

Then we have, for , the inverse Mellin transform

which implies the Mellin transform and Newton series, for ,

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