Differential Ops, Special Polynomials, Binomial Transforms, Interpolation, and the Inverse Mellin Transform: (pdf, under construction. ) Relations between differential operators represented in the two basis sets and and the underlying binomial transforms of the associated coefficients of the pairs of series are sketched as well as the relations among the Newton-Gauss interpolation of these coefficients, the action of the differential operators, and an associated modified Mellin and inverse Mellin transform pair. The associated Laguerre polynomials and the Bell / Touchard / Exponential polynomials are examined in this light. Connections of the associated Laguerre polynomials to the Witt Lie algebra and modular forms are established.

Errata:

At the top of pg. 8, should be .

In the lower set of equations on pg. 12, should be replaced by .

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Yep. I’m not so familiar with the incomplete Bell polynomials, only the Bell polynomials of OEIS A036040 and their related morphs such as A036039.