## Differential Ops, Special Polynomials, Binomial Transforms, Interpolation, and the Inverse Mellin Transform

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 $x^k (\frac{d}{dx})^k$ and $x^k (\frac{d}{dx})_{x=0}^k$ 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, $\phi_n(x(m).)$  should be $\phi_n((m).)$ .

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