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 .

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One Response to Differential Ops, Special Polynomials, Binomial Transforms, Interpolation, and the Inverse Mellin Transform

  1. Tom Copeland says:

    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.

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