Given a compositional inverse pair and , i.e.,

,

with with , , and , construct the binomial Sheffer sequence with the exponential generating function

.

Then the associated formal group law (FGL) may be expressed as

Given a compositional inverse pair and , i.e.,

,

with with , , and , construct the binomial Sheffer sequence with the exponential generating function

.

Then the associated formal group law (FGL) may be expressed as

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Posted in Math
Tagged Binomial Sheffer sequences, Composition, Creation and annihilation operators, Differential operators, Expansion of FGL, Finite operator calculus, Formal group laws FGL, Inversion, Ladder, Power series, Raising and lowering, Reversion, Symmetric polynomials, Umbral calculus
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(This is a duplicate of a Mathoverflow question posed in Oct 2014 that ran the gauntlet of the OCD cadre there–the demonstrative ones I assume avoid stepping on cracks in the sidewalk and become obstructive, hostile, and/or jealous at popular questions they can’t immediately circumscribe within their own rigid, limited views on how and what math should be done. See also this blog post and this MO-Q. In a couple of days the question got around 800 views, 9 upvotes, 7 downvotes, two or three close requests, several unconstructive/petty comments on rewriting/formatting, but no answers. Recognizing, from experience on MO, the futility of further refining the question and having pursued a plausible partial answer myself, I deleted most of it. Here is the full text so that I and others can easily follow related references.)

**Prelude on Two Threads**

Last month (Sept. 2014) the workshop New Geometric Structures in Scattering Amplitudes was hosted by CMI with the following partial overview:

“*Recently, remarkable mathematical structures have emerged in the study of scattering amplitudes, revealing deep links to algebraic geometry, arithmetic and combinatorics. There have been many exciting dual representations of amplitudes including*

*polytopes, twistor diagrams and the positive grassmannian leading to the concept of the amplituhedron,**polylogs, multizeta values and multiloop integrals,**…*

*Each of these disparate ideas and methods have their own remarkable successes, and indeed have led to important progress in related areas of mathematics. They also face obstacles that they need to overcome in order to become important tools in resolving the most important open conjectures in the field and for the explicit construction of amplitudes.*”

When considering operator inverses, one usually considers multiplicative inverses. As noted earlier in several entries, particularly, “Bernoulli and Blissard meet Stirling … ” (BBS), we see compositional inverse pairs of operators playing an important role in making associations among important integer arrays and combinatorics.

Using the arguments in BBS as a template, let

and , then, at any point satisfying the inverse relations,

.

And, if these relations are satisfied about the origin, i.e.,

, and , then these ratios serve as the e.g.f.s of the moments of the Appell sequences

and .

From Wikipedia on the LRT, with

,

.

Letting and ,

and , giving

,

the Lagrange inversion formula about the origin, whose expansion in the Taylor series coefficients of is discussed in OEIS A248927. See also A134685. For connections to free probability, free cumulants and moments, Appell sequences, noncrossing partitions, and other combinatorics, see A134264.

Let be the inverse of w.r.t. to . Then

, or

, and

.

The solution for the inverse of this last type is also presented in the post on the inviscid Burgers’ equation and the post Generators, Inversion, and Matrix, Binomial, and Integral Transforms.

The Laplace transform (LPT) argument in Appendix II of the Generators pdf can be briefly extended to derive the last form of the LRT above. (See below.)

Posted in Math
Tagged Compositional inverse, Lagrange inversion, Lagrange reversion theorem
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The Pincherle derivative is implicitly used in Eqn. 2.19 page 13 of “Mastering the master field” by Gopakumar and Gross. The raising and creation operators in the paper are analogous to those for a Laplace-dual Appell sequence, or Dirac-Appell sequence, comprised of the Dirac delta function and its derivatives, formed by taking the inverse Laplace transform of the polynomials of an Appell polynomial sequence.

The fundamental D-A sequence can be defined as the sequence with and and e.g.f. . Another example is provided by OEIS A099174 with the D-A sequence where are the modified Hermite polynomials listed in the Example section of the entry. The modified Hermite polynomials can be characterized several ways:

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Tagged Appell polynomial sequences, Appell sequences, Conjugation of operators, Creation and annihilation operators, Differential operators, Dirac delta function, Dirac-Appell sequence, Generalized Appell sequence, Inverse Laplace transform, Ladder operators, Modified Hermite polynomials, Operator calculus, Pincherle derivative, Raising and lowering operators, Umbral calculus
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