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# Category Archives: Math

## An Intriguing Tapestry: Number triangles, polytopes, Grassmannians, and scattering amplitudes

(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 … Continue reading

## Compositional Inverse Operators and Sheffer Sequences

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 … Continue reading

## The Lagrange Reversion Theorem and the Lagrange Inversion Formula

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 … Continue reading

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Tagged Compositional inverse, Lagrange inversion, Lagrange reversion theorem
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## Dirac-Appell Sequences

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, … Continue reading

Posted in Math
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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## The Pincherle Derivative and the Appell Raising Operator

The raising and lowering operators and for a sequence of functions , with and , defined by and have the commutator relation with respect to action on the space spanned by this sequence of functions. If for any particular natural number , … Continue reading