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# Tag Archives: Operator calculus

## 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

## 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

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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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## Fractional calculus and interpolation of generalized binomial coefficients

Draft Interpolation of the generalized binomial coefficients underlie the representation of a particular class of fractional differintegro operators by convolution integrals and Cauchy-like complex contour integrals.

## Goin’ with the Flow: Logarithm of the Derivative

Goin’ with the Flow: Logarithm of the Derivative Operator is a pdf set of notes under construction on the relations between the commutator of the logarithm of the derivative operator, the Pincherle derivative, Lie operator derivatives, and the two umbrally inverse … Continue reading

Posted in Math
Tagged Bell polynomials, Binomial Sheffer sequences, Commutatator, Confluent hypergeometric functions, Differential operators, Factorial polynomials, Falling factorial, Flow equations, Fractional calculus, Generalized Laguerre functions, Inversion, Laguerre polynomials, Lie derivatives, Logarithm of the derivative operator, Matrix representations, Operator calculus, Pincherle derivative, Rising factorial, Sheffer sequences, Simplices, Stirling numbers, Vandermonde matrix
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## Fractional Calculus, Interpolation, and Traveling Waves

Fractional Calculus, Interpolation, and Traveling Waves: A note (pdf file) on the fractional integro-derivative (FID) regarded as an interpolation of the integral derivatives of the function acted upon–to be more precise, a sinc function interpolation, as in the Whittaker-Shannon interpolation … Continue reading

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Tagged Fractional calculus, Interpolation of operators, Operator calculus
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## Infinigens, the Pascal Triangle, and the Witt and Virasoro Algebras

Infinitesimal Generators, the Pascal Pyramid, and the Witt and Virasoro Algebras: A short set of notes sketching some relationships among infinitesimal generators represented as differential operators and infinite-dimensional matrices, the Pascal triangle / pyramid, conformal transformations, the Witt and Virasoro … Continue reading

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Tagged Associated Laguerre polynomials, Dedekind eta function, Infinitesimal generators, Iterated derivatives, Lie groups and algebra, Lie Witt algebra, Linear fractional transformations, Lower triangular matrices, Mobius transformations, Number triangles, Operator calculus, Pascal triangle, SL2, Vector fields
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## Lagrange à la Lah

Lagrange à la Lah Part I and Lagrange à la Lah Part II are a set of notes on partition polynomials derived from binomial Sheffer sequences via umbral refinement, their relation to compositional inversion via the Laplace transform, and their characterization by umbral … Continue reading

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Tagged Bell polynomials, Cayley trees, Composition, Compositional inverse, Differential operators, Faa di Bruno formula, Falling factorial, Forests, Generalized shift operator, Generlized Taylor series, Lagrange inversion, Lagrange partition polynomials, Lah polynomials, Operator calculus, Partition polynomials, Rising factorial, Sheffer sequences, Special functions, Special polynomials, Stirling numbers, Stirling polynomials, Tree graphs, Trees, Umbral calculus, Umbral operator trees
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