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# Tag Archives: Raising and lowering operators

## Differintegral Ops and the Bernoulli and Reciprocal Polynomials

A short pdf on differintegral operators that generate the Bernoulli polynomials and their elegant consorts the Reciprocal polynomials, which form an inverse pair under umbral composition (mostly reprising notes in earlier posts): Differintegral Ops and the Bernoulli and Reciprocal Polynomials … Continue reading

Posted in Math
Tagged Appell polynomial sequences, Bernoulli numbers, Bernoulli polynomials, Commutator, Creation and annihilation operators, Pincherle derivative, Raising and lowering operators, Reciprocal polynomials, Riemann zeta function, Sheffer Appell polynomials, Umbral calculus, umbral compositional inverse
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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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## Fractional Calculus, Gamma Classes, the Riemann Zeta Function, and an Appell Pair of Sequences

The background info and comments for the MSE question Lie group heuristics for a raising operator for and the MO question Riemann zeta function at positive integers and an Appell sequence of poylnomials introduce an Appell sequence of polynomials containing … Continue reading

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Tagged Appell sequences, Associated Laguerre polynomials, Bell polynomials, Confluent hypergeometric functions, Convolution operators, Creation and annihilation operators, Cycle index polynomials, Differential operators, Digamma function, Falling factorials, Fractional calculus, Gamma classes, Gamma genus, Infinitesimal generators, Inverse Mellin transform, Mellin transform, Psi function, Raising and lowering operators, Riemann zeta function, Rising factorials, Umbral calculus, Umbral compositional inverse pair
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## Mathemagical Forests

The set of notes Mathemagical Forests is an expansion of the May notes and discusses some connections between rooted trees, derivative operators, Lagrange inversion, the Legendre transformation, the Faa di Bruno formula, Sheffer sequences and umbral calculus, and the infinite dimensional Witt … Continue reading

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Tagged Cayley, Comtet, Connes-Muscovi, Creation and annihilation operators, Eulerian numbers, Faa di Bruno formula, integer arrays, Iterated derivatives, Lagrange inversion, Legendre transformation, Lie Witt algebra, Number triangles, Operator ordering, Raising and lowering operators, Rooted trees, Sheffer sequences, triangular matrices, Umbral calculus
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