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# Tag Archives: Differential operators

## The Riemann Zeta and the Calculus

(Under construction: Reprising investigations over several years.) By virtue of the relation between the values of the Riemann zeta function at the negative integers, , and the Bernoulli numbers and between the Bernoulli polynomials and the partial sums of the … Continue reading

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Tagged annihilation operator, Bernoulli numbers, Bernoulli polynomials, Beta integral, complex Cauchy contour integral, Confluent hypergeometric functions, convolution integral, creation operators, Differential operators, Digamma function, Dirac delta function, Fractional calculus, gamma function, Generalized Laguerre functions, generalized Laguerre polynomials, harmonic numbers, infinigen, infinitesimal generator, lowering operator, Mellin convolution, Mellin transform, operational calculus, Raising operators, Riemann zeta function, Sheffer Appell polynomials
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## Pre-Lie algebras, Cayley’s analytic trees, and mathemagical forests

Referring to week 299 of John Baez’s old blog or the Pre-Lie Algebra entry of nLab, a left pre-Lie algebra satisfies the associative relation, (AR), . To see the relation to Cayley’s work of 1857 as described in my pdf … Continue reading

## Formal group laws and binomial Sheffer sequences

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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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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## 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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