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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A generalized Taylor series operator: Umbral shift, binomial transform, and interpolation

A generalized Taylor series operator represented as two different infinite sums of differential operators related by a binomial transform can provide some intuition on umbral substitution and interpolation of umbral coefficients. (These notes reprise those in other earlier posts.) Consider … Continue reading

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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

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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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Witt Differential Generator for Special Jack Symmetric Functions / Polynomials

Exploring some relations among the multinomial coefficients of OEIS A036038 and the compositional inversion formulas of A134264, A248120, and A248927, related to numerous combinatorial structures and areas of analysis, including noncrossing partitions and free probability,  I came across the Jack … Continue reading

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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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Generators, Inversion, and Matrix, Binomial, and Integral Transforms

Generators, Inversion, and Matrix, Binomial, and Integral Transforms is a belated set of notes (pdf) on a derivation of a generating function for the row polynomials of  OEIS-A111999 from its relation to the compositional inversion (a Lagrange inversion formula, LIF) presented … Continue reading

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