Category Archives: Math

A Diorama of the Digamma

(Under construction) This series is divergent, so we may be able to do something with it. — Heaviside The divergent series for the pole of the Riemann zeta function is Lets’s use Mellin transform interpolation (essentially the master’s (Ramanujan) master … Continue reading

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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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Jumpin’ Riemann !_ !__ !___ ! !_____ ! Mangoldt–da mon–got it !___!_!

The magic of Mangoldt summoning Riemann’s miraculous miniscules-the nontrivial zeros. In response to observations initiated by Matt McIrvin of a sum of exponentials of the imaginary part of the non-trivial zeroes of the Riemann zeta function, assuming the Riemann hypothesis … Continue reading

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In the Realm of Shadows: Umbral inverses and associahedra, noncrossing partitions, symmetric polynomials, and similarity transforms

In the earlier post Compositional Inverse Operators and Sheffer Sequences, I constructed relations among a generic power series, call it , or ordinary generating function (o.g.f.), its compositional inverse and four sets of Sheffer polynomial sequences–two Appell sequences and and … Continue reading

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Squaring Triangles

This post illustrates what Feynman praised as a beautiful facet of mathematics–abstraction from the concrete–as well as the fascinating synergy at one of its crossroads–that of algebra and enumerative geometry. One day last fall in a class, several curious 12-th … Continue reading

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Commutators, matrices and an identity of Copeland

The arXiv “Commutators, matrices and an identity of Copeland” by Darij Grinberg proves and extends an identity I proposed for a matrix computation of the partition polynomials generated by iterated multiplication of a tangent vectorwhere and is a function or … 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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