# Category Archives: Math

## More on Formal Group Laws, Binomial Sheffer Sequences, and Linearization Coefficients

A formula for computing the structure, or linearization, constants for reducing products of pairs of polynomials of a binomial Sheffer sequence, , is presented in terms of the umbral compositional inverses of the polynomials, . To say the pair are … Continue reading

## 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

## 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

## 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

## 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

## 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

## 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