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# 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

Posted in Math
Tagged Digamma function, Fractional calculus, gamma function, generalized harmonic numbers, harmonic numbers, infinigen, infinitesimal generator, Inverse Mellin transform, logarithmic derivative, Mellin integral transform, Mellin interpolation, Newton interpolation, power sum polynomials, reciprocal integers, Riemann zeta function, Rising factorial, Taylor series
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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

Posted in Math
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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## 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

Posted in Math
Tagged Chebyshev function, Digamma function, Dirac delta function, Euler's product formula. Hadamard's product formula, Fourier transform, Inverse Mellin transform, Landau Xi function, logarithmic derivative, Mangoldt function, Mangoldt summatory function, Riemann jump function, Riemann Xi function, Riemann zeta function nontrivial zeros, Riemann's prime counting function
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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

Posted in Math
Tagged Appell polynomial sequences, Associahedra, binomial Sheffer polynomial sequences, complete homogeneous symmetric polynomials, Compositional inversion, conjugation, elementary symmetric polynomials, matrix inverses, multiplicative inversion, Noncrossing partitions, refined Lah polynomials, similarity transformation, Stasheff polytopes, Symmetric polynomials, Umbral composition, Umbral inverse
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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

## 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

Posted in Math
Tagged algebraic combinatorics, Algebraic geometry, Appell polynomial sequences, Associahedra, Butcher series, Combinatorics, commutators, Compositional inversion, convex polytopes, Dyck paths, enumerative geometry, Evolution equations, free cumulants and moments, Free probability, generalized permutahedra, Hirzebruch criterion, Hopf algebra of monoids, inviscid Burgers' equation, iterated convolutions, Kdv equation, Koszul duality, matices, Noncrossing partitions, phylogenetic trees, polygon dissections, Pre-Lie algebra, rooted Cayley trees, Todd class
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