Author Archives: Tom Copeland

Differintegral Ops and the Bernoulli and Reciprocal Polynomials

A short pdf on differintegral operators that generate the Bernoulli polynomials and their elegant consorts the Reciprocal polynomials, which form an inverse pair under umbral composition (mostly reprising notes in earlier posts): Differintegral Ops and the Bernoulli and Reciprocal Polynomials … Continue reading

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Juggling Zeros in the Matrix (Example II)

This is a sequel to my last post Skipping over Dimensions, Juggling Zeros in the Matrix with a second example: Laguerre polynomials of order  -1/2,  OEIS A176230. Juggling Zeros in the Matrix (Example II)  

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Skipping over Dimensions, Juggling Zeros in the Matrix

Slipping between dimensions can often simplify a problem. A fundamental example is side-stepping off the real line to the complex plane to find the zeros of a polynomial. In the cases I have in mind, this jump amounts to aerating … Continue reading

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The Faber Appells

The Newton-Waring-Girard identities are core constructs in the theory of symmetric polynomials/functions (see the Newton’s Identities entry of Wikipedia for a compilation of some symmetric polynomials and Gould for some history). One identity involves the Faber partition polynomials of OEIS … Continue reading

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Mellin Convolution for Generalized Hadamard Product of Functions/Power Series

From representations of umbral substitution by differential operators, this note derives an inverse Mellin transform and its associated Mellin convolution giving a generalized Hadamard product of two functions or power series. Mellin convolution for generalized Hadamard product of functions/power series

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Jumpin’ Riemann!…..!..!.!.Mangoldt–da mon–got it!….!..!

The magic of Mangoldt summoning Riemann’s miraculous miniscules-the nontrivial zeros. (Originally published in Sept. 2019. Inadvertanly deleted in April) In response to observations initiated by Matt McIrvin of a sum of exponentials of the imaginary part of the non-trivial zeroes … Continue reading

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Hacking Reality [Official Film]

Fun, well-done, elementary intro to E8 in physics (and dynamics and propaganda of the physics/math/Internet community). A variation on the motif Shadows of Simplicity.

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Witt-Lie algebra, Associated Groups, and the Right and Left Generalized Factorials

With , the Witt vectors when exponentiated give rise to the action for . (Cf. the MathOverflow question Motivation of the Virasoro algebra.) The function represents a Lie group under composition with respect to the parameter . That is, with … Continue reading

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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

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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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