Tag Archives: umbral compositional inverse

Composition, Conjugation, and the Umbral Calculus–Part I

Posted on November 20, 2021 by Tom Copeland

Relationships among a set of Appell and binomial Sheffer sequences derived from the function , its compositional inverse, and their multiplicative inverses are re-explored in the pdf below, and a simple formula for calculating the Bernoulli numbers from the Stirling … Continue reading →

Posted in Math | Tagged Abel polynomials, Appell matrices, Appell sequences, Bell polynomials, Bernoulli polynomials, Compositional inverse, Falling factorials, Functional composition, Functional iteration, Hyperbinomial polynomials, Lambert W-function, Matrix cojugation, Multiplicative inverse, Pascal matrix, Reciprocal polynomials, Sheffer matrices, Sheffer polynomials, Stirling polynomials of the first kind, Stirling polynomials of the second kind, Umbral calculus, umbral compositional inverse, Umbral conjugation | Leave a comment

Cycles and Heat: Hermite-Sheffer Evolution Equations

Posted on July 1, 2021 by Tom Copeland

The pdf below relates the basic (Graves-Pincherle-Lie-) Heisenberg-Weyl algebra to partial differential equations–evolution equations–defining the exponential generating functions (e.g.f.s) of sequences of functions that have associated ladder ops–a raising / creation op, , and a lowering / destruction / … Continue reading →

Posted in Math | Tagged Appell sequences, Baker-Campbell-Hausdorff-Dynkin expansion, Binomial Sheffer sequences, Chebyshev Hermite polynomials, Composition partition polynomials, Conjugation of operators, Creation and annihilation operators, Cycle index polynomials of symmetric group, Deformed diffusion equation, Disentangling operator relation, Evolution equations, Gaussian function, Heat equation, hypertetrahedrons, hypertriangles, Ladder operators, Laguerre polynomials, Lie-Heisenberg-Weyl group / algebra, n-simplices, normal ordering, Odd double factorials, Partial differential equations, Partition polynomials, Perfect matchings, Raising and lowering operators, Sheffer polynomials, Stirling partition polynomials of the first kind, Umbral calculus, umbral compositional inverse | Leave a comment

The Heat Equation, the Quantum Harmonic Oscillator, the Hermite Polynomials, and Gaussian Integral Transforms

Posted on February 26, 2021 by Tom Copeland

The Appell polynomial operator calculus of the Hermite polynomials provides a way to quickly derive, collate, and connect various properties of the Gaussian to important constructs in math and physics–the heat, or diffusion, equation; the quantum mechanical harmonic oscillator; the … Continue reading →

Posted in Math | Tagged Appell polynomial sequences, central limit theorem, Conjugation of operators, creation and annihilation / destruction operators, Cumulants, diffusion equation, Fourier transform, Gaussian distribution / probability density function, Heat equation, heat operator, Hermite polynomials, Ladder operators, Laplace transform, Lie-Heisenberg-Weyl group / algebra, Mellin transform, Mellin transform interpolation, quantum mechanical harmonic oscillator, Raising and lowering operators, Segal-Bargmann transform, umbral compositional inverse | 2 Comments

Appell Matrices

Posted on September 23, 2020 by Tom Copeland

A summary of observations recorded in diverse sites over the years (stemming from OEIS A133314) on the general matrix algebra of the coefficient matrices of Appell polynomial sequences and the connections of that algebra to umbral compositional inversion, binomial convolution, … Continue reading →

Posted in Math | Tagged Appell polynomial sequences, Convolution, Exponential generating functions, Infinigens, Infinitesimal generators, Matrix inverse, Matrix representations, Multiplicative inverse, Sheffer Appell polynomials, Umbral calculus, umbral compositional inverse | Leave a comment

Reciprocity and Umbral Witchcraft: An Eve with Stirling, Bernoulli, Archimedes, Euler, Laguerre, and Worpitzky

Posted on August 11, 2020 by Tom Copeland

Motivated by the appearance of the Eulerian polynomials in algebraic geometry, geometric combinatorics, and in some derivations of the Baker-Campbell-Hausdorff-Dynkin (BCHD) expansion, identities are generated using umbral Sheffer calculus couplings of the iconic inverse pair and , naturally relating the … Continue reading →

Posted in Math | Tagged Appell sequences, Bernoulli numbers, Bernoulli polynomials, Binomial Sheffer sequences, Compositional inverse, Differential operators, Eulerian numbers, Eulerian polynomials, f polynomials, face polynomials, Grassmannian, H polynomials, Multiplicative inverse, Permutahedra, Sheffer sequences, Stellahedra, Stirling numbers of the first kind, Stirling numbers of the second kind, Stirling polynomials of the first and second kinnds, Swiss knife polynomials, Umbral calculus, umbral compositional inverse, Umbral inverse pair, Worpitzky Identity | Leave a comment

Differintegral Ops and the Bernoulli and Reciprocal Polynomials

Posted on July 27, 2020 by Tom Copeland

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 →

Posted in Math | Tagged Appell polynomial sequences, Bernoulli numbers, Bernoulli polynomials, Commutator, Creation and annihilation operators, Pincherle derivative, Raising and lowering operators, Reciprocal polynomials, Riemann zeta function, Sheffer Appell polynomials, Umbral calculus, umbral compositional inverse | Leave a comment

Juggling Zeros in the Matrix (Example II)

Posted on July 15, 2020 by Tom Copeland

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)

Posted in Math | Tagged Appell polynomials, Diffferential operators, Exponential generating functions, Hermite polynomials, Inverse matrices, Matrices, Special functions, Umbral calculus, umbral compositional inverse | Leave a comment

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

Posted on January 1, 2020 by Tom Copeland

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 →

Posted in Math | Tagged Associahedra, Binomial Sheffer sequences, compositional partition polynomials, Formal group laws FGL, Lagrange inversion, linearization coefficients, products of Sheffer polynomials, refined Lah polynomials, umbral compositional inverse | Comments Off

Compositional Inverse Operators and Sheffer Sequences

Posted on November 6, 2016 by Tom Copeland

When considering operator inverses, one usually considers multiplicative inverses. As noted earlier in several entries, particularly, “Bernoulli and Blissard meet Stirling … ” (BBS), we see compositional inverse pairs of operators playing an important role in making associations among important … Continue reading →

Posted in Math | Tagged Appell polynomials, Bernoulli polynomials, Compositional inversion, Operator calculus, Sheffer polynomials, Stirling number polynomials, Umbral calculus, umbral compositional inverse, Umbral derivative | 2 Comments

Fractional calculus and interpolation of generalized binomial coefficients

Posted on August 15, 2015 by Tom Copeland

Draft Interpolation of the generalized binomial coefficients underlie the representation of a particular class of fractional differintegro operators by convolution integrals and Cauchy-like complex contour integrals.

Posted in Math | Tagged Bell polynomials, Differential operators, Fourier transform, Fractional calculus, Generalized binomial coefficients, Inverse Mellin transform, Mellin transform, Newton interpolation, Operator calculus, Sinc function interpolation, umbral compositional inverse | 2 Comments

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