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# Tag Archives: Compositional inverse

## The Lagrange Reversion Theorem and the Lagrange Inversion Formula

From Wikipedia on the LRT, with , . Letting and , and , giving , the Lagrange inversion formula about the origin, whose expansion in the Taylor series coefficients of is discussed in OEIS A248927. See also A134685. For connections … Continue reading

Posted in Math
Tagged Compositional inverse, Lagrange inversion, Lagrange reversion theorem
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## The Elliptic Lie Triad: KdV and Riccati Equations, Infinigens, and Elliptic Genera

The Elliptic Lie Triad: Riccati and KdV Equations, Infinigens, and Elliptic Genera (This site was not correctly updating, so the notes were transcribed to this pdf.) Addendum to The Elliptic Lie Triad

Posted in Math, Uncategorized
Tagged Bernoulli numbers, Burgers' equation, Chebyshev polynomials, Combinatorics, Compositional inverse, Differential operators, Elliptic curves, Elliptic formal group law, Elliptic genera, Enumerative combinatorics, Euler numbers, Eulerian numbers, Evolution equations, Faber polynomials, Geodesics of Virasoro-Bott group, Grassmann polynomials, Grassmannian, Heat equation, Hyperbolic tangent, Infinigens, Infinitesimal generators, KdV, Korteweg-de Vries equation, L-genus, Lie algebra, Lucas (Cardan) polynomials, Refined Eulerian numbers, Riccati, Schwarzian derivative, SL2, Solitons, Swiss knife polynomials, Viscous Burgers equation
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## Appell polynomials, cumulants, noncrossing partitions, Dyck lattice paths, and inversion

The raising op for any Appell sequence is determined by the derivative of the log of the e.g.f. of the basic number sequence, connecting the op to the combinatorics of the cumulant expansion OEIS-127671 of the moment generating function and … Continue reading

Posted in Math
Tagged Appell sequences, Associahedra, Bernoulli, Bernoulli polynomials, Catalan numbers, Compositional inverse, Cumulants, Dyck lattce paths, Eulerian numbers, Free cumulants, Free probability, Hirzebruch Todd class criterion, Lagrange inversion, Noncrossing partitions, Permutohedra, Raising operators, Riemann zeta
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## Bernoulli, Blissard, and Lie meet Stirling and the simplices: State number operators and normal ordering

A set of identities that encapsulates relations among the Bernoulli numbers, the Stirling numbers of the first and second kinds, and operators related to the umbral calculus of Blissard and his contemporaries: Decoding:

Posted in Math
Tagged Appell sequences, Bernoulli polynomials, Compositional inverse, Conjugation and derivation, Differential operators, Finite difference operator, formal group laws, Hurwitz zeta function, Lie derivatives, Lowering operators, Multiplicative inverse, Raising operators, Riemann zeta function, Sheffer sequences, Stirling numbers, Umbral calculus

## Compositional Inverse Pairs, the Inviscid Burgers-Hopf Equation, and the Stasheff Associahedra

Compositional Inverse Pairs, the Inviscid Burgers-Hopf Equation, and the Stasheff Associahedra: A brief note (pdf) on some relations among these topics, including the Catalan and Fuss-Catalan numbers. References are provided linking the analysis with the distribution of eigenvalues of random … Continue reading

## Lagrange à la Lah

Lagrange à la Lah Part I and Lagrange à la Lah Part II are a set of notes on partition polynomials derived from binomial Sheffer sequences via umbral refinement, their relation to compositional inversion via the Laplace transform, and their characterization by umbral … Continue reading

Posted in Math
Tagged Bell polynomials, Cayley trees, Composition, Compositional inverse, Differential operators, Faa di Bruno formula, Falling factorial, Forests, Generalized shift operator, Generlized Taylor series, Lagrange inversion, Lagrange partition polynomials, Lah polynomials, Operator calculus, Partition polynomials, Rising factorial, Sheffer sequences, Special functions, Special polynomials, Stirling numbers, Stirling polynomials, Tree graphs, Trees, Umbral calculus, Umbral operator trees
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## Flipping Functions with Permutohedra

Flipping Functions with Permutohedra : A short note on forming the multiplicative and compositional inverses of functions using the refined Eulerian h-polynomials OEIS-A145271 and refined face polynomials of permutohedra OEIS-A049019.

## Short Note on Lagrange Inversion

Short Note on Lagrange Inversion is an addendum to OEIS-A134685.