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# Tag Archives: Lagrange inversion

## 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

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Tagged Compositional inverse, Lagrange inversion, Lagrange reversion theorem
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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

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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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## The Hirzebruch criterion for the Todd class

The Hirzebruch criterion for the Todd class is given in “The signature theorem: reminiscences and recreations” by Hirzebruch. The formal power series that defines the Todd class must satisfy . The e.g.f. for the Bernoulli numbers uniquely satisfies this criterion. … Continue reading

## The Bernoulli polynomials and Hirzebruch’s generalized Todd class

Let’s connect the Bernoullis, using their basic operational definition rather than their e.g.f., to the Todd genus and more through formal group laws (FGL, see note at bottom) and associated Lie ops and, therefore, compositional inversion. [This is done through … Continue reading

## Bernoulli Appells

The defining characteristic of the Bernoulli numbers operationally is that they are the basis of the unique Appell sequence, the Bernoulli polynomials, that “translate” simply under the generalized binomial transform (Appell property) and satisfy (for an analytic function, such as … Continue reading

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Tagged Appell sequences, Cycle index polynomials, Differential operators, Faulhaber's formula, formal group laws, Hurwitz zeta function, Lagrange inversion, Lah polynomials, Mellin transform, Pincherle derivative, Raising operators, Riemann zeta function, Sheffer sequences, Simplices, Stirling numbers, Symmetric polynomials, umbral compositional inverse
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