Tag Archives: Lagrange inversion

An lnfinite Wronskian Matrix, Binomial Sheffer Polynomials, and the Lagrange Reversion Theorem

Form the infinite Wronskian matrix with elements . A generating function for this matrix is with . If , then also , where is a binomial Sheffer sequence of polynomials. Then in this particular case, and so is the product … Continue reading

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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Generators, Inversion, and Matrix, Binomial, and Integral Transforms

Generators, Inversion, and Matrix, Binomial, and Integral Transforms is a belated set of notes (pdf) on a derivation of a generating function for the row polynomials of  OEIS-A111999 from its relation to the compositional inversion (a Lagrange inversion formula, LIF) presented … Continue reading

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

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