Tag Archives: Umbral calculus

Formal group laws and binomial Sheffer sequences

Given a compositional inverse pair and , i.e., , with with , , and ,  construct the binomial Sheffer sequence with the exponential generating function . Then the associated formal group law (FGL) may be expressed as Advertisements

Posted in Math | Tagged , , , , , , , , , , , , , | 1 Comment

Compositional Inverse Operators and Sheffer Sequences

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 , , , , , , , , | Leave a comment

Dirac-Appell Sequences

The Pincherle derivative  is implicitly used in Eqn. 2.19 page 13 of “Mastering the master field” by Gopakumar and Gross. The raising and creation operators in the paper are analogous to those for a Laplace-dual Appell sequence, or Dirac-Appell sequence, … Continue reading

Posted in Math | Tagged , , , , , , , , , , , , , , | 1 Comment

Fractional Calculus, Gamma Classes, the Riemann Zeta Function, and an Appell Pair of Sequences

The background info and comments for the MSE question Lie group heuristics for a raising operator for and the MO question Riemann zeta function at positive integers and an Appell sequence of poylnomials introduce an Appell sequence of polynomials containing … Continue reading

Posted in Math | Tagged , , , , , , , , , , , , , , , , , , , , , | Leave a comment

A Class of Differential Operators and the Stirling Numbers

The differential operator with can easily be expanded in terms of the operators by considering its action on

Posted in Math | Tagged , , , , , , , , , , , , , , , | Leave a comment

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 , , , , , , , , , , , , , , ,

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 , , , , , , , , , , , , , , , , , , , , , , , , | Leave a comment