# Tag Archives: Stirling numbers

## 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

## 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:

## Goin’ with the Flow: Logarithm of the Derivative

Goin’ with the Flow: Logarithm of the Derivative Operator is a pdf set of notes under construction on the relations between the commutator of the logarithm of the derivative operator, the Pincherle derivative, Lie operator derivatives, and the two umbrally inverse … 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