Infinigens, the Pascal Triangle, and the Witt and Virasoro Algebras

Infinitesimal Generators, the Pascal Pyramid, and the Witt and Virasoro Algebras: A short set of notes sketching some relationships among infinitesimal generators represented as differential operators and infinite-dimensional matrices, the Pascal triangle / pyramid, conformal transformations, the Witt and Virasoro algebras, the Hermite and generalized Laguerre polynomials of order , the Dedekind eta function, and combinatorics of the underlying integer matrices, inspired by Alexander Givental’s presentation of Virasoro operators in “Gromov-Witten invariants and quantization of quadratic Hamiltonians.” Some potential connections to knots, flows, dynamics, and Poincare-Maass series are also mentioned.

Depressed Equations and Generalized Catalan Numbers

Discriminating Deltas, Depressed Equations, and Generalized Catalan Numbers is a set of notes on the the relation of generating functions of the generalized Catalan numbers, e.g., OEIS-A001764, to the compositional inverse of and the tangent envelope of associated discriminant curves.

Riemann’s Jump Function J(x) for the Primes

Dirac’s Delta Function and Riemann’s Jump Function J(x) for the Primespresents Riemann’s jump fct., as introduced in H. M. Edward’s Riemann’s Zeta Function (Dover, 2001), couched in terms of the Dirac delta function and the inverse Mellin transform.

A Generalized Dobinski Relation and the Confluent Hypergeometric Fcts.

The Inverse Mellin Transform, Bell Polynomials, a Generalized Dobinski Relation, and the Confluent Hypergeometric Functions   presents a generalized Dobinski relation umbrally incorporating the Bell / Touchard / Exponential polynomials that is defined operationally through the action of the operator  f(x d/dx) on a modified inverse Mellin transform. Relations to the Dirac delta function/operator and, through an appropriate choice of f, the confluent hypergeometric functions, one set of which are the generalized Laguerre functions, are sketched and finally some exercises presented.

Note on the Inverse Mellin Transform and the Dirac Delta Function.

The use of the Dirac Delta Function allows simple derivations of many common Mellin Transforms. The Mellin Transform and the Dirac Delta Function is a quick note that addresses a question posed on MathOverflow.

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 operator trees–the

1. Bell / Bruno / Touchard partition polynomials
2. Lah / Laguerre partition polynomials  (order -1)
3. Stirling / cycle index polynomials (#s of the first kind / symmetric group)
4. Lagrange partition polynomials

Addendum to “Mathemagical Forests”

Addendum to Mathemagical Forests presents Sheffer polynomials for the generators of the infinite-dimensional Witt Lie algebra discussed in the paper “Mathemagical Forests”.