Cycles and Heat: Hermite-Sheffer Evolution Equations

 

The pdf below relates the basic (Graves-Pincherle-Lie-) Heisenberg-Weyl algebra to partial differential equations–evolution equations–defining the exponential generating functions (e.g.f.s) of sequences of functions that have associated ladder ops–a raising / creation op, , and a lowering / destruction / annihilation op, . Such ops are an integral component of quantum theory. The probabilist’s, or Chebyshev, family of Hermite polynomials, whose moments–the aerated odd double factorials of matching theory–are those of a Gaussian function, play a central role in the analysis. 

The ladder ops of the Sheffer polynomial sequences, with its subgroups of Appell and binomial sequences, are developed along with other aspects of the umbral operator calculus characterizing these sequences. These ladder ops form the spatial component of the Hermite-Sheffer evolution equations. 

These evolution equations are then used to construct the heat / diffusion equation on the real line, deformed versions of the equation, and their solutions,  presented as e.g.f.s of the Hermite polynomials composed with the cycle index polynomials of the symmetric groups , a.k.a. the Stirling partition polynomials of the first kind.

At the heart of the analysis is the generalized raising op , which is shown to reduce to the raising op of the Hermite polynomials under appropriate conjugation with differential shift ops containing the Stirling partition polynomials. These two ops are characterized several ways, analytically and combinatorially.

Cycles and Heat: Hermite-Sheffer Evolution Equations (pdf)

This is chiefly related to OEIS A344678.

Added 2/22/22:

While reviewing some material in “The Theory of Linear Operators” by Davis (1936), I came across a section from pp. 194-200 on powers of the the sum of two noncommuting operators, referencing even older material beginning with Charles Graves. He does not mention even earlier work by Scherk