In RobinHankin/weyl: The Weyl Algebra

set.seed(0)
knitr::opts_chunk$set(echo = TRUE)
library("weyl")
library("multicool")
options("digits" = 5)

![](`r system.file("help/figures/weyl.png", package = "weyl")`){width=10%}

This short document shows how Stirling numbers appear when working with the Weyl algebra. Stirling numbers of the second kind, written $S(n,k)$, count the number of equivalence relations on a set of $n$ elements having precisely $k$ equivalence classes.

Using Weyl algebra terminology, we consider powers of $x\partial$, that is $(x\partial)^i$ for $i=1,2,\ldots$. Taking $i=2$ as an example, we have $(x\partial)^2=(x\partial)(x\partial)=x(\partial x)\partial=x(x\partial+1)\partial=x^2\partial^2+x\partial$. It can be proved (but the margins are too small) that $(x\partial)^n=\sum_{i=1}^nS(n,k)\partial^kx^k$. Taking $n=7$ as an example we have

(xd7 <- (x*d)^7)
options(polyform = TRUE)
xd7

and it is possible to decompose this expression as follows:

f <- function(w){
   jj <- numeric(nterms(w)) 
   jj[index(w)] <- coeffs(w)
   jj
}
f(xd7)

We can compare this with results from the multicool package:

 multicool::Stirling2(7,1:7)

giving identical results.

RobinHankin/weyl documentation built on April 14, 2025, 11:49 a.m.