The algebra for formal power series in elements of the Weyl algebra.
In particular it allows Taylor expansions of expressions like exp(z*(a^2-a'^2)/2) where a and a' don't commute and [a,a']=1.
- Example0: Wick's theorem and wiring diagrams
- Example1: Bernoulli numbers and Euler-Maclaurin summation
- Example2: Umbral calculus with Laguerre and Hermite polynomials
- Example3: Example from QM - properties of squeezed states
- Example4: Investigating various rational functions of X and D
Many of these results can be explained by "Combinatorial Models of Creation-Annihilation", Blasiak & Flajolet https://arxiv.org/abs/1010.0354
This is a completed project, but incomplete code. The code isn't perfect but it allowed me to perform certain computations I wanted to experiment with.
Some things that could be improved
- format of output
- division is (of course) a partial function. In this case we have left-division and right-division and I haven't made any effort to characterise when exactly a division is valid.
- Showing two series share the first few coefficients isn't a proof that the series are equal. Except it is(!!!) if you can prove some other properties first. Maybe one day I'll do this. See https://en.wikipedia.org/wiki/Holonomic_function