I have started a sandbox page:
http://axiom-wiki.newsynthesis.org/SandBoxPolynomialSequencesMatrix
In order to avoid boredom skip down to "Examples" (i.e. ^F Examples).
The material above that the is just preliminary matrix operations that
seem to be missing. People are welcome to improve the routines but
please leave them obvious!
This is intended to illustrate how to generate coefficient matrices for
polynomial sequences directly from the Generating Function via. a matrix
interpretation of the Generating function.
The base ideas come from generating functions like
f(t)*exp(x*g(t))
From Roman's "Umbral Calculus" and
The Matrices of Pascal and Other Greats
Lidia Aceto & Donato Trigiante
http://www.academia.edu/22095557/The_Matrices_of_Pascal_and_Other_Greats
A, somewhat tedious, introduction to the mathematics is at:
https://www.dropbox.com/sh/i2f7lehirme848p/AAA3jUgIkLNshPK88HuOR4MJa?dl=0
H-generating-9.pdf
If you are knowledgeable, skip to Theorem 6 for anything new (?). It's
titled for Sheffer Sequences but Theorem 9 illustrates the extension to
things like:
(1 + G(t) − x · K(t))^a
The techniques makes a lot proven (and reproven (and reproven...))
polynomial sequence properties obvious and uniform. But I haven't
organized my thoughts well enough to put up.
I will be continuing some additions and editing as time goes on (and if
I feel like it). Anybody having a request or comment please let me know.
Ray
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