Generating polynomials for a multistep method

[dynamicskillingme]

Hi, I'm struggling to understand how the generating polynomials work and are implemented in the difference equation for a general ODE y' = f(t,y)
Difference Equation

Generating polynomials

"Coefficients are normalized either by a_k = 1 or sigma(1) = 1

 

[Simon Bridge]

You can gain understanding by doing examples: lots and lots of examples.
Do you have a specific question?

 

[dynamicskillingme]

Sorry if I wasn't clear enough, I don't understand the concept behind the generating polynomials. My notes state the examples of the three theta methods but I can't understand how they are obtained (e.g. implicit euler is sigma(w) = w )

 

[Simon Bridge]

You notes just say there are these things called generating polynomials that can be used t help solve ODE's but does not tell you what they generate or how the method is motivated?

Have you seen:
https://en.wikipedia.org/wiki/Generating_function