Hm, I do not even know the best formulation for my question in the header. It is not for the math but for the proper writing/terminology. I've come across the term "base change" recently but the context was a bit different, so I don't know whether this is possibly correct/incorrect here.
As I've said in the subject, I have two expressions of the same function; one time I express it as a power series in x, say $\small f(x) = a + bx + cx^2+dx^3 + ... $, and in a certain article I find the same problem handled, but with a formula like $\small g(x)=A + B(1-x) + C(1-x)(2-x) + D(1-x)(2-x)(3-x) + ... $ (the actual coefficients don't matter here)
If I expand $\small g(x) $ and collect like powers of x to make a power series of it, it is expected, that that power series has the same coefficients as f(x) (or: "they are identical"). (There is a problem in it, that the expansion leads to divergent sums for x but that need not be discussed here). Let's assume, I'm correct and the series come out to be identical. Btw, I know that the transformation behind this involves the Stirling numbers 1st kind.
My question is: how do I write in a small article, that the series f(x) is expressible by g(x) and vice versa? Perhaps "g(x) is a Stirling-transformation on f(x)" ? or "We do a change-of-basis from f(x) to g(x)" ?