I've been doing some work in SAGE with a linear operator on a polynomial algebra T: F[x] --> F[x] (here F is some field, x is a variable, and T is the operator), storing the data of T as a list, so that T[n] keeps track of T(x^n). Here n is a nonnegative integer, of course. This has been worked pretty well for me.
Now I would like to do similar computations in a space where x is invertible. That is, I now have a linear operator T: F[x,x^{-1}] --> F[x, x^{-1}]. I would love to store the data of T as a signed-integer-indexed list, so that T[n] again keeps track of T(x^n), but now n can be any (positive or negative) integer.
Is there a way to implement this easily in Python/SAGE? Right now I am dragging T around as a pair of lists, T1[n] := T(x^n) and T2[n] := T(x^{-n}). This is annoying and inelegant and inefficient (I constantly have to take three cases -- n positive, negative, or zero), but it does work. But is there something better out there?
I am not a good programmer in any sense, and I won't have time to work very hard on this. But I hope that I am missing something clean and simple. Thanks in advance for any help.
