Science Talk

leox · Guest

For the series
P(t)=a_0+a_1 t+a_2\,t^2+.....+ a_n\,t^n+......
there exist a way to extract a subseries of the following form
P_k(t)=a_0+a_k\,t^k+a_{2k}\,t^{2k}+a_{3k}\,t^{3k}+...
To do it we should act on the series by the transformation

F_{k}(t^n)=1/k \sum_{i=0}^{k-1} (\zeta^i t)^n,

Here \zeta - primitive k-th root of unity.
As result of the transformation all terms t^n for n <> 0 mod k are
vanished and we will get exact the series P_k(t).

Question. How generalise the technigue for series of two and more
variables? For example, suppose we have the series
P(t,s)=\sum_{i,j\geq 0} a_{i,j} t^i s^j
and we want to extract the subseries of the form

P_{k,l}=a_{0,0}+a_{k,l} t^k s^l+a_{2k,2l} (t^k s^l)^2+a_{3k,3l} (t^k
s^l)^3+....
What kind of manipulations lead to solution?
Unfortunatelly, the composition of the trasformations F_{k} and F_{l}
( for every variables) doesnt work.