I have two functions $A(x), B(x)$, given in a special power series form:
$A(x)=1-x^{2}\left(\frac{a}{10}-\sum_{k=1}^{9}b_{k}\left(\frac{(x^{2}-1)}{r}\right)^{k}\right)$
$B(x)=1-x^{2}\left(\frac{c}{10}-\sum_{k=1}^{9}d_{k}\left(\frac{(x^{2}-1)}{r}\right)^{k}\right)$
where $a,b_{k},c,d_{k}$ are real numbers, and $r\in\mathbb{R}^{+}$.
What I want to do, is to find the composition of these series $A(B(x))$ in a similar form, i.e. find such $u=u(a,b_{k},c,d_{k},r)$ and $v_{k}=v_{k}(a,b_{k},c,d_{k},r)$ so that
$A(B(x))=1-x^{2}\left(\frac{u}{10}-\sum_{k}v_{k}\left(\frac{(x^{2}-1)}{r}\right)^{k}\right)$
I tried defining these series by brute force:
A[x_] := 1 - x^2*((a/10) + Sum[b[k]*((x^2 - 1)/r)^k, {k, 1, 9}]);
B[x_] := 1 - x^2*((c/10) + Sum[d[k]*((x^2 - 1)/r)^k, {k, 1, 9}]);
And then calling A[B[x]]//Expand
but it took forever long to run. Is there an efficient way to get an explicit power series expansion for A[B[x]]
?
Then I can just get the desired form by simple equating coefficients, but what my question is: how to get an output for A[B[x]]
in the standard power series form?
Thanks! (this is not homework)