# An unknown operation on sequences

Given two sequences $a_n$ and $b_n$ I want to compute a third sequence $c_n$. It is a mathematical operation. The first three terms look like this:

$c_ 1 = {\frac {a_{{1}}b_{{0}}}{b_{{1}}a_{{0}}}}$

$c_2 = {\frac {a_{{2}}b_{{0}}}{b_{{2}}a_{{0}}}}-\frac12 {\frac {{a_{{1}}}^{2}{b_{{0}}}^{2}}{{b_{{1}}}^{2}{a_{{0}}}^{2}}}$

$c_3 = {\frac {a_{{3}}b_{{0}}}{b_{{3}}a_{{0}}}} - \frac13 {\frac {a_{{1}}{b_{{0}}}^{2}a_{{2}}}{b_{{1}}{a_{{0}}}^{2}b_{{2}}}} -\frac13 {\frac {{b_{{0}}}^{2} \left( 2a_{{2}}a_{{0}}{b_{{1}}}^{2} -{a_{{1}}}^{2}b_{{0}}b_{{2}} \right) a_{{1}}}{b_{{2}}{a_{{0}}}^{3}{b_{{1}}}^{3}}}$

These formulas are given by Maple in symbolic form after a longer computation. I assume that they can be generated by a simple algorithm. Yet I was unable to identify this algorithm from the formulas nor do I know what Maple really did. Perhaps these are well known formulas and someone can recognize them?

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But what did you ask Maple to do? –  Henning Makholm Dec 15 '12 at 20:37
@Henning I would like to understand the operation on purely formal grounds, say comparable to the convolution product; therefore the genesis seems irrelevant here. Moreover it would be too complicated to explain here. –  Julia Primes Dec 15 '12 at 20:50
@Julia: It's not irrelevant, for two different reasons. First, it might help us find the pattern in the $c_i$, and second, there are an infinity of algorithms that will produce these three coefficients but differ on subsequent coefficients, and it may not be obvious which of them is the most "simple", so the question isn't well-defined unless you specify the entire sequence. –  joriki Dec 16 '12 at 8:07