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Suppose that I have two sequences $(a_0,a_1,a_2,\dots)$ and $(b_0,b_1,b_2,\dots)$ and the corresponding generating functions $A(x)=a_0+a_1x+a_2x^2+\dots$ and $B(x)=b_0+b_1x+b_2x^2+\dots$. Letting $c_i=\frac{a_i}{b_i}$, is there a way to retrieve the generating function for the sequence $c_0,c_1,c_2,\dots$?

GaussJordan
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    IMHO, there is no connection between generating functions of the $a_n$ and the $b_n$ on one side and the $c_n=a_n/b_n$ on the other side. – Jean Marie May 19 '23 at 11:45
  • Related but not quite a duplicate https://math.stackexchange.com/questions/4744/formal-power-series-coefficient-multiplication – ronno May 19 '23 at 12:12

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