Reversion of Power Series

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Theorem

Let $\displaystyle y = \sum_{n \mathop = 1}^\infty c_n x^n$ be a power series.


Then:

$\displaystyle x = \sum_{n \mathop = 1}^\infty C_n y^n$

is also a power series, where:

\(\displaystyle c_1 C_1\) \(=\) \(\displaystyle 1\)
\(\displaystyle {c_1}^3 C_2\) \(=\) \(\displaystyle -c_2\)
\(\displaystyle {c_1}^5 C_3\) \(=\) \(\displaystyle 2 {c_2}^2 - c_1 c_3\)
\(\displaystyle {c_1}^7 C_4\) \(=\) \(\displaystyle 5 c_1 c_2 c_3 - 5 {c_2}^2 - {c_1}^2 c_4\)
\(\displaystyle {c_1}^9 C_5\) \(=\) \(\displaystyle 6 {c_1}^2 c_2 c_4 + 3 {c_1}^2 {c_3}^2 - {c_1}^3 c_5 + 14 {c_2}^4 - 21 c_1 {c_2}^2 c_3\)
\(\displaystyle {c_1}^{11} C_6\) \(=\) \(\displaystyle 7 {c_1}^3 c_2 c_5 + 84 c_1 {c_2}^3 c_3 + 7 {c_1}^3 c_3 c_4 - 28 {c_1}^2 c_2 {c_3}^2 - {c_1}^4 c_6 - 28 {c_1}^2 {c_2}^2 c_4 - 42 {c_2}^5\)


Proof


Sources