The Lagrange inversion theorem is a result in combinatorial mathematics and algebra that provides a formula for finding the coefficients of a power series that is the inverse of another power series. It is particularly useful when dealing with formal power series and can be applied in various areas including combinatorics, algebraic geometry, and differential equations.
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Over a field of characteristic zero, let be the unique formal power series with constant coefficient zero satisfying , where . Lagrange inversion gives, for ,Here the brackets denote coefficient extraction. The theorem obtains coefficients of an implicitly defined series without solving for it explicitly.
For and , it gives , the Catalan numbers for . This is useful when a recursive counting construction naturally gives an implicit generating function.