Editing Bell polynomials (section)

===Reversion of series===
{{main|Lagrange inversion theorem}}

Let two functions ''f'' and ''g'' be expressed in formal [[power series]] as

:<math>f(w) = \sum_{k=0}^\infty f_k \frac{w^k}{k!}, \qquad \text{and}  \qquad  g(z) = \sum_{k=0}^\infty g_k \frac{z^k}{k!},</math>

such that ''g'' is the compositional inverse of ''f'' defined by ''g''(''f''(''w'')) = ''w'' or ''f''(''g''(''z'')) = ''z''. If ''f''<sub>0</sub> = 0 and ''f''<sub>1</sub> ≠ 0, then an explicit form of the coefficients of the inverse can be given in term of Bell polynomials as{{sfn|Charalambides|2002|p=437|loc=Eqn (11.43)}}

:<math> g_n = \frac{1}{f_1^n} \sum_{k=1}^{n-1} (-1)^k n^{\bar{k}} B_{n-1,k}(\hat{f}_1,\hat{f}_2,\ldots,\hat{f}_{n-k}), \qquad n \geq 2, </math>

with <math> \hat{f}_k = \frac{f_{k+1}}{(k+1)f_{1}},</math> and <math>n^{\bar{k}} = n(n+1)\cdots (n+k-1) </math> is the rising factorial, and <math>g_1 = \frac{1}{f_{1}}. </math>