# How to invert this type of infinite series?

If have a function $$f$$ given by a series $$f(z) = \sum_{n,m = 0}^\infty u_{n,m} z^{n + m t}$$ for some $$t\in\mathbb{R}^+$$.

Is there an straightforward way (something similar to the inversion formula for Taylor series) to derive a similar series for $$f^{-1}(z)$$?

• Lagrange inversion? en.wikipedia.org/wiki/Lagrange_inversion_theorem – Lord Shark the Unknown Mar 22 at 22:08
• I'm not sure how that would work in this case, since it's not exactly a Taylor series – Dark Malthorp Mar 22 at 22:10
• What is the inversion formula for Taylor series ? – Yves Daoust Mar 22 at 22:11
• With $w:=z^t$, you in fact have a bivariate series $f(z,w)=\sum u_{n,m}z^nw^m$. – Yves Daoust Mar 22 at 22:14
• OK. Does that help me invert it? – Dark Malthorp Mar 23 at 1:04