Lagrange inversion

2023-06-01 Improve this page

#Lagrange inversion

Lagrange inversion is a method to extract formal power series coefficients, from the functional inverse of a power series, see [Wil94]. A proof by induction can be found in [SW23].

Theorem

Let \(f(u)\) and \(\phi(u)\) be formal power series, with \(\phi(0)=1.\) Then there is a unique formal power series \(u(t)\) solving \(u(t)= t\cdot\phi(u(t)).\) Moreover, \[[t^n] f(u(t)) = \frac{1}{n}[u^{n-1}]\{ f'(u)\phi(u)^n \}.\]

One particular consequence is the following (when taking \(f(x)=x\)):

Theorem

Let \(u(x)\) satisfy the equation \(u(x) = x \cdot R(u(x)).\) Then \[[x^n]u = \frac{1}{n} [x^{n-1}] R(x)^n.\]

This allows us to find the generating functions (and closed-form formulas) for the Fuss–Catalan numbers.

Bibliography

[SW23]Erlang Surya and Lutz Warnke. Lagrange inversion formula by induction. arXiv:2305.17576, 2023.

.bib

@article{SuryaWarnke2023x,
Author = {Erlang Surya and Lutz Warnke},
Title = {Lagrange Inversion Formula by Induction},
Year = {2023},
Eprint = {2305.17576},
  url = {https://arxiv.org/abs/2305.17576},
journal = {arXiv e-prints}
}

[Wil94]Herbert S. Wilf. Generatingfunctionology. Elsevier, 1994.

.bib

@book{Wilf1994,
	doi = {10.1016/c2009-0-02369-1},
	url2 = {https://doi.org/10.1016%2Fc2009-0-02369-1},
	year = {1994},
	author = {Herbert S. Wilf},
	publisher = {Elsevier},
	title = {generatingfunctionology}
}