How to get inverse function(series) of a series that with unknown power I am facing a problem in my research:
If a series takes form of:
$$
y=a_1x^{n_1}+a_2x^{n_2}+\cdots+a_jx^{n_j}+\cdots
$$
How to get its inverse function(series)?
Maybe its inverse series takes form of:
$$
x=b_1y^{m_1}+b_2y^{m_2}+\cdots+b_jy^{m_3}+\cdots
$$
$n_1,n_2,\cdots,m_1,m_2,\cdots$ are arbitrary rationals, arbitrary reals but not integers completely. Actually, it takes form of
$$
n_i(or\ m_i)=3l_iz+k_i
$$
where $l_i$ and $k_i$ are arbitrary positive integers but z is arbitrary reals.
But I don't know how to get it. At least I want to get the first few terms.
 A: 1.) Rational exponents
Transform your series $y$ of $x$ to a power series $z$ of $t$ by substituting $x\to t^p$, where $p$ is the least common multiple of the denominators of $n_1,n_2,...\ $. Consider rule 4.2.19 on page 70 of [Abramowitz/Stegun 1970].
Transform $z$ by subtracting its constant term.
The inversion will be made by Lagrange inversion:
You can create the usual system of equations that is used for inverting power series and solve the system term by term. See e.g. [Chernoff 1947].
Example for your original series where all exponents are integer:
$$f(x)=a_1x^{n_1}+a_2x^{n_2}+\cdots+a_jx^{n_j}+\cdots$$
$$f^{-1}(y)=b_1y^{m_1}+b_2y^{m_2}+\cdots+b_jy^{m_3}+\cdots$$
$$f(f^{-1}(x))=x:$$
$$a_1\left(b_0+b_1x^{m_1}+b_2x^{m_2}+b_3x^{m_3}+\cdots\right)^{n_1}+a_2\left( b_0+b_1x^{m_1}+b_2x^{m_2}+b_3x^{m_3}+\cdots\right)^{n_2}+a_3\left(b_0+b_1x^{m_1}+b_2x^{m_2}+b_3x^{m_3}+\cdots\right)^{n_3}+\cdots=x$$
Expand this expression and collect all terms of the same degree of $x$. Solve the equation system successively for $m_1$, $m_2$ and so on.
Or apply one of the general formulas of Lagrange inversion.
2.) Real exponents
Maybe the following helps:
Loeb, D.: Series with general exponents. 1995
Maybe the binomial theorem for real exponents together with [Xiao-Xiong Gan 2021] helps. Apply the equation from 1.), but now with real exponents.
$\ $
[Abramowitz/Stegun 1970] Abramowitz, M.; Stegun, I.: Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables. National Bureau of Standard 1970
[Chernoff 1947] Chernoff, H.: A Note on the inversion of power series. Math. Comp. 2 (1947) 331-335
[Xiao-Xiong Gan 2021] Xiao-Xiong Gan: Formal Analysis. An Introduction. De Gruyter, 2021: 9 Formal series and general exponents. 9.4 The real exponents of formal power series
