Questions tagged [inverse-series]

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Property of an integer sequence related to series reversion

Thinking of some questions of homotopical algebra for operads, I ended up with a following question, perhaps someone will recognize something here: Let $\{a_n\}_{n\ge 2}$ be a sequence of nonnegative ...
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2 votes
0 answers
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Functional inverse of $z=1+w+\cdots+w^{n-1}$

Migrated from the MSE. I am interested in the functional inverse of $$ z=1+w+\cdots+w^{n-1},\quad w\geq0,\ n>1. $$ This function is strictly increasing on $w\geq0$ and thus admits an inverse. By ...
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3 votes
1 answer
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"Lagrange inversion" around an extremum

Cross-posted from Math Stackexchange. In an older question to which I provided an answer it was asked how to compute a particular limit involving the roots of a transcedental function around its ...
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3 votes
1 answer
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Inverse of the incomplete elliptic integral of the second kind

The incomplete elliptic integral of the second kind $E(\varphi \, | \,k)$ is defined as follows: $$E(\varphi \, | \,k) = \int_0^\varphi \sqrt{1-k^2\sin^2\theta} \, \mathrm{d}\theta $$ Where $0<k^...
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2 votes
1 answer
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Correction terms in the asymptotic expansion of hypergeometric function

I am interested in obtaining the asymptotic expansion of $r(\rho)$ (which is the inverse of $\rho$ below), $$\rho=\frac{2b}{1-q}\left(1-\left(\frac br\right)^{1-q}\right)^{1/2}\left(_2F_1\left(\frac{1}...
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2 votes
1 answer
416 views

Is $\sigma(n)$ has explicit compositional inverse formula since it has series representation?

let $\sigma_x(n)$ be a power of sum divisor function such that : $\sigma_x(n)=\sum_{d|n} d^x$ . My question is: Does $\sigma$ has explicit compositional inverse formula since it has series ...
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1 vote
1 answer
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Can I apply Lagrange inversion theorem? [closed]

I want to invert the equation $$\eta = g(x)\sqrt{1+g'(x)^2}$$ to get $x$ as a function of $\eta$. Assume $g(0)=0$, $g'(0)=0$ and $g'(x)>0$ for $x>0$ (Think $g(x) = x^p$ for $p\geq 2$ integer). ...
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17 votes
1 answer
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Positivity of coefficients of the inverse of a certain power series

Consider the unique formal power series $g(z)$ with $g(0)=0$ and $g'(0)=1$ satisfying the equation $$ g(z)-g(z)^8+g(z)^{15}=z, $$ that is the inverse of $$ z-z^8+z^{15} $$ in the group of formal ...
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