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9 votes
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Conjectured closed form of $\int_0^1 \frac{\text{Li}_2\left(\frac{x}{4}\right)}{4-x}\,\log\left(\frac{1+\sqrt{1-x}}{1-\sqrt{1-x}}\right)\,dx$

As Henri Cohen remarked, the identity to prove is equivalent to $$\sum_{n=1}^\infty \frac{H_n^{(2)}}{(n+1)(2n+1)\binom{2n}{n}}=\frac{\pi^4}{972}.\tag{1}$$ In turn, this follows readily from the OP's ...
GH from MO's user avatar
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9 votes
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Nicer expression for 2.1369288...?

Apply Lagrange reversion to @TheSimpliFire’s equation: $$\frac{k}{(k-1)^2}+1=e^k\iff k=1+\sqrt{\frac k{e^k-1}}\\\implies k=1+\sum_{n=1}^\infty\frac1{n!}\left.\frac{d^{n-1}}{dx^{n-1}}\left(\frac x{e^x-...
Tyma Gaidash's user avatar
12 votes

Nicer expression for 2.1369288...?

With a few substitutions, we find that $$c=\frac{k^3}{k^2-k+1}\quad\text{where}\quad1+\frac k{(1-k)^2}=e^k.$$ The solution for $k$ requires a more advanced function than Lambert $W$.
TheSimpliFire's user avatar
2 votes
Accepted

Functional equations based on composition

There are no solutions, even if we only assume that $f$ has a derivative at every point. Note that if $f(x)>x$ for all $x$, then $f(f(x))>f(x)>x$ etc, thus $\sum a_k f^k(x)\geqslant (\sum a_k)...
Fedor Petrov's user avatar
1 vote

Functional equations based on composition

As it was pointed in the comments, the case of a linear function $px$ should be excluded. Let's investigate the given sum on a linear function $f(x)=px+q$ with $q\ne0$. We have $$\sum_{k=0}^n a_k (px+...
Max Alekseyev's user avatar

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