Is it possible to solve for $y$ in this equation? $$-y^{-x}+y-1=0$$ People have mentioned the use of the Lambert W function or other non-elementary functions, but I haven't been able to make use of them. I'd preferably like to find a closed form expression using elementary functions, but if that isn't possible any other equation with $y$ isolated is fine.
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$\begingroup$ What are $x$ and $y$? $\endgroup$– user7427029Commented Jul 13, 2022 at 17:57
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1$\begingroup$ there is no closed-form solution in terms of elementary functions (and I would not know how to solve it in term of some known special function either) $\endgroup$– Carlo BeenakkerCommented Jul 13, 2022 at 18:40
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$\begingroup$ related answers: math.stackexchange.com/questions/257455/… math.stackexchange.com/questions/4491579/… mathoverflow.net/questions/116276/…*1hef901*_gaNTExMTAxNjQ0LjE2NjY1NTA3MDE._ga_S812YQPLT2*MTY3MDAwNTcwMy4xMTQuMS4xNjcwMDA2NzMxLjAuMC4w $\endgroup$– IV_Commented Dec 2, 2022 at 19:31
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1 Answer
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Yes, it is. In fact, this is a trinomial equation. A closed form expression can be obtained using confluent Fox-Wright Function $\ _1\Psi_1^*(\zeta)$. See here
A linearly convergent series can be worked out for $x>0$ starting from this MO link, here
$$y = 1+\sum_{n=1}^{\infty}\frac{1}{nx+1}\cdot \binom{(nx+1)/(x+1)}{n}$$ where binomials must be expressed in terms of Gamma Function.
You can look at these previous answers in MO here and here as well.
Since Fox-Wright function is a special case of Fox-H function, you can work with the closed form solution using last versions of Wolfram's Mathematica that have implemented Fox-H. Some steps in this line are found in the former links.
answered Jul 14, 2022 at 1:21