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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What are $x$ and $y$? – user7427029 Jul 13 '22 at 17:57
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1there 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) – Carlo Beenakker Jul 13 '22 at 18:40
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related answers: https://math.stackexchange.com/questions/257455/inverse-function-of-y-frac-lnx1-ln-x https://math.stackexchange.com/questions/4491579/is-it-possible-to-solve-the-equation-x-1-x-y-explicitly/4492364#4492364 https://mathoverflow.net/questions/116276/what-is-fx-if-f-1x-frac-lnx1-ln-x?_gl=11hef901_gaNTExMTAxNjQ0LjE2NjY1NTA3MDE._ga_S812YQPLT2*MTY3MDAwNTcwMy4xMTQuMS4xNjcwMDA2NzMxLjAuMC4w – IV_ Dec 02 '22 at 19:31
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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.
Jorge Zuniga
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