This seems such a simple question that I fear I must have missed some elementary maths.


I am looking for a way to solve $x+x^a = y$ by reference to an already defined function, $a,x,y > 0$ are real.


Failing that a reasonable approximation for $a$ in $(0,1)$.


Many thanks!



Is there a specific named function that is the inverse of $x+x^a$ for $x$ real?
The answer is yes indeed. It is a special case of Fox-H function, a variation of the confluent Fox-Wright $\_{1}\Psi\_{1}$ function (a generalization of the confluent hypergeometric function $\_{1}F\_{1}$) providing the inverse function. See a previous answer [here](https://mathoverflow.net/questions/406250/transcendental-formulas-for-roots-of-polynomials/406340#406340) for details and references. For this particular case solution is


(Setting $\alpha = a$), for **$\alpha>1$**


$$x = y\cdot\,\_{1}\Psi\_{1}([1,\alpha];[2,\alpha-1];-y^{\alpha-1})$$ which can be set as $$x=y+\sum\_{n=1}^\infty\binom{n\alpha}{n-1}\frac{(-1)^ny^{n(\alpha-1)+1}}{n}$$ whose convergence region is $|y^{\alpha-1}|<|(\alpha-1)^{\alpha-1}\alpha^{-\alpha}|$. For non integer $\alpha>1$ binomials must be set in terms of $\Gamma$ function.


Since Fox-Wright's generalized function can be expressed in terms of [Fox-H function](https://gitlab.com/RZ-FZJ/hypergeom/-/blob/master/Hypergeom.pdf) we have $$\,\_{1}\Psi\_{1}([1,\alpha];[2,\alpha-1];-y^{\alpha-1})=H\_{1,2}^{1,1}([(0,\alpha)];[(0,1),(-1,\alpha-1)];y^{\alpha-1})$$ $$\,\_{1}\Psi\_{1}([1,\alpha];[2,\alpha-1];-y^{\alpha-1})=H\_{2,1}^{1,1}([(1,1),(2,\alpha-1)];[(1,\alpha)];y^{1-\alpha})$$ for this particular case, [Wolfram's Mathematica 12.3](https://reference.wolfram.com/language/ref/FoxH.html) provides an explicit inverse as


**$x=y\cdot$FoxH[{{{0,$\alpha$}},{{}}},{{{0,1}},{{-1,$\alpha$-1}}},$y^{\alpha-1}$]**


For **$0<\alpha<1$** the solution is $$x = y\cdot(1-\,\_{1}\Psi\_{1}([1,\alpha^{-1}];[2,\alpha^{-1}-1];-y^{\alpha^{-1}-1}))$$


and the above relationships are turned in replacing $\alpha$ by $\alpha^{-1}$ and Fox-Wright function $\,\_{1}\Psi\_{1}$ by $1-\,\_{1}\Psi\_{1}$. In this case Mathematica's expression is


**$x=y\cdot(1-$FoxH[{{{0,$\alpha^{-1}$}},{{}}},{{{0,1}},{{-1,$\alpha^{-1}$-1}}},$y^{\alpha^{-1}-1}$])**


Finally, just to complement this answer, general trinomial equation solutions are developed in section 4 of the following


**Reference**


Miller A.R., Moskowitz I.S. Reduction of a Class of Fox-Wright Psi Functions for Certain Rational Parameters. Computers Math. Applic. Vol. 30, No. 11, pp. 73-82, (1995). Pergamon


A preprint can be found [here](https://apps.dtic.mil/sti/pdfs/ADA465435.pdf). (Document has mis-embedded fonts, isolated commas must be replaced by $\Gamma$ symbol)


