Solving $\frac{L-E}{(I-L)^\alpha} = AN^{-a}+BD^{-b}$ for $L$

Question

I want to show how to solve for $L$ in the following equation:

$$ \frac{L-E}{(I-L)^\alpha} = AN^{-a}+BD^{-b} $$

where $a,b,N,D>0$.

I think that this has no closed solution, but I want to make sure I am not overlooking anything. If this does not have a close solution, can I somehow approximate the solution in closed form?

Answer 1

$$\frac{L-E}{(I-L)^\alpha}=c$$

We see, for rational $\alpha$, your equation is related to a polynomial equation and we can use the solution theory of polynomial equations.

But for non-rational $\alpha$, your equation is related to a polynomial equation in dependence of more than one algebraically independent monomials ($L,(I-L)^\alpha$) and with no univariate factor. We therefore don't know how to rearrange the equation for $L$ by applying only finite numbers of elementary functions (operations) we can read from the equation.

$$\frac{L-E}{(I-L)^\alpha}=c$$ $$L-E=c\ (I-L)^\alpha$$ $$L-E-c\ (I-L)^\alpha=0$$ $L\to I-x^\frac{1}{\alpha}$:
for real $\alpha,x$: $$I-x^\frac{1}{\alpha}-E-cx=0$$ $$cx+x^\frac{1}{\alpha}+(E-I)=0$$

We see, this equation is a trinomial equation with real exponents. We can transform it to a form like in equation 8.1 of [Belkic 2019]. Solutions in terms of Bell polynomials, Pochhammer symbols or confluent Fox-Wright Function $\ _1\Psi_1$ can be obtained therefore.

[Belkić 2019] Belkić, D.: All the trinomial roots, their powers and logarithms from the Lambert series, Bell polynomials and Fox–Wright function: illustration for genome multiplicity in survival of irradiated cells. J. Math. Chem. 57 (2019) 59-106

[Szabó 2010] Szabó, P. G.: On the roots of the trinomial equation. Centr. Eur. J. Operat. Res. 18 (2010) (1) 97-104

Answer 2

Let $a=\alpha$ for easier MathJax and apply Lagrange reversion: $$\frac{L-E}{(I-L)^a}=b\implies L=1-e^\frac{2\pi i k}a\left(\frac{L-E}b\right)^\frac1a=1+\sum_{n=1}^\infty\frac{e^\frac{2\pi i k n}a}{(-\sqrt[a]b)^nn!}\left.\frac{d^{n-1}}{dz^{n-1}}(z-E)^\frac1a\right|_1$$ Therefore: $$L_k=E+(1-E)\sum_{n=0}^\infty\frac{\left(\frac na\right)!}{\left(\left(\frac1a-1\right)n+1\right)!n!}\left(-\frac{(1-E)^{\frac1a-1}}{\sqrt[a]b}e^\frac{2\pi ik}a\right)^n $$ shown here. So the Fox Wright function, a special case of Fox H, appears:

$$\bbox[3px,border: 4px ridge cyan]{\frac{L-E}{(1-L)^a}=b\implies L_k=E+(1-E)\,_1\Psi_1\left(^{\ \,\left(1,\frac1a\right)}_{\left(2,\frac1a-1\right)};-\frac{(1-E)^{\frac1a-1}}{\sqrt[a]b}e^\frac{2\pi ik}a\right)}$$