AsymptoticSolve and Series not working in limit $to infty$. How to solve this functional polynomial relationship?

I want to solve for $J_d$ as a function of $n_d$ for $etagg1$ and $etaggeta_0$ in the following equations by eliminating $eta$ from the two equations.
begin{equation}
J_d = J_0cdotleft[frac{eta^{frac{d-1}{t}+1}}{Gammaleft(frac{d-1}{t}+2right)}-frac{left(eta-eta_0right)^{frac{d-1}{t}+1}}{Gammaleft(frac{d-1}{t}+2right)}right]
end{equation}
begin{equation}
n_d = n_0cdot left[frac{eta^{frac{d}{t}}}{Gammaleft(frac{d}{t}+1right)}+frac{left(eta-eta_0right)^{frac{d}{t}}}{Gammaleft(frac{d}{t}+1right)}right]
end{equation}
Here $J_0$, $n_0$, $eta_0$, $d$ and $t$ are constants.
I know I can ignore $eta_0$ in the second equation and solve for $eta$ in terms of $n_d$ and then substitute it in first equation to get $J_d$ in terms of $n_d$. Is there a way of solving it where I don’t ignore $eta_0$ and still get a good approximation for $J_d$ as a function of $n_d$.

So in order to get $n_d$ in terms of $eta$ basically I tried to use AsymptoticSolve but it is giving me error.

AsymptoticSolve[
 nd - n0*((η)^(d/t)/Gamma[d/t + 1] + (η - η0)^(d/t)/
      Gamma[d/t + 1]) == 0, nd, η -> Infinity]

But clearly this syntax works as shown here https://reference.wolfram.com/language/ref/AsymptoticSolve.html

Now I tried to take series of the RHS of equation 2 in limit $etatoinfty$,it again shows the same problem. What should I do in order to solve this simple analytical problem.