I intend to demonstrate that a wing with an elliptical chord distribution, without torsion ($\alpha(z)=\alpha=\text{const}$) and constant section ($\beta(z)=\beta=\text{const}$, and $a_{2\text{D}}(z)=a_{2\text{D}}=\text{const}$), can induce an elliptical circulation distribution.
Well, for an elliptical chord distribution, the chord is defined by:
$$c(z)=c_C\sqrt{1-\left(\frac{z}{s}\right)^2}$$
where $c_C$ is a constant.
On the other hand, the circulation distribution is defined by:
$$\Gamma(z)=\frac{1}{2}a_{2\text{D}}V_\infty c(z)\left[\alpha+\beta+\alpha_i(z)\right]$$
So, by substituting $c(z)$ on $\Gamma(z)$, we get:
$$\Gamma(z)=\frac{1}{2}a_{2\text{D}}V_\infty \left[c_C\sqrt{1-\left(\frac{z}{s}\right)^2}\right]\left[\alpha+\beta+\alpha_i(z)\right]$$
which can be written as: $$\Gamma(z)=\Gamma_C(z)\sqrt{1-\left(\frac{z}{s}\right)^2}$$
where $\Gamma_C(z)=\frac{1}{2}a_{2\text{D}}V_\infty c_C\left[\alpha+\beta+\alpha_i(z)\right]$
So, to demonstrate that an elliptical chord distribution can induce an elliptical circulation distribution, we would also need to demonstrate that $\alpha_i(z)=\alpha_i=\text{const}$, to make $\Gamma_C$ a constant.
I don't know how to do this last step. How would you do it?
Note that:
$s:=$ semispan of the wing,
$z:=$ transversal coordinate of the wing, it goes from $-s$ to $s$,
$\alpha:=$ geometric angle of attack,
$\alpha_i:=$ induced angle of attack,
$\beta:=$ symmetric of the null lift (geometric) angle of attack,
$a_{2\text{D}}:=$ airfoil lift coefficient slope,
$V_\infty:=$ freestream airspeed.