Reflected light

I figured out the general expressions for the total flux visible from each polynomial term in the case that the disk is bounded by an elliptical terminator segment of semi-minor axis b. Assuming this is our geometry,

the expressions for the case where the planet is between new phase and half phase are

[ericagol]

And, as you foresaw, the first two terms are straightforward:

I_0 = \tfrac{1}{2} \left(-b\sqrt{1-b^2}+\cos^{-1} b\right)

and

I_1 = \tfrac{1}{3} (1-b^2)^{3/2}.

[ericagol]

@rodluger Let:

I_j(b) = \int_b^1 \alpha^j (1-\alpha^2)^{1/2} d\alpha

Then, I find:

I_j = \frac{1}{j+2} \left[ b^{j-1}(1-b^2)^{3/2} + (j-1) I_{j-2}\right]

which is your recursion relation! The first integral integrates to the Gamma function term which you gave above.