I have an equation
$$ -\frac{1}{2}y''(x)-b~\text{sech}^2(ax)~y=-\frac{k^2}{2}y $$
and I know that it has solution in terms of Legendre polynomials:
$$ y_1(x)=P_s^\epsilon(\zeta) $$ $$ y_2(x)=Q_s^\epsilon(\zeta) $$
where
$$ s=\frac{1}{2} \left(\sqrt{\frac{8 b}{a^2}+1}-1\right) $$
$$ \epsilon =\frac{k}{a} $$
$$ \zeta =\tanh (a x) $$
I also know that the solution can be written in the form of Hypergeometric function:
$$ y_3=\left(1-\zeta ^2\right)^{\epsilon /2} \, _2F_1\left(\epsilon -s,s+\epsilon +1;\epsilon +1;\frac{1-\zeta }{2}\right) $$
Since there are are only two linear independent solutions, $y_3$ must can be expressed in terms of $y_1$ and $y_2$, so what is the relationship between them? And what is the fourth solution $y_4$ which is linear independent with respect to $y_3$?
Here are some mathematica code to verify the solution:
eqs = -(1/2) y''[x] - b (Sech[a x])^2 y[x] == -(k^2/2) y[x];
ζ = Tanh[a x]; s = 1/2 (Sqrt[1 + (8 b)/a^2] - 1); ϵ = k/a;
y1 = LegendreP[s , ϵ, ζ];
y2 = LegendreQ[s , ϵ, ζ];
y3 = (1 - ζ^2)^(ϵ/2)Hypergeometric2F1[ϵ - s, ϵ + s + 1, ϵ + 1, 1/2 (1 - ζ)];
eqs /. y -> Function[{x}, Evaluate[y1]] // FullSimplify
eqs /. y -> Function[{x}, Evaluate[y2]] // FullSimplify
eqs /. y -> Function[{x}, Evaluate[y3]] // FullSimplify
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