be written as $$\begin{split}
  & \Psi_{N,n}(r, \varphi) = {\cal N}_{N,n} Z^{(2n+a+b)}_N (z) \Phi^{(a,b)}_n (\varphi), \\
  & Z^{(2n+a+b)}_N (z) = \left(\frac{z}{\omega}\right)^{\left(n + \frac{a+b}{2}\right) k} L^{((2n+a+b)k)}_N
          (z) e^{- \frac{1}{2} z}, \qquad z = \omega r^2, \\
  & \Phi^{(a,b)}_n (\varphi) = \cos^a k\varphi \sin^b k\varphi P^{\left(a - \frac{1}{2}, b - \frac{1}{2}\right)}_n
          (\xi), \qquad \xi = - \cos 2k\varphi,
\end{split}  \label{eq:w-f}$$ in terms of Laguerre and Jacobi polynomials. In (\[eq:w-f\]), ${\cal N}_{N,n}$ denotes a normalization constant, which can be easily calculated from some known properties of these polynomials [@gradshteyn] and is given by $${\cal N}_{N,n} = \bar{\cal N}_{N,n} {\cal N}^{(a,b)}_{n}$$ with $$\begin{split}
  & \bar{\cal N}_{N,n} = (-1)^N \left(\frac{2\, \omega^{(2n+a+b)k + 1} N!}{\Gamma(N + (2n+a+b)k +1)}\right)
        ^{1/2}, \\
  & {\cal N}^{(a,b)}_{n} = \left(\frac{2k\, n! (2n+a+b) \Gamma(a+b+n)}{\Gamma\left(a + n + \frac{1}{2}\right)
        \Gamma\left(b + n + \frac{1}{2}\right)}\right)^{1/2}.
\end{split}$$ Observe that the optional phase factor $(-1)^N$ in $\bar{\cal N}_{N,n}$ has been introduced to get positive matrix elements for the ${\rm sp}(2, {{\mathbb{R}}})$ generators $K_{\pm}$ in conformity with the conventional choice.

After multiplication by the fermionic vacuum state $|0\rangle$ (i.e., $b_x |0\rangle = b_y |0\rangle = {\bar{b}}_x |0\rangle = {\bar{b}}_y |0\rangle = 0$), the wavefunctions (\[eq:w-f\]) yield eigenstates of the supersymmetrized TTW Hamiltonian (\[eq:super-TTW\]) with eigenvalues $${\cal E}_{N,n} = 