## 2 A Necessary but Not Sufficient Condition
### Theorems  
Theorem 4 applies to \(2^{n}\) equations of (3) and gives a recurrence relation needed to compute polynomial solutions in the neighbourhood of an ordinary point. Theorems 5 and 6 give the polynomial solutions for the \(2^{n-1}+2^{n-2}\) equations from (3) about a regular singular point. The proofs of Theorems 4 and 5 may be found in the Appendix A, while the proof of Theorem 6 follows closely to the proof of Theorem 5 and is therefore omitted. Theorem 4: \(\boldsymbol{(\alpha_{0}\
eq 0)}\)  
_For \(\alpha_{0}\
eq 0\), the second-order linear differential Equation (3) admits the solution \(y(r)=\mathcal{P}_{m}(r)=\sum_{i=0}^{m}\mathcal{C}_{i}\,r^{i}\) in the neighbourhood of the ordinary point \(r  
_for which the recurrence relation (54) reduces to:_  
\[\Big{[}\alpha_{2}\left(\ell+s\right)\left(\ell+s-1\right)+\beta_{1}\left(\ell+s \right)-\tau_{0}\Big{]}\mathcal{C}_{\ell}=0\,,\quad\ell=0,\,1,\,\ldots,\,m\,. \tag{56}\]  
_Clearly when \(\ell=m\) and if \(\mathcal{C}_{m}\
eq 0\) then Equation (56) gives the necessary condition:_  
\[\alpha_{2}\left(m+s\right)\left(m+s-1\right)+\beta_{1}\left(m+s\right)-\tau_{0 }=0\,,\qquad\mathcal{C}_{m}\
eq 0\,, \tag{57}\]  
_as expected for the solutions of the Cauchy-Euler equation._  
The Mathematica\({}^{\text{\  
\[\left[\alpha_{4}\left(\ell-2\right)\left(\ell-3\right)+\beta_{3}\left( \ell-2\right)-\tau_{2}\right]\mathcal{C}_{\ell-2}+\left[\alpha_{3}\left(\ell-1 \right)\left(\ell-2\right)+\beta_{2}\left(\ell-1\right)-\tau_{1}\right] \mathcal{C}_{\ell-1}\] \[\quad+\left[\alpha_{2}\,\ell\left(\ell-1\right)+\beta_{1}\,\ell- \tau_{0}\right]\mathcal{C}_{\ell}+\left[\alpha_{1}\left(\ell+1\right)\ell+ \beta_{0}\left(\ell+1\right)\right]\mathcal{C}_{\ell+1}+\left[\alpha_{0}\left( \ell+2\right)\left(\ell+1\right)\right]\mathcal{C}_{\ell+2}=0\,, \tag{59}\]  
for \(\ell=0,\,1,\,2,\,\ldots,\,m+2\) and \(C_{j}=0\) for \(j<0\). For the zero-degree (\(m=0\)) polynomial solution, \(\mathcal{P}_{0}(r)=1,\,\ell=0,\,1,\,2\). Noting that \(\mathcal{C}_{j}=0\), for all \(j\
eq 0\), it follows for \(\ell=0\) and \(\ell=1\) that \(\tau_{0}=0\) and \(\tau_{1}=0\) respectively, while for \(\ell=2\), the necessary condition \(\tau_{2}=0\) follows. For the first-degree (\(m=1\)) polynomial solution, \(\mathcal{P}_{1}(r)=\mathcal{C}_{0}+\mathcal{C}_{1}\,r\), \(\ell=0,\,1,\,2,\,3\). Noting that \(\mathcal{C}_{j}=0\), for all \(j\
eq 0\), \(1\), we have for \(\ell=0\) that \(-\tau_{0}\,\mathcal{C}_{0}+\beta_{0}\,\mathcal{C}_{1}=0\), while for \(\ell=1\) and \(\ell=2\) we have \(-\tau_{1}\,\mathcal{C}_{0}+\left(\beta_{1}-\tau_{0}\right)\mathcal{C}_  
[PERSON] et al. [16] considered an electron moving in a Coulombic field, \(A_{0}=Z\,e/r\), with a constant homogeneous magnetic field described by \(A_{1}=-B\,y/2\) and \(A_{2}=B\,x/2\). With this choice of potential and for \(r>0\), (64) becomes  
\[\left(i\,\partial_{t}-\left[ \begin{array}{cc}\mathcal{M}&-e^{-i\,\phi}( \partial_{r}-\partial_{\phi}+\frac{eB\,r}{2})\\ e^{i\,\phi}(\partial_{r}+\partial_{\phi}-\frac{eB\,r}{2})&-\mathcal{M}\end{array}  \right]-\frac{Z\,e^{2}}{r}\right)\Psi(t,\vec{r})=0\,, \tag{65}\]  
where polar coordinates, the identities \(P_{x}\pm i\,P_{y}=-i\,e^{\pm i\,\phi}(\partial_{r}\pm(\partial_{\phi}-(e\,B\,r /2)))\), \(\partial_{x}=\cos\phi\,\partial_{r}-[\sin(\phi)/r]\,\partial_{\phi}\), and \(\partial_{y}=\sin(\phi)\,\partial_{r}+[\cos(\phi)/r]\,\partial_{\phi}\) were employed [16]. Using separation of variables  
\[\Psi(t,\vec{r})=\frac{1}{\sqrt{r}}\,e^{-i\,Et}\,\left( \begin{array}{cc}F(r) \,e^{i\,l\,\phi}\\ G(r)\,e^{i\,(l+1)\,\phi}\end{array}