## 2 A Necessary but Not Sufficient Condition
* In the case of \(m=1\) and \(h=1\), Equation (20) reads: \[-\frac{d^{2}\,\psi(r)}{dr^{2}}+\left(-\frac{v}{\sqrt{r}}+\frac{l\,(l+1)}{r^{2}} \right)\psi(r)=E\,\psi(r)\,,\qquad v>0\,,\quad r>0\,,\quad\psi(0)=\psi(\infty)=0\,, \tag{8}\]6 (26)and the corresponding Equation (25) with \(q_{2,0}=\alpha_{1}\), \(q_{2,b=1}=\alpha_{2}\), \(q_{1,0}=\beta_{0}\), \(q_{1,b=1}=\beta_{1}\), \(q_{0,b=1}=\gamma_{0}\) and \(q_{0,0}=0\) reads \[-\frac{d^{2}\,\psi(r)}{dr^{2}}+\left(-\frac{v}{\sqrt{r}}+\frac{l\,(l+1)}{r^{2}} \right)\psi(r)=E\,\psi(r)\,,\qquad v>0\,,\quad r>0\,,\quad\psi(0)=\psi(\infty)=0\,, \tag{8}\]7 (27) The two-term recurrence formula for the solution of Equation (27) is \[-\frac{d^{2}\,\psi(r)}{dr^{2}}+\left(-\frac{v}{\sqrt{r}}+\frac{l\,(l+1)}{r^{2}} \right)\psi(r)=E\,\psi(r)\,,\qquad v>0\,,\quad r>0\,,\quad\psi(0)=\psi(\infty)=0\,, \tag{8}\]8 (28) where \(\lambda\) are the zeros of the indicial equation \(\alpha_{1}\,\lambda\,(\lambda-1)+\beta_{0}\,\lambda=0\), namely, \(\lambda=0\), and \(\lambda=1-\beta_{0}/\alpha_{1}\). The closed form solutions in terms of the Gauss hypergeometric functions, respectively, are: \[-\frac{d^{2}\,\psi(r)}{dr^{2}}+\left(-\frac{v}{\sqrt{r}}+\frac{l\,(l+1)}{r^{2}} \right)\psi(r)=E\,\psi(r)\,,\qquad v>0\,,\quad r>0\,,\quad\psi(0)=\psi(\infty)=0\,, \tag{8}\]9 \[z^{2}\,\psi^{\prime\prime}(z)-z\,\psi^{\prime}(z)+\left(\left(\frac{\sqrt{2}\,v }{\kappa^{3/2}}+\frac{E}{\kappa^{2}}z\right)z^{3}-4\,l\,(l+1)\right)\psi(z)=0 \,,\qquad\psi(0)=\psi(\infty)=0\,. \tag{9}\]0 (29) and \[z^{2}\,\psi^{\prime\prime}(z)-z\,\psi^{\prime}(z)+\left(\left(\frac{\sqrt{2}\,v }{\kappa^{3/2}}+\frac{E}{\kappa^{2}}z\right)z^{3}-4\,l\,(l+1)\right)\psi(z)=0 \,,\qquad\psi(0)=\psi(\infty)=0\,. \tag{9}\]1 \[z^{2}\,\psi^{\prime\prime}(z)-z\,\psi^{\prime}(z)+\left(\left(\frac{\sqrt{2}\,v }{\kappa^{3/2}}+\frac{E}{\kappa^{2}}z\right)z^{3}-4\,l\,(l+1)\right)\psi(z)=0 \,,\qquad\psi(0)=\psi(\infty)=0\,. \tag{9}\]2 (30)
* In this case of \(m=2\) and \(h=2\), Equation (20) reads \[z^{2}\,\psi^{\prime\prime}(z)-z\,\psi^{\prime}(z)+\left(\left(\frac{\sqrt{2}\,v }{\kappa^{3/2}}+\frac{E}{\kappa^{2}}z\right)z^{3}-4\,l\,(l+1)\right)\psi(z)=0 \,,\qquad\psi(0)=\psi(\infty)=0\,. \tag{9}\]3 (31) and the corresponding Equation (25) reads \[z^{2}\,\psi^{\prime\prime}(z)-z\,\psi^{\prime}(z)+\left(\left(\frac{\sqrt{2}\,v }{\kappa^{3/2}}+\frac{E}{\kappa^{2}}z\right)z^{3}-4\,l\,(l+1)\right)\psi(z)=0 \,,\qquad\psi(0)=\psi(\infty)=0\,. \tag{9}\]4 (32) The two-term recurrence relation for the solutions of Equation (32) reads \[z^{2}\,\psi^{\prime\prime}(z)-z\,\psi^{\prime}(z)+\left(\left(\frac{\sqrt{2}\,v }{\kappa^{3/2}}+\frac{E}{\kappa^{2}}z\right)z^{3}-4\,l\,(l+1)\right)\psi(z)=0 \,,\qquad\psi(0)=\psi(\infty)=0\,. \tag{9}\]5 (33) where, \(\lambda\) are the roots of the indicial equation \(\lambda\,(\lambda-1)=0\), \(\alpha_{0}\
eq 0\), with closed form solutions \[z^{2}\,\psi^{\prime\prime}(z)-z\,\psi^{\prime}(z)+\left(\left(\frac{\sqrt{2}\,v }{\kappa^{3/2}}+\frac{E}{\kappa^{2}}z\right)z^{3}-4\,l\,(l+1)\right)\psi(z)=0 \,,\qquad\psi(0)=\psi(\infty)=0\,. \tag{9}\]6 \[z^{2}\,\psi^{\prime\prime}(z)-z\,\psi^{\prime}(z)+\left(\left(\frac{\sqrt{2}\,v }{\kappa^{3/2}}+\frac{E}{\kappa^{2}}z\right)z^{3}-4\,l\,(l+1)\right)\psi(z)=0 \,,\qquad\psi(0)=\psi(\infty)=0\,. \tag{9}\]7 (34) and \[z^{2}\,\psi^{\prime\prime}(z)-z\,\psi^{\prime}(z)+\left(\left(\frac{\sqrt{2}\,v }{\kappa^{3/2}}+\frac{E}{\kappa^{2}}z\right)z^{3}-4\,l\,(l+1)\right)\psi(z)=0 \,,\qquad\psi(0)=\psi(\infty)=0\,. \tag{9}\]8 \[z^{2}\,\psi^{\prime\prime}(z)-z\,\psi^{\prime}(z)+\left(\left(\frac{\sqrt{2}\,v }{\kappa^{3/2}}+\frac{E}{\kappa^{2}}z\right)z^{3}-4\,l\,(l+1)\right)\psi(z)=0 \,,\qquad\psi(0)=\psi(\infty)=0\,. \tag{9}\]9 (35)  
**Remark 4**. : _For the admissible values of the equation parameters, the recurrence relations (28) and (33) are generic formulas for the two-term recurrence relations for the series solutions of the differential Equations (27)and (32) respectively.