## 3 Expansions of the IG Modes by the HG and LG Modes at Small Values of \(p\)
### Cases with the Characteristic Polynomial of Degree 3
 \tag{25}\]  
with \(m=0,1,2\) and  
\[A_{0}=A_{0}(m,\varepsilon)=\frac{(\lambda_{m}^{\varepsilon}-9)(\lambda_{m}^{ \varepsilon}-25)}{2\varepsilon^{2}}-\frac{5}{2},\ A_{1}=A_{1}(m,\varepsilon)= \frac{\lambda_{m}^{\varepsilon}-25}{\varepsilon}.\]  
For the limiting cases, the normalized IG modes are shown in Table 5. \begin{table}
\begin{tabular}{c c c} \hline  & \(\varepsilon\approx\mathbf{0}\) & \(\varepsilon\to\infty\) \\ \hline \(\lambda_{0}^{\varepsilon}\) & \(1+3\varepsilon-\varepsilon^{2}+\ldots\) & \(-3\varepsilon+13-\frac{18}{\varepsilon}+\frac{18}{\varepsilon^{2}}+\ldots\) \\ \(\lambda_{1}^{\varepsilon}\) & \(9+\frac{11\varepsilon^{2}}{16}+\frac{3\varepsilon^{3}}{8}+\ldots\) & \(\varepsilon+17+\frac{5}{2}-\frac{48}{\varepsilon}+\ldots\) \\ \(\lambda_{2}^{\varepsilon}\) & \(25+\frac{5\varepsilon^{2}}{16}+\ldots\) & \(5\varepsilon+5+\frac{10}{\varepsilon}+\frac{30}{\varepsilon^{2}}+\ldots\) \\ \hline \end{tabular}
\end{table} 
Table 4: Asymptotic expansions of the roots (24). Equation (22) establishes relationships between the roots of the characteristic polynomials: \(\lambda_{m}^{\sigma}(\varepsilon)=\lambda_{m}^{\varepsilon}(-\varepsilon)\). Therefore, as per Equation (21), odd IG modes are obtained from the even IG modes by swapping the indices of the HG modes, replacing \(\varepsilon\) with \(-\varepsilon\), and adding a multiplier \((-1)^{m}\):  
\[ \begin{array}{l}\mathcal{I}_{5,2m+1}^{\sigma}(\mathbf{r},\varepsilon)=64\sqrt {2}\{2\text{Im}\mathcal{L}_{0,5}(\mathbf{r})-B_{1}Im\mathcal{L}_{1,3}( \mathbf{r})+B_{0}\text{Im}\mathcal{L}_{2,1}(\mathbf{r})\}\\ =(B_{0}+3B_{1}+5)\mathcal{H}_{4,1}(\mathbf{r})+2(B_{0}+B_{1}-5)\mathcal{H}_{2,3}(\mathbf{r})+(B_{0}-B_{1}+1)\mathcal{H}_{0,5}(\mathbf{r}),\end{array}  \tag{26}\]  
with \(m=0,1,2\) and  
\[B_{0}=B_{0}(m,\varepsilon)=\frac{(\lambda_{m}^{\sigma}-9)(\lambda_{m}^{\sigma }-25)}{2\varepsilon^{2}}-\frac{5}{2},\ B_{1}=B_{1}(m,\varepsilon)=\frac{ \lambda_{m}^{\sigma}-25}{\varepsilon}.\]  
Similarly to the above considered case for \(p=4\), for many values of \(\varepsilon\), the roots of the characteristic polynomials can be found without using the Cardano formula. For instance, for the even IG modes, the following values can be chosen: \(\varepsilon=\left\{1,\ \frac{3}{2},\ 3,\ 5,\ \frac{21}{4},\ 8\right\}\), whereas for the odd IG modes--the values \(\varepsilon=\left\{\frac{3}{4},\ \overline{7},\ \overline{5},\ 3,\ \frac{88}{18},\ \frac{45}{4}\right\}\). In particular, for \(\varepsilon=3\), we obtain the roots \(\lambda_{m}^{\varepsilon}=\left\{0,\ 16,\ 28\right\}\) and \(\lambda_{m}^{\sigma}=\left\{8-2\sqrt{97},\ 10,\ 8+2\sqrt{97}\right\}\). Therefore, for example,  
\[ \begin{array}{l}\mathcal{I}_{5,1}^{\varepsilon}(\mathbf{r},3)=48\{\mathcal{H }_{5,0}(\mathbf{r})+10\mathcal{H}_{3,2}(\mathbf{r})+15\mathcal{H}_{1,4}( \mathbf{r})\},\\ \mathcal{I}_{5,3}^{\sigma}(\mathbf{r},3)=48\{-\mathcal{H}_{0,5}(\mathbf{r})+1 0\mathcal{H}_{2,3}(\mathbf{r})+5\mathcal{H}_{4,1}(\mathbf{r})\}.\end{array}  \tag{27}\]  
There is one more case with the cubic characteristic equation, the case \(p=6\) for the odd IG modes:  
\[\lambda_{6}^{\sigma}(\lambda,\varepsilon)=\lambda^{3}-56\lambda^{2}+16(49- \varepsilon^{2})\lambda-384(6-\varepsilon^{2})=0. \tag{28}\]  
Then  
\[\lambda_{0}^{\varepsilon}=\frac{56}{3}-2\varepsilon_{6},\ \lambda_{1}^{ \varepsilon}=\frac{56}{3}+\varepsilon_{6}-\sqrt{3}\varepsilon_{6},\ \lambda_{2}^{ \varepsilon}=\frac{56}{3}+\varepsilon_{6}+\sqrt{3}\varepsilon_{6}, \tag{29}\]  
with  
\[ \begin{array}{l}c_{6}=\frac{4}{3}\sqrt{49+3\varepsilon^{2}}\cos\phi,\ s_{6}= \frac{4}{3}\sqrt{49+3\varepsilon^{2}}\sin\phi,\ \phi=\frac{1}{3}\arccos\left(-\frac{143-18 \varepsilon^{2}}{\left(49+3\varepsilon^{2}\right)^{3/2}}\right).\end{array} \]  
Correspondingly, the odd, non-normalized IG modes are given by  
\[