## 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  
For mode \(\mathbf{IG}_{4,2m}^{\varepsilon}(\mathbf{r},\varepsilon)\) with \(m=0,1,2\), we obtain the following characteristic equation:  
\[\chi_{4}^{\varepsilon}(\lambda,\varepsilon)=\lambda^{3}-20\lambda^{2}+16(4- \varepsilon  
whose roots \(\lambda_{m}^{\varepsilon}(\varepsilon)\) can be found by using the Cardano formula. In ascending order, they read as follows:  
\[\lambda_{0}^{\varepsilon}=\frac{20}{3}-2\varepsilon_{4},\ \lambda_{1}^{ \varepsilon}=\frac{20}{3}+c_{4}-s_{4}\sqrt{3},\ \lambda_{2}^{\varepsilon}=\frac{20}{3}+c_{4}+s_{4}\sqrt{3} \tag{18}\]  
with  
\[c_{4}=\frac{4}{3}\sqrt{13+3\varepsilon^{2}}\cos\phi,\ s_{4}=\frac{4}{3}\sqrt{ 13+3\varepsilon^{2}}\sin\phi,\ \phi=\frac{1}{3}\arccos\left(-\frac{35-18 \varepsilon^{2}}{\left(13+3\varepsilon^{2}\right)^{3/2}}\right).\]  
Asymptotic expansions of all three roots at small and large values of \(\varepsilon\) are given in Table 2.  
These formulae are very useful for investigating the limiting cases \(\varepsilon\to 0\) and \(\varepsilon\to\infty\). For \(p=4\), the even, non-normalized IG modes are given by  
\[ \begin{array}{l}\mathcal{I}_{4,2m}^{\varepsilon}(\mathbf{r},\varepsilon)=32 \{2\text{Re}\mathcal{L}_{0,4}(\mathbf{r})-A_{1}\text{Re}\mathcal{L}_{1,2}( \mathbf{r})+A_{0}\mathcal{L}_{2,0}(\mathbf{r})\}\\ =(A_{0}+A_{1}+1)\mathcal{H}_{4,0}(\mathbf{r})+(2A_{0}-6)\mathcal{H}_{2,2}( \mathbf{r})+(A_{0}-A_{1}+1)\mathcal{H}_{0,4}(\mathbf{r}),\end{array}  \tag{19}\]  
with \(m=0,1,2\) and  
\[A_{0}=A_{0}(m,\varepsilon)=\frac{(\lambda_{m}^{\varepsilon}-4)(\lambda_{m}^{ \varepsilon}-16)}{4\varepsilon^{2}}-1,\ A_{1}=A_{1}(m,\varepsilon)=\frac{ \lambda_{m}^{\varepsilon}-16}{\varepsilon}. \tag{20}\]  
We do not present the normalized version in order to avoid cumbersome fractions, and we give the normalized IG modes only in the limiting cases (Table 3). For numerical computation of the IG modes, when the ellipticity parameter is neither very small nor very large, the Cardano formula with expansions (19) should be employed, with the normalizing multiplier added if necessary. If, however, the ellipticity parameter is close to the limiting values, then the asymptotic expressions of the characteristic roots are more convenient. For instance, if \(m=2\) and \(\varepsilon\to 0\), we obtain \(A_{0}=-\frac{1}{144}\varepsilon^{2}+O(\varepsilon^{4})\), \(A_{1}=\frac{1}{3}\varepsilon+O(\varepsilon^{3})\), and, thus, \(\mathcal{I}_{4,4}^{\varepsilon}(\mathbf{r},\varepsilon)=64\text{Re}\mathcal{L }_{0,4}(\mathbf{r})-\frac{32}{3}\varepsilon\text{Re}\mathcal{L}_{1,2}( \mathbf{r})+O(\varepsilon^{2})\). For many values of \(\varepsilon\), the roots of the Equation (17) can be easily found without using the Cardano formula; for example, this is the case for \(\varepsilon=\left\{\frac{1}{2}\sqrt{5},\ 2,\ \frac{6}{5}\sqrt{5},\ \frac{3}{2}\sqrt{5},\ \sqrt{10},\ 2 \sqrt{5}\right\}\). It is enough to choose some integer \(\lambda\) as a root and find which \(\varepsilon\) it corresponds to (relative to \(\varepsilon\), Equation (17) is quadratic). In particular, for \(\varepsilon=2\), we obtain \(\lambda_{0}^{\varepsilon}=6-2\sqrt{33}\), \(\lambda_{1}^{\varepsilon}=8\), and \(\lambda_{2}^{\varepsilon}=6+2\sqrt{33}\). \begin{table}
\begin{tabular}{c c c} \hline \hline  & \(\varepsilon\to\infty\) & \(\varepsilon\to\infty\) \\ \hline \(\lambda_{0}^{\varepsilon}\) & \(-3\varepsilon^{2}+\frac{33}{16}\varepsilon^{4}-\ldots\) & \(-4\varepsilon+4-\frac{6}{16}+\frac{12}{\varepsilon^{2}}-\ldots\) \\ \(\lambda_{1}^{\varepsilon}\) & \(4+\frac{8\varepsilon^{2}}{3}-\frac{56\varepsilon^{4}}{432}+\ldots\) & \(12-\frac{24}{3}-\frac{24}{3}+\ldots\) \\ \(\lambda_{2}^{\varepsilon}\) & \(16+\frac{5}{3}-\frac{56\varepsilon^{4}}{432}+\ldots\) & \(4\varepsilon+4+\frac{6}{\varepsilon}+\frac{12}{\varepsilon^{2}}+\ldots\) \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Asymptotic expansions of the roots (18).