## 3 Expansions of the IG Modes by the HG and LG Modes at Small Values of \(p\)
### Even IG Modes at \(p=6\)  
Now we consider the case \(p=6\) for the even IG modes. The characteristic polynomial is quartic:  
\[\chi_{6}^{e}(\lambda,\varepsilon)=\lambda^{4}-56\lambda^{3}+8(98-5\varepsilon^ {2})\lambda^{2}-96(24-17\varepsilon^{2})\lambda-144\varepsilon^{2}(96- \varepsilon^{2})=0. \tag{32}\]  
Its roots \(\lambda_{m}^{e}(\varepsilon)\) can be obtained by the Ferrari formula. We use the variant of this formula presented in [19]. Let us introduce an auxiliary variable \(h=\frac{4}{3}(49+5\varepsilon^{2})+C_{6}+S_{6}\sqrt{3}\) with  
\[ \begin{array}{l}C_{6}=\frac{4}{3}\sqrt{889+178\varepsilon^{2}+13 \varepsilon^{4}}\cos\phi,\ S_{6}=\frac{4}{3}\sqrt{889+178\varepsilon^{2}+13 \varepsilon^{4}}\sin\phi\\ \phi=\frac{1}{3}\arccos\left(-\frac{24.013-6.033\varepsilon^{2}+1317\varepsilon ^{4}+35\varepsilon^{6}}{\left(889+178\varepsilon^{2}+13\varepsilon^{4}\right) ^{3/2}}\right).\end{array} \]  
Then, four roots of Equation (32) are expressed in ascending order via \(h\), as follows:  
\[ \begin{array}{l}\lambda_{0}^{e}=14-\sqrt{h}-\sqrt{4(49+5 \varepsilon^{2})-h-\frac{64(9-2\varepsilon^{2})}{\sqrt{h}}},\\ \lambda_{1}^{e}=14-\sqrt{h}+\sqrt{4(49+5\varepsilon^{2})-h-\frac{64(9-2 \varepsilon^{2})}{\sqrt{h}}},\\ \lambda_{2}^{e}=14+\sqrt{h}-\sqrt{4(49+5\varepsilon^{2})-h+\frac{64(9-2 \varepsilon^{2})}{\sqrt{h}}},\\ \lambda_{3}^{e}=14+\sqrt{h}+\sqrt{4(49+5\varepsilon^{2})-h+\frac{64(9-2 \varepsilon^{2})}{\sqrt{h}}}.\end{array}  \tag{33}\]  
Asymptotic expansions of all roots at small and large values of \(\varepsilon\) are written in Table 6.  
Then, the non-normalized IG modes are expressed via the LG and HG modes in the following way:  
\[ \begin{array}{l}\mathcal{I}_{6,2m}^{e}(\mathbf{r},\varepsilon)=128\{4 \text{Re}\mathcal{L}_{0,6}(\mathbf{r})-2A_{2}\text{Re}\mathcal{L}_{1,A}( \mathbf{r})\ +\ 2A_{1}\text{Re}\mathcal{L}_{2,2}(\mathbf{r})-3A_{0}\mathcal{L}_{3,0}( \mathbf{r})\}\\ =(A_{0}+A_{1}+A_{2}+1)\mathcal{H}_{6,0}(\mathbf{r})+(3A_{0}+A_{1}-5A_{2}-15) \mathcal{H}_{4,2}(\mathbf{r})\\ +(3A_{0}-A_{1}-5A_{2}+15)\mathcal{H}_{2,4}(\mathbf{r})+(A_{0}-A_{1}+A_{2}-1) \mathcal{H}_{0,6}(\mathbf{r}),\end{array}  \tag{34}\] \begin{table}
\begin{tabular}{c c c} \hline \hline  & \(\varepsilon\approx\mathbf{0}\) & \(\varepsilon\rightarrow\infty\) \\ \hline \(\lambda_{0}^{e}\) & \(-6\varepsilon^{2}+\frac{129\varepsilon^{4}}{16}-\ldots\) & \(-6\varepsilon+6-\frac{15}{2}+\frac{60}{2}-\ldots\) \\ \(\lambda_{1}^{e}\) & \(4+\frac{31\varepsilon^{2}}{6}-\frac{280\varepsilon^{6}}{456}+\ldots\) & \(-2\varepsilon+22-\frac{5}{24}-\frac{60}{2}+\ldots\) \\ \(\lambda_{2}^{e}\) & \(16+\frac{8\varepsilon^{2}}{15}+\frac{3150\varepsilon^{4}}{54,100}-\ldots\) & \(2\varepsilon+22+\frac{21}{2}-\frac{60}{2}-\ldots\) \\ \(\lambda_{3}^{e}\) & \(36+\frac{30}{10}+\frac{34}{16,000}+\ldots\) & \(6\varepsilon+6+\frac{15}{\varepsilon}+\frac{60}{\varepsilon^{2}}+\ldots\) \\ \hline \hline \end{tabular}
\end{table} 
Table 6: Asymptotic expansions of the roots (33). with \(m=0,1,2,3\) and  
\[ \begin{array}{l}A_{0}=\frac{(\lambda_{m}^{e}-4)(\lambda_{m}^{e}-16)(\lambda_{m} ^{e}-36)}{(12\lambda^{3})}-\frac{4(\lambda_{m}^{e}-24)}{3\varepsilon},\\ A_{1}=\frac{(\lambda_{m}^{e}-16)(\lambda_{m}^{e}-36)}{2\varepsilon^{2}}-3,A_{2}= \frac{\lambda_{m}^{e}-36}{\varepsilon}.\end{array} \]  
For the limiting cases, the normalized IG modes are shown in Table 7.  
It is obvious that when index \(p\), which determines the degree of the characteristic polynomial, increases, finding all of its roots in a simple form and without numerical methods becomes increasingly impractical. Nevertheless, it is easy to observe that for \(\varepsilon=\sqrt{19}\), one obtains the roots \(\lambda_{m}^{e}=\left\{-22,18-4\sqrt{6},18+4\sqrt{6},42\right\}\), whereas the value \(\varepsilon=3\sqrt{3}\) yields the roots \(\lambda_{m}^{e}=\left\{2-8\sqrt{13},26-4\sqrt{22},2+8\sqrt{13},26+4\sqrt{22}\right\}\). Thus, for example,  
\[