## 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
\begin{table}
\begin{tabular}{c c c c} \hline \hline  & \(\varepsilon\to-\infty\) & \(\varepsilon=0\) & \(\varepsilon\to+\infty\) \\ \hline \(\mathbf{IG}_{4,0}^{\varepsilon}(\mathbf{r},\varepsilon)\) & \(+\mathbf{HG}_{4,0}(\mathbf{r})\) & \(\mathbf{LG}_{2,0}(\mathbf{r})\) & \(+\mathbf{HG}_{0,4}(\mathbf{r})\) \\ \(\mathbf{IG}_{4,2}^{\varepsilon}(\mathbf{r},\varepsilon)\) & \(+\mathbf{HG}_{2,2}(\mathbf{r})\) & \(\sqrt{2}\text{Re}\mathcal{LG}_{1,2}(\mathbf{r})\) & \(-\mathbf{HG}_{2,2}(\mathbf{r})\) \\ \(\mathbf{IG}_{4,4}^{\varepsilon}(\mathbf{r},\varepsilon)\) & \(+\mathbf{HG}_{0,4}(\mathbf{r})\) & \(\sqrt{2}\text{Re}\mathcal{LG}_{0,4}(\mathbf{r})\) & \(+\mathbf{HG}_{4,0}(\mathbf{r})\) \\ \hline \hline \end{tabular}
\end{table} 
Table 3: Limiting cases of the normalized even IG modes at \(p=4\). In addition, it can be shown that a simple interrelation exists between the IG modes constructed for the parameters \(\varepsilon\) and \(-\varepsilon\):  
\[ \begin{array}{l}\mathbf{IG}_{p,2m}^{\varepsilon}(x,y,-\varepsilon)=(-1)^{m} \mathbf{IG}_{p,2m}^{\varepsilon}(y,x,\varepsilon),\\ \mathbf{IG}_{p,2m}^{\varepsilon}(x,y,-\varepsilon)=(-1)^{m+1}\mathbf{IG}_{p,2 m}^{\varepsilon}(y,x,\varepsilon),\\ \mathbf{IG}_{p,2m+1}^{\varepsilon}(x,y,-\varepsilon)=(-1)^{m}\mathbf{IG}_{p, 2m+1}^{\varepsilon}(y,x,\varepsilon).\end{array}  \tag{21}\]  
The first two formulae here are written for the even value of \(p\), whereas the last is written for an odd value of \(p\). This result demonstrates once again that the case of negative \(\varepsilon\) is no less important in investigating the IG modes than the case of positive \(\varepsilon\). In particular, Equation (21) indicates that the IG modes with negative values of the parameter \(\varepsilon\) are also orthogonal to each other, similarly to the modes with positive values of \(\varepsilon\). The proof of the relationships (21) is based upon the following property of the characteristic polynomials:  
\[ \begin{array}{l}\chi_{p}^{(\varepsilon,\rho)}(\lambda,\varepsilon)=\chi_{p }^{(\varepsilon,\rho)}(\lambda,-\varepsilon),\text{ if $p$ is even,}\\ \chi_{p}^{\rho}(\lambda,\varepsilon)=\chi_{p}^{\rho}(\lambda,-\varepsilon), \text{ if $p$ is odd.}\end{array}  \tag{22}\]  
Now, we consider the case wherein \(p=5\). For the even IG modes, the characteristic equation is given by  
\[\chi_{5}^{\varepsilon}(\lambda,\varepsilon)=\lambda^{3}-(35+3\varepsilon) \lambda^{2}+(259+102\varepsilon-13\varepsilon^{2})\lambda-225-675\varepsilon +205\varepsilon^{2}+15\varepsilon^{3}=0. \tag{23}\]  
We again apply the variant of the Cardano formula wherein the cubic equation has three real-valued roots, and these roots in the ascending order read as follows:  
\[\lambda_{0}^{\varepsilon}=\left(\frac{35}{3}+\varepsilon\right)-2c_{5},\ \lambda_{1}^{ \varepsilon}=\left(\frac{35}{3}+\varepsilon\right)+c_{5}-s_{5}\sqrt{3},\ \lambda_{2}^{ \varepsilon}=\left(\frac{35}{3}+\varepsilon\right)+c_{5}+s_{5}\sqrt{3}, \tag{24}\]  
where  
\[ \begin{array}{l}c_{5}=\frac{4}{3}\sqrt{28-6\varepsilon+3\varepsilon^{2}} \cos\phi,\ s_{5}=\frac{4}{3}\sqrt{28-6\varepsilon+3\varepsilon^{2}}\sin\phi, \\ \phi=\frac{1}{3}\arccos\left(-\frac{80+9\varepsilon-18\varepsilon^{2}}{\left(2 8-6\varepsilon+3\varepsilon^{2}\right)^{3/2}}\right)\end{array} \]  
Asymptotic expansions of all three roots at small and large values of \(\varepsilon\) are given in Table 4.  
For \(p=5\), even IG modes (without normalization) are given by  
\[ \begin{array}{l}\mathcal{I}_{5,2m+1}^{\varepsilon}(\mathbf{r},\varepsilon)= 64\sqrt{2}\{2\text{Re}\mathcal{L}_{0,5}(\mathbf{r})-A_{1}\text{Re}\mathcal{L }_{1,3}(\mathbf{r})+A_{0}\text{Re}\mathcal{L}_{2,1}(\mathbf{r})\}\\ =(A_{0}+A_{1}+1)\mathcal{H}_{5,0}(\mathbf{r})+2(A_{0}-A_{1}-5)\mathcal{H}_{3, 2}(\mathbf{r})+(A_{0}-3A_{1}+5)\mathcal{H}_{1,\mathcal{A}}(\mathbf{r}),\end{array}