## 3 Expansions of the IG Modes by the HG and LG Modes at Small Values of \(p\)
### Numerical Methods and IG Modes  
It is known that for all characteristic polynomials \(\chi_{p}^{(e\rho)}(\lambda,\varepsilon)\), regardless of the index \(p\) and the parity flag \((e,o)\), the coefficient for any term of the form \(\lambda^{v}\varepsilon^{u}\) is an integer. Therefore, the series expansions of the roots \(\lambda_{m}^{(e\rho)}(\varepsilon)\) by powers of \(\varepsilon\) (both at \(\varepsilon\to 0\) and at \(\varepsilon\to\pm\infty\)) have expansion coefficients that are rational numbers (see the above formulae as examples of such expansions). Consequently, asymptotic expansions of the IG modes by the LG and HG modes have coefficients that can be represented as series of the powers of \(\varepsilon\) (if \(\varepsilon\to 0\), then by powers \(\varepsilon,\varepsilon^{2},\ldots\); if \(\varepsilon\to\pm\infty\), then by the negative powers), and the coefficients in these series are also rational numbers.  
For instance, let us consider mode \(\mathbf{IG}_{3,1}^{\varepsilon}(\mathbf{r},\varepsilon)\), writing it in the following form:  
\[\mathbf{IG}_{3,1}^{\varepsilon}(\mathbf{r},\varepsilon)=\frac{1}{2\sqrt{2\pi }}\cdot\frac{(A_{0}+1)\mathcal{H}_{3,0}(\mathbf{r})+(A_{0}-3)\mathcal{H}_{1,2 }(\mathbf{r})}{\sqrt{\big{(}A_{0}+1\big{)}^{2}3+\big{(}A_{0}-3\big{)}^{2}}}. \tag{36}\]  
Here, \(2\sqrt{2\pi}=\|\mathcal{H}_{1,2}(\mathbf{r})\|=4\|\mathcal{L}_{1,1}(\mathbf{ r})\|\) and  
\[A_{0}=\frac{\lambda_{0}^{e}-9}{\varepsilon}=\frac{-4+\varepsilon-2\sqrt{4-2 \varepsilon+\varepsilon^{2}}}{\varepsilon}.\]  
If \(\varepsilon\to 0\) then  
\[\begin{array}{l}\mathbf{IG}_{3,1}^{\varepsilon}(\mathbf{r},\varepsilon)= \frac{1}{2\sqrt{2\pi}}\cdot\Big{\{}\Big{(}-\frac{1}{2}+\frac{\varepsilon}{16 }+\frac{7\varepsilon^{2}}{256}+\frac{11\xi^{3}}{2048}-\ldots\Big{)}\,\mathcal{H }_{3,0}(\mathbf{r})\\ +\Big{(}-\frac{1}{2}-\frac{35}{16}-\frac{9\varepsilon^{2}}{256}+\frac{15\xi^{ 3}}{2048}+\ldots\Big{)}\,\mathcal{H}_{1,2}(\mathbf{r})\Big{\}}.\end{array} \tag{37}\]  
In particular, as \(-\frac{1}{2}\{\mathcal{H}_{3,0}(\mathbf{r})+\mathcal{H}_{1,2}(\mathbf{r})\}=4 \sqrt{2}\text{Re}\mathcal{L}_{1,1}(\mathbf{r})\), then, as a limiting case, we obtain an identity that is already well known: \(\mathbf{IG}_{3,1}^{\varepsilon}(\mathbf{r},0)=\sqrt{2}\text{Re}\mathbf{LG}_{1, 1}(\mathbf{r})\).  
\begin{table}
\begin{tabular}{c c c c} \hline  & \(\varepsilon\to-\infty\) & \(\varepsilon=\mathbf{0}\) & \(\varepsilon\to+\infty\) \\ \hline \(\mathbf{IG}_{6,0}^{\varepsilon}(\mathbf{r},\varepsilon)\) & \(-\mathbf{HG}_{6,0}(\mathbf{r})\) & \(\mathbf{LG}_{3,0}(\mathbf{r})\) & \(-\mathbf{HG}_{0,6}(\mathbf{r})\) \\ \(\mathbf{IG}_{6,2}^{\varepsilon}(\mathbf{r},\varepsilon)\) & \(-\mathbf{HG}_{4,2}(\mathbf{r})\) & \(\sqrt{2}\text{Re}\mathbf{LG}_{2,2}(\mathbf{r})\) & \(+\mathbf{HG}_{2,4}(\mathbf{r})\) \\ \(\mathbf{IG}_{6,4}^{\varepsilon}(\mathbf{r},\varepsilon)\) & \(-\mathbf{HG}_{2,4}(\mathbf{r})\) & \(\sqrt{2}\text{Re}\mathbf{LG}_{1,4}(\mathbf{r})\) & \(-\mathbf{HG}_{4,2}(\mathbf{r})\) \\ \(\mathbf{IG}_{6,6}^{\varepsilon}(\mathbf{r},\varepsilon)\) & \(-\mathbf{HG}_{0,6}(\mathbf{r})\) & \(\sqrt{2}\text{Re}\mathbf{LG}_{0,6}(\mathbf{r})\) & \(+\mathbf{HG}_{6,0}(\mathbf{r})\) \\ \hline \end{tabular}
\end{table}
Table 7: Limiting cases of normalized even IG modes at \(p=6\).  
If \(\varepsilon\to+\infty\) then  
\[\begin{array}{l}\mathbf{IG}_{3,1}^{\varepsilon}(\mathbf{r},\varepsilon)=\frac{1 }{2\sqrt{2\pi}}\Big{\{}\Big{(}-\frac{1}{2\varepsilon^{2}}+\frac{1}{16\varepsilon ^{3}}+\frac{19}{16\varepsilon^{4}}+\dots\Big{)}\mathcal{H}_{3,0}(\mathbf{r}) \\ +\Big{(}-1+\frac{3}{8\varepsilon^{2}}+\frac{3}{4\varepsilon^{3}}+\frac{45}{12 8\varepsilon^{4}}+\dots\Big{)}\mathcal{H}_{1,2}(\mathbf{r})\Big{\}}.\end{array} \tag{38}\]  
Here, in the limiting case, we obtain a HG mode: \(\mathbf{IG}_{3,1}^{\varepsilon}(\mathbf{r},\infty)=-\mathbf{HG}_{1,2}( \mathbf{r})\).