## 4 Applying the Pade Approximants for Approximate Computation of IG Modes  
In this section, we demonstrate using a concrete example how the Pade approximants can be employed to compute the IG modes. In the limiting case, IG mode reduces to a HG mode: \(\mathbf{IG}_{3,1}^{\varepsilon}(\mathbf{r},\infty)=-\mathbf{HG}_{1,2}( \mathbf{r})\) (third row in Table 1). To obtain an approximate expansion of mode \(\mathbf{IG}_{3,1}^{\varepsilon}(\mathbf{r},\varepsilon)\) from the HG modes that is suitable for the whole range \(\varepsilon\in[0,\infty)\), we use the Pade approximants [20]. The Pade approximant of some function \(f(z)\) is given by the expression  
\[f(z\big{|}[L/M])=\frac{a_{0}+a_{1}z+\dots+a_{L}z^{L}}{1+b_{1}z+\dots+b_{M}z^{M}}, \tag{39}\]  
where the coefficients \(a_{k}\) and \(b_{k}\) are chosen so that at small values \(z\), it has several first terms of the Taylor expansion by the powers of \(z\), exactly the same as those of the function \(f(z)\), whereas at large \(z\), several first terms of the asymptotic expansion by the powers \((1/z)\) are the same as those of the function \(f(z)\). Collecting Equations (37) and (38) into the one, we get  
\[\mathbf{IG}_{3,1}^{\varepsilon}(\mathbf{r},\varepsilon)=\frac{1}{2\sqrt{2\pi} }\cdot\{C_{3,0}(\varepsilon)\mathcal{H}_{3,0}(\mathbf{r})+C_{1,2}(\varepsilon )\mathcal{H}_{1,2}(\mathbf{r})\}, \tag{40}\]  
where exact representations of the functions of \(\varepsilon\) are given by Equation (36), whereas their approximations in the form of Equation (39) depend on the choice of the parameters \(L\) and \(M\). Since  
\[C_{3,0}(0)=C_{1,2}(0)=-\frac{1}{2},\;C_{3,0}(\varepsilon)\underset{\varepsilon \to\infty}{\approx}-\frac{1}{2\varepsilon},\;C_{1,2}(\infty)=-1,\]  
then the Pade approximants of the functions \(C_{3,0}(\varepsilon)\) and \(C_{1,2}(\varepsilon)\) should have the following form:  
\[ \begin{array}{l}C_{3,0}(\varepsilon\big{|}[L/(L+1)]\,)=\frac{-\frac{1}{2}+a _{1}\varepsilon+\dots+a_{L}z^{L}}{1+b_{1}\varepsilon+\dots+b_{L}z^{L}-2a_{L} \varepsilon^{L+1}},\\ C_{1,2}(\varepsilon\big{|}[L/L])=\frac{-\frac{1}{2}+\frac{3}{2}\varepsilon+ \dots+\tilde{a}_{L}z^{L}}{1+b_{1}\varepsilon+\dots+b_{L}-1\varepsilon^{L}- \tilde{a}_{L}z^{L}}.\end{array}  \tag{41}\]  
For each fixed value of the parameter \(L\), the coefficients of the rational functions in Equation (41) can be found by solving systems of equations. These equations are obtained if we expand functions (41) into series in powers of \(\varepsilon\) at \(\varepsilon\to 0\) and in powers of \(1/\varepsilon\) at \(\varepsilon\to+\infty\), and then make equal these expansions to those already known from Equations (37) and (38). This approach is easy to implement in modern computer algebra systems. We here give only the simplest results when the obtained fractions are not very cumbersome yet:  
\[ \begin{array}{l}C_{3,0}(\varepsilon\big{|}[2/3]\,)=\frac{-\frac{1}{2}-\frac{ 1}{212}\varepsilon-\frac{20}{16\varepsilon^{2}}}{1+\frac{200\varepsilon^{2}} {1+\frac{100\varepsilon^{2}}{16\varepsilon^{2}}+\frac{40}{16\varepsilon^{2}} }{1+\frac{200\varepsilon^{2}}{16\varepsilon^{2}}+\frac{40}{16\varepsilon^{2}} }},\\ C_{1,2}(\varepsilon\big{|}[2/2]\,)=\frac{-\frac{1}{2}-\frac{1}{2}-\frac{1}{2} -\frac{1}{2}\varepsilon-\frac{20}{16\varepsilon^{2}}}{1+\frac{3}{2} \varepsilon+\frac{275}{16\varepsilon^{2}}},\\ C_{1,2}(\varepsilon\big{|}[3/3]\,)=\frac{-\frac{1}{2}-\frac{200\varepsilon^{2}} {16\varepsilon^{2}}-\frac{133111}{50\varepsilon^{2}}-\frac{93}{66\varepsilon^{ 3}}}{1+\frac{932\varepsilon^{2}}{16\varepsilon^{2}}+\frac{1351\varepsilon^{2}} {16\varepsilon^{2}}+\frac{275}{16\varepsilon^{3}}}.\end{array}