## 4 Taylor and Pade Expansions of Mie Coefficients  
It is rather obvious that the aforementioned straightforward but complicated Mie coefficients (Equations (1) and (2)) cannot actually offer direct physical interpretation on the studied problem, mostly due to their mathematical form. Several remedies have been proposed based on approximative methods, such as the Taylor expansion of the Mie coefficients [53; 60; 71; 77; 78; 79; 80; 81], the Weierstrass approximation [82], and the Pade approximation [63; 83; 84; 85]. The main objective of this section is to review some simple, yet intuitive, expressions regarding the dependencies governing the behavior of the Mie coefficients by expanding both \(a_{n}\) and \(b_{n}\) in a Taylor (Mie-Taylor) and Pade (Mie-Pade) manner. A short analysis for the position and width of the resonances, as well as the depolarization and radiative damping processes will be given as a function of the size parameter for both plasmonic and dielectric resonances. We begin by approximating Equations (1) and (2) with a Taylor series expansion for small \(x\), i.e.,  
\[a_{1}^{T}\approx-i\frac{2}{3}\frac{\varepsilon_{1}-1}{\varepsilon_{1}+2}x^{3}- i\frac{1}{5}\frac{\varepsilon_{1}^{2}(1+\mu_{1})-6\varepsilon_{1}+4}{( \varepsilon_{1}+2)^{2}}x^{5}+O[x]^{6} \tag{15}\]  
\[b_{1}^{T}\approx-i\frac{2}{3}\frac{\mu_{1}-1}{\mu_{1}+2}x^{3}-i\frac{1}{5} \frac{\mu_{1}^{2}(1+\varepsilon_{1})-6\mu_{1}+4}{(\mu_{1}+2)^{2}}x^{5}+O[x]^{6} \tag{16}\]  
A similar type of expressions can be readily found in many textbooks [16; 18; 19], and are valid for a general homogeneous sphere possessing both magnetic and dielectric properties. In the case of a magnetically inert (\(\mu_{1}=1\)) sphere, the above terms reduce to a much simpler form, viz.,  
\[a_{1}^{T}\approx-i\frac{2}{3}\frac{\varepsilon_{1}-1}{\varepsilon_{1}+2}x^{3}- i\frac{2}{5}\frac{(\varepsilon_{1}-2)(\varepsilon_{1}-1)}{(\varepsilon_{1}+2)^{ 2}}x^{5}+O[x]^{6} \tag{17}\]  
\[b_{1}^{T}\approx-i\frac{1}{45}(\varepsilon_{1}-1)x^{5}+O[x]^{7} \tag{18}\]The first term of the electric Mie coefficient represents the extracted polarizability for the electrostatic problem. Therefore, by truncating the orders higher than \(x^{3}\), we deduce that  
\[a_{1}^{T}=-i\frac{2}{9}\alpha x^{3} \tag{19}\]  
implying that the first Mie coefficient contains the same polarization enhancing character as the electrostatic polarizability. Although the Taylor expansion delivers a straightforward connection with the electrostatic problem, and therefore with the plasmonic enhancement, it offers very little insight regarding the dielectric resonances. This can be explained by the fact that the dielectric resonances asymptotically occur at extremely large permittivity contrast values for small spheres, as can be seen in Figure 2. Likewise, Equation (18) does not capture any resonances (poles of a system). Therefore, an alternative approximation stratagem is necessary to gain these pieces of information. The form of \(a_{n}\) and \(b_{n}\) suggests the usage of a rational approximation capable in describing all features of the modeled system (zeros and poles). Such expansions, fulfilling all the above requirements, are the Pade approximants. The Pade approximants are a special type of rational approximation constructed by expressing a function in a rational form and expanding each of the rational terms in power series [86; 87]. Here, the Pade expansion that corresponds to the fractional expression of two polynomials \(\frac{P(x)}{Q(x)}\) of order \(x^{L}\) and \(x^{M}\), is denoted as \([L/M]\), following the notation of [86]. As an example, a set of Pade approximants for the spherical Bessel and Hankel functions can be found in Table 1.  
Although the application of the Pade approximants for scattering problems was previously utilized for the Rayleigh regime [83; 84], recent studies shed light on the scattering phenomena in a systematic way, especially with respect to the resonant conditions and other enabled mechanisms [63; 72; 85].