## 3 Special Cases of Weight Functions
\begin{array}{ll}T_{f}(s)&\in\{\frac{1}{2}(\frac{3s+1}{3s-1}),\frac{1 }{8}(5+\frac{3}{s^{2}}),\frac{1}{32}\big{(}6s+1+\frac{125}{6s-1}\big{)},\frac{2 }{8}-3s+\frac{9}{8}s^{2}\},\\ L_{f}(s)&\in\{\frac{1}{4}(\frac{3s+1}{3s-1})^{2},\frac{1}{2}(\frac{3}{s}-1), \frac{2}{3s-1},\frac{2s}{5s-3},\frac{1}{4}(7-3s^{2}),\frac{4k^{2}}{7s-3},\frac {1}{4}(1+\frac{3}{s^{2}}),\frac{4}{1+3s^{2}},\frac{5-3s}{2},\frac{2(2-s)}{(1+s) }\}.\end{array} \right. \tag{20}\]  
**Remark 2**. : _Existing studies with combinations of weight functions \((T_{f},L_{f})\in\{\big{(}\frac{1}{2}(\frac{3s+1}{3s-1}),\frac{1}{4}(\frac{3s+1 }{3s-1})^{2}\big{)},\big{(}\frac{1}{2}(\frac{3s+1}{3s-1}),\frac{2}{3s-1}\big{)},\big{(}\frac{1}{8}(5+\frac{3}{s^{2}}),\frac{1}{2}(\frac{3}{s}-1)\big{)}, \big{(}\frac{1}{2}(\frac{3s+1}{3s-1}),\frac{1}{2}(\frac{3}{s}-1)\big{)}\}\) can be found, respectively, in [3,8,9,22]. These existing methods are denoted by \(EM_{1},EM_{2},EM_{3}\), and \(EM_{4}\), respectively, for use below. Note that selecting \(\alpha=\beta=2\) in (16) readily yields \(T_{f}(s)=\frac{2s}{3s-1}\)._  
**Case B: \(\gamma=1\)**  
\[\left\{ \begin{array}{ll}T_{f}(s)\in\{\frac{2s}{3s-1},\frac{2s}{5s-3},\frac{1 }{2}(3-s),-\frac{1}{2}s(3s-5),\frac{1+s}{2s},\frac{3s-1}{2s^{2}},\frac{2}{1+s} \frac{5-6}{3s-5},\frac{-2}{s(3-5)},\frac{4}{3}(3+\frac{1}{s^{2}}),\frac{4k^{2}} {5s-1},\frac{4}{s^{2}+3},\\ \frac{1}{4}(5-s^{2}),(\frac{3s+1}{4})^{2},\frac{(s-5)^{2}}{16},4(\frac{3s-5}{s -1})^{2},(\frac{4s}{5s-1})^{2},\frac{16}{(s+3)^{2}},\frac{1}{4}(\frac{3s+1}{s +1})^{2}\},\\ L_{f}(s)\in\{\frac{7-8+3s^{2}}{2},\frac{1}{2}(1+\frac{1}{s^{2}}),\frac{1}{2} (s-2+\frac{3}{s}),\frac{4s+1}{3s-1},\frac{1}{34}(4s+1+\frac{81}{4s-1}),\frac{1} {48}(32-7s+\frac{96}{7s-3})\}.\end{array} \right. \tag{21}\]  
**Remark 3**. : _Existing studies with combinations of weight functions \((T_{f},L_{f})\in\{\big{(}\frac{1+s}{2s},\frac{7-8-8+3s^{2}}{2}\big{)},\big{(} \frac{2}{1+s},\frac{s+1}{3s-1}\big{)},\big{(}\frac{1+s}{2s},\frac{1}{2}(1+\frac {1}{s^{2}})\big{)}\}\) can be found, respectively, in [7,22,23]. These existing methods are denoted by \(EM_{5},EM_{6}\) and \(EM_{7}\), respectively, for use below._  
In view of (20) and (21), we can select a total of 154 special pairs of second-order rational weight functions \((T_{f}(s),L_{f}(s))\). Excluding known studies, the following pairs of weight functions \((T_{f}(s),L_{f}(s))\) may be of great interest to us. The corresponding methods to such pairs of weight functions \((T_{f}(s),L_{f}(s))\) are denoted by \({\color[rgb]{0,0,0}{\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0} \pgfsys@color@rgb@stroke{0}{0}{0}{\pgfsys@color@  
**Definition 1**. : _(Computational Convergence Order (COC) and Approximated Computational Convergence Order (ACOC)) Assume that theoretical asymptotic error constant \(\eta=\lim_{n\to\infty}\frac{|\epsilon_{n}|}{|\epsilon_{n-1}|^{p}}\) and convergence order \(p\geq 1\) are known. Define \(p_{n}=\big{|}\frac{\log|\epsilon_{n}/\eta|}{\log|\epsilon_{n-1}|}\big{|}\) as the computational convergence order. Note that \(\lim_{n\to\infty}p_{n}=p\). Approximated computational convergence order \(\bar{p}_{n}\) is defined as \(\bar{p}_{n}\)=\(\big{|}\frac{\log(|\epsilon_{n}-\epsilon_{n-1}|/|\epsilon_{n-1}-\epsilon_{n-2}| )}{\log(|\epsilon_{n-1}-\epsilon_{n-2}|/|\epsilon_{n-2}-\epsilon_{n-3}|)}\big{|}\) requiring knowledge at four points \(x_{n},x_{n-1},x_{n-2},x_{n-3}\)._