## 4 Error Analysis
This shows that, while holding \(\alpha\) fixed, the bounding constant \(\vartheta_{a,\lambda}\) displays a unimodal behavior: it rises from 0 as \(\lambda\) increases from \(-1/2^{+}\), attains a maximum at \(\lambda^{*}\approx-0.1351\), and subsequently decays monotonically for \(\lambda>\lambda^{*}\). Figure 3 highlights how \(\vartheta_{a,\lambda}\) decays with increasing \(\alpha\) while exhibiting varying sensitivity to \(\lambda\), with the \(\lambda=\lambda^{*}\) case (red curve) representing the parameter value that maximizes \(\vartheta_{a,\lambda}\) for fixed \(\alpha\). Figure 2: (**Left**) Comparison of the functions \(T_{2}\) (blue curve) and \(T_{3}\) (red curve) over the interval \(\mathbb{R}_{-1/2^{+}}^{-}\). (**Right**) The product \(T_{2}T_{3}\) (purple), showing the combined behavior of the two functions. All plots demonstrate the dependence on the parameter \(\lambda\) in the negative domain near zero. It is important to note that the bounding constant \(\theta_{a,\lambda}\) modulates the error bound without altering its super-exponential decay as \(n\to\infty\). Near \(\lambda=-1/2\), \(\theta_{a,\lambda}\to 0\), which shrinks the truncation error bound, improving accuracy despite potential sensitivity in the SG polynomials. At \(\lambda=\lambda^{*}\), the maximized \(\theta_{a,\lambda}\) widens the bound, though it still vanishes as \(n\to\infty\). Beyond \(\lambda=\lambda^{*}\), the overall error bound decreases monotonically, despite a slower \(n\)-dependent decay for larger positive \(\lambda\). The super-exponential term \(\big{(}\frac{e}{4n}\big{)}^{n}\) ensures rapid convergence in all cases. The decreasing \(\theta_{a,\lambda}\) after \(\lambda^{*}\), combined with the fact that the truncation error bound (excluding \(\theta_{a,\lambda}\)) is smaller for \(-1/2<\lambda\leq 0\) compared to its value for \(\lambda>0\), further suggests that \(\lambda=0\) appears \"optimal\" in practice for minimizing the truncation error bound \(\forall n\) in a stable numerical scheme, given the SG polynomial instability near \(\lambda=-1/2\). However, for relatively small or moderate values of \(n\), other choices of \(\lambda\) may be optimal, as the derived error bounds are asymptotic and apply only as \(n\to\infty\). In what follows, we analyze the truncation error of the quadrature Formula (21), and demonstrate how its results complement the preceding analysis. **Theorem 3**. : _Let \(j\in\mathbb{J}_{n}^{+},t\in\boldsymbol{\Omega}_{1}\), and assume that \(\hat{G}_{j}^{\lambda}\Big{(}t(1-y^{1/\alpha})\Big{)}\) is interpolated by the SG polynomials with respect to the variable \(y\) at the SGG nodes \(\hat{t}_{n_{q}\uparrow 0:n_{q}}^{\lambda_{q}}\). Then \(\exists\,\eta=\eta(y)\in\boldsymbol{\Omega}_{1}^{\circ}\) such that the truncation error, \(\Xi_{j,n_{q}}^{\lambda_{q}}(\eta)\), in the quadrature approximation (21) is given by_  
\[\Xi_{j,n_{q}}^{\lambda_{q}}(\eta)=\frac{(-1)^{n_{q}+1}\hat{\chi}_{j,n_{q}+1}^{ \lambda}}{(n_{q}+1)!\hat{\kappa}_{n_{q}+1}^{\lambda_{q}}}\bigg{(}\frac{t}{ \alpha}\bigg{)}^{n_{q}+1}\eta^{\frac{(n_{q}+1)(1-\alpha)}{\alpha}}\mathcal{G} _{j-n_{q}-1}^{\lambda+n_{q}+1}\Big{(}t\Big{(}1-\eta^{\frac{1}{\alpha}}\Big{)} \Big{)}\,\mathcal{I}_{1}^{(y)}\mathcal{G}_{n_{q}+1}^{\lambda_{q}}\cdot\mathbb{ I}_{j\geq n_{q}+1}, \tag{34}\]  
_where \(\hat{\chi}_{n,m}^{\lambda}\) is defined by_  
\[\left[\frac{(1+\lambda)\operatorname{csch}\!\left(\tfrac{1}{1+2 \lambda}\right)}{1+2\lambda}\right]^{\frac{1}{2}+\lambda}\left[\alpha\sinh \!\left(\frac{1}{\alpha}\right)\right]^{-\alpha/2}\!\left[(1+\lambda)\sinh \!\left(\frac{1}{1+\lambda}\right)\right]^{\frac{1+\lambda}{2}}. \tag{28}\]0  
Proof. : Let \(\hat{G}_{j}^{\lambda,m}\) denote the \(m\)th-derivative of \(\hat{G}_{j}^{\lambda}\,\forall j\in\mathbb{J}_{n}^{+}\). Ref.