## 2 An Overview on the SGCPs
### Some Fundamental Formulas of the SGCPs  
The SGCPs denoted by \(G_{n}^{(\sigma)}(z)\) are orthogonal polynomials on \([0,\mathcal{L}]\) whose orthogonality relation is ([48]):  
\[\int_{0}^{\mathcal{L}}G_{n}^{(\sigma)}(z)\;G_{m}^{(\sigma)}(z)\,w(z)dz=h_{n}^ {\sigma}\,\delta  
\[DG_{j}^{\sigma}(x)=\frac{j! }{\mathscr{L}}\sum_{s=0}^{j-1}\frac{(2+\sigma)^{-s+j}( s+j)_{s-j+1}}{s!\left(j-s-1\right)!\left(-1+\frac{1}{2+\sigma}+j\right)_{s-j+1}} \,{}_{3}F_{2}\left( \begin{array}{c}\frac{1+\sigma}{2+\sigma}-j,-m,2-2j+m\\ 1-2j,2-\frac{1}{2+\sigma}-j\end{array} \bigg{|}1\right)G_{s}^{\sigma}(x). \tag{5}\]  
Proof. : The series representation of \(G_{j}^{\sigma}(x)\) enables one to write the following:  
\[DG_{j}^{\sigma}(x)=\frac{(2+\sigma)^{j-1}(1+\sigma-(2+\sigma)j)}{\mathscr{L}^{ j-r}}\sum_{r=0}^{j-1}\frac{(j-r)\binom{j}{r}\Big{(}2-\frac{1}{2+\sigma}-j \Big{)}_{-1+r}}{(2j-r)_{r}}\,x^{j-r-1}. \tag{6}\]  
Formula (4) converts (6) into the following:  
\[ \begin{split} DG_{j}^{\sigma}(x)=&\frac{(2+\sigma)^{ j-1}(1+\sigma-(2+\sigma)j)}{\mathscr{L}}\sum_{r=0}^{j-1}\frac{(\frac{j}{r})(j-r)! \,\Gamma\Big{(}-1+\frac{1}{2+\sigma}+j-r\Big{)}\Big{(}2-\frac{1}{2+\sigma}-j \Big{)}_{r-1}}{(2j-r)_{r}}\times\\ &\sum_{t=0}^{j-r-1}\frac{(2+\sigma)^{1-j+r+t}}{(j-r-t-1)!\,t!\, \Gamma\Big{(}-1+\frac{1}{2+\sigma}+j-r-t\Big{)}(1+2(-1+j-r)-2t)_{t}}G_{j-r-t -1}^{\sigma}(x).\end{split}  \tag{7}\]  
After some computations, (7) can be written as follows:  
\[ \begin{split} DG_{j}^{\sigma}(x)=&\frac{(1-\sigma (-1+j)-2j)}{\mathscr{L}}\sum_{m=0}^{j-1}\frac{(2+\sigma)^{m}}{(j-m-1)!\,\Gamma \Big{(}-1+\frac{1}{2+\sigma}+j-m\Big{)}}\times\\ &\sum_{r=0}^{m}\frac{(\frac{j}{r})(j-r)!\,\Gamma\Big{(}-1+\frac{ 1}{2+\sigma}+j-r\Big{)}\Big{(}2-\frac{1}{2+\sigma}-j\Big{)}_{-1+r}}{(m-r)!\,( -1+2j-2m)_{m-r}(2j-r)_{r}}\,G_{j-m-1}^{\sigma}(x).\end{split}  \tag{8}\]  
The last formula is equivalent to the following:  
\[ \begin{split} DG_{j}^{\sigma}(x)=&\frac{j!\,\Gamma \Big{(}-1+\frac{1}{2+\sigma}+j\Big{)}}{\sqrt{\pi}\,\mathscr{L}}\sum_{m=0}^{j- 1}\frac{2^{2(-1+j-m)}(2+\sigma)^{1+m}\Gamma\Big{(}-\frac{1}{2}+j-m\Big{)}}{ \Gamma\Big{(}-1+\frac{1}{2+\sigma}+j-m\Big{)}m!\,(2j-m-2)!}\times\\ &{}_{3}F_{2}\left(\frac{\frac{1+\sigma}{2+\sigma}-j,-m,2-2j+m}{1- 2j,2-\frac{1}{2+\sigma}-j}\bigg{|}1\right)G_{j-m-1}^{\sigma}(x),\end{split}  \tag{9}\]  
and, thus, the following formula may be obtained:  
\[ \begin{split} DG_{j}^{\sigma}(x)=&\frac{j!}{\mathscr{ L}}\sum_{s=0}^{j-1}\frac{(2+\sigma)^{-s+j}(s+j)_{s-j+1}}{s!\left(j-s-1 \right)!\,\Big{(}-1+\frac{1}{2+\sigma}+j\Big{)}_{s-j+1}}\times\\ &{}_{3}F_{2}\left(\begin{array}{c}\frac{1+\sigma}{2+\sigma}-j, 1-s-j,s-j+1\\ 1-2j,2-\frac{1}{2+\sigma}-j\end{array}\bigg{|}1\right)G_{s}^{\sigma}(x).\end{split}  \tag{10}\]  
This proves Theorem 1. Now, the OMDs of the SGCPs can be constructed. Consider the vector \(\mathbf{G}(z)\) with the following components:  
\[\mathbf{G}(z)=[G_{0}^{\sigma}(z),G_{1}^{\sigma}(z),\dots,G_{N}^{\sigma}(z)]^{T}. \tag{11}\]Using Formula (5), we can write the following:  
\[\frac{d\,G(z)}{dz}=\Delta\,G(z). \tag{12}\]  
The matrix \(\Delta\) is called the OMD of the SGCPs, and its elements can be expressed in the following form:  
\[\Delta_{j,s}= \begin{cases}m_{s,j},&\text{if }s>j,\\ 0,&\text{otherwise},\end{cases}  \tag{13}\]  
with  
\[m_{s,j}=\frac{(2+\sigma)^{-s+j}j!\left(s+j\right)_{s-j+1}}{\mathscr{L}\,s! \left(j-s-1\right)!\left(-1+\frac{1}{2+\sigma}+j\right)_{s-j+1}}\,{}_{3}F_{2} \left( \begin{array}{c}\frac{\sigma+1}{\sigma+2}-j,1-s-j,s-j+1\\ 1-2j,2-\frac{1}{2+\sigma}-j\end{array} \right|1\right).\]  
For example, the matrix \(\Delta\) for \(N=5\) and \(\sigma=3\) has the following form:  
\[\mathcal{M}=\frac{1}{\mathscr{L}}\left( \begin{array}{cccccc}0&0&0&0&0&0\\ 5&0&0&0&0&0\\ -10&10&0&0&0&0\\ \frac{57}{2}&-6&15&0&0&0\\ -\frac{284}{27}&\frac{230}{7}&-\frac{36}{7}&20&0&0\\ \frac{250}{3}&-\frac{70}{3}&40&-\frac{100}{21}&25&0\end{array} \right).\]  
**Remark 1**.