Appendix
Appendix A: derivation of Eq (6) using Eq. (3)
Employing Eq. (3), we have
$$\begin{aligned} M_{p,,0}^{1} & = - \frac{p!}{{\left( {p + 2} \right)!}}\left[ {\left( {\mathop \sum \limits_{i = 0}^{p} x_{2}^{i + 1} x_{0}^{p - i} - \mathop \sum \limits_{i = 0}^{p} x_{0}^{i} x_{1}^{p + 1 - i} } \right)y_{0} + \left( {\mathop \sum \limits_{i = 0}^{p} x_{0}^{i + 1} x_{1}^{p - i} } \right.} \right. \\ & \quad \left. {\left. { - \mathop \sum \limits_{i = 0}^{p} x_{1}^{i} x_{2}^{p + 1 - i} } \right)y_{1} + \left( {\mathop \sum \limits_{i = 0}^{p} x_{1}^{i + 1} x_{2}^{p - i} - \mathop \sum \limits_{i = 0}^{p} x_{2}^{i} x_{0}^{p + 1 - i} } \right)y_{2} } \right],\;p = 0,1, \ldots ,\infty \\ M_{0,q}^{1} & = - \frac{q!}{{\left( {q + 2} \right)!}}\left[ {\left( {\mathop \sum \limits_{j = 0}^{q} y_{0}^{j} y_{1}^{q + 1 - j} - \mathop \sum \limits_{j = 0}^{q} y_{2}^{j + 1} y_{0}^{q - j} } \right)x_{0} + \left( {\mathop \sum \limits_{j = 0}^{q} y_{1}^{j} y_{2}^{q + 1 - j} } \right.} \right. \\ & \quad \left. {\left. { - \mathop \sum \limits_{j = 0}^{q} y_{0}^{j + 1} y_{1}^{q - j} } \right)x_{1} + \left( {\mathop \sum \limits_{j = 0}^{q} y_{2}^{j} y_{0}^{q + 1 - j} - \mathop \sum \limits_{j = 0}^{q} y_{1}^{j + 1} y_{2}^{q - j} } \right)x_{2} } \right],q = 0,1, \ldots ,\infty . \\ \end{aligned}$$
(A.1)
Using the relation \(\sum_{i=0}^{p}{A}^{i+1}{B}^{p-i}=\sum_{i=0}^{p}{A}^{p+1-i}{B}^{i}\) in which, each \(A\) and \(B\) can be replaced with one of the variables \({x}_{0}, {x}_{1}, {x}_{2}, {y}_{0}, {y}_{1}, {y}_{2},\) Eq. (A.1) can be simplified as below
$$\begin{aligned} M_{p,0}^{1} & = - \frac{p!}{{\left( {p + 2} \right)!}}\left[ {y_{0} \mathop \sum \limits_{i = 0}^{p} \left( {x_{2}^{p + 1 - i} - x_{1}^{p + 1 - i} } \right)x_{0}^{i} + y_{1} } \right. \\ & \quad \left. { \times \mathop \sum \limits_{i = 0}^{p} \left( {x_{0}^{p + 1 - i} - x_{2}^{p + 1 - i} } \right)x_{1}^{i} + y_{2} \mathop \sum \limits_{i = 0}^{p} \left( {x_{1}^{p + 1 - i} - x_{0}^{p + 1 - i} } \right)x_{2}^{i} } \right],p = 0,1, \ldots ,\infty \\ M_{0,q}^{1} & = - \frac{q!}{{\left( {q + 2} \right)!}}\left[ {x_{0} \mathop \sum \limits_{j = 0}^{q} \left( {y_{1}^{q + 1 - j} - y_{2}^{q + 1 - j} } \right)y_{0}^{j} + x_{1} } \right. \\ & \quad \left. { \times \mathop \sum \limits_{j = 0}^{q} \left( {y_{2}^{q + 1 - j} - y_{0}^{q + 1 - j} } \right)y_{1}^{j} + x_{2} \mathop \sum \limits_{j = 0}^{q} \left( {y_{0}^{q + 1 - j} - y_{1}^{q + 1 - j} } \right)y_{2}^{j} } \right], \\ q & = 0,1, \ldots ,\infty \\ \end{aligned}$$
(A.2)
Making use of the relation \(\sum_{i=0}^{p}{(A}^{p+1-i}-{B}^{p+1-i}){C}^{i}=\left(A-B\right)\sum_{r+s+t=p}{C}^{r}{B}^{s}{A}^{t},\) in which, each of \(A,B\) and \(C\) can be replaced with one of the variables \({x}_{0},{x}_{1},{x}_{2},{y}_{0},{y}_{1},{y}_{2},\) Eq. (A.2) can be more simplified. In this relation, the counting numbers \(r,s,t\le p\) are chosen such that \(r+s+t=p.\) These relations can be proved by the mathematical induction. The summation \(\sum_{r+s+t=p}{x}_{0}^{r}{x}_{1}^{s}{x}_{2}^{t}\) runs over all monomials \({x}_{0}^{r}{x}_{1}^{s}{x}_{2}^{t},\) of degree \(p.\) Finally, employing the simplified form of Eq. (A.2) and viewing the first equation of (1) results in Eq. (6).
Appendix B
Since the coordinates \({x}_{0},{x}_{1},{x}_{2}\) and \({y}_{0},{y}_{1},{y}_{2}\) are the roots of Eq. (11), respectively, then
$$\begin{aligned} & x_{0}^{p} - X_{1} x_{0}^{p - 1} + X_{2} x_{0}^{p - 2} - X_{3} x_{0}^{p - 3} = 0 \\ & x_{1}^{p} - X_{1} x_{1}^{p - 1} + X_{2} x_{1}^{p - 2} - X_{3} x_{1}^{p - 3} = 0 \\ & x_{2}^{p} - X_{1} x_{2}^{p - 1} + X_{2} x_{2}^{p - 2} - X_{3} x_{3}^{p - 3} = 0 \\ & y_{0}^{p} - Y_{1} y_{0}^{p - 1} + Y_{2} y_{0}^{p - 2} - Y_{3} y_{0}^{p - 3} = 0 \\ & y_{1}^{p} - Y_{1} y_{1}^{p - 1} + Y_{2} y_{1}^{p - 2} - Y_{3} y_{1}^{p - 3} = 0 \\ & y_{2}^{p} - Y_{1} y_{2}^{p - 1} + Y_{2} y_{2}^{p - 2} - Y_{3} y_{3}^{p - 3} = 0 \\ \end{aligned}$$
(B.1)
Summing up the first and second triple sets of the above equations, we find
$$\begin{aligned} S_{p} & = X_{1} S_{p - 1} - X_{2} S_{p - 2} + X_{3} S_{p - 3} ,p \ge 3 \\ R_{p} & = Y_{1} R_{p - 1} - Y_{2} R_{p - 2} + Y_{3} R_{p - 3} \\ \end{aligned}$$
(B.2)
where \({S}_{p}={x}_{0}^{p}+{x}_{1}^{p}+{x}_{2}^{p}\) and \({R}_{p}={y}_{0}^{p}+{y}_{1}^{p}+{y}_{2}^{p}.\) Furthermore, \({S}_{0}=3,{S}_{1}={X}_{1},{S}_{2}={({X}_{1})}^{2}-2{X}_{2}\) and \({R}_{0}=3,{R}_{1}={Y}_{1},{R}_{2}={({Y}_{1})}^{2}-2{Y}_{2}.\) Also, we can easily prove that the set of variables \({x}_{1}{x}_{2}, {x}_{0}{x}_{2}, {x}_{0}{x}_{1}\) and \({y}_{1}{y}_{2}, {y}_{0}{y}_{2}, {y}_{0}{y}_{1}\) are the zeros of the equation \({z}^{3}-{X}_{2}{z}^{2}+{X}_{3}{X}_{1}z-{\left({X}_{3}\right)}^{2}=0,\) and \({w}^{3}-{Y}_{2}{w}^{2}+{Y}_{3}{Y}_{1}w-{\left({Y}_{3}\right)}^{2}=0,\), respectively. Therefore, by following up the similar procedure of obtaining Eq. (B.2), we have
$$\begin{aligned} D_{p} & = X_{2} D_{p - 1} - X_{3} X_{1} D_{p - 2} + \left( {X_{3} } \right)^{2} D_{p - 3} \\ E_{p} & = X_{2} E_{p - 1} - X_{3} X_{1} E_{p - 2} + \left( {X_{3} } \right)^{2} E_{p - 3} \\ \end{aligned}$$
(B.3)
where \({D}_{p}={x}_{0}^{p}{x}_{1}^{p}+{x}_{1}^{p}{x}_{2}^{p}+{x}_{0}^{p}{x}_{2}^{p}\) and \({E}_{p}={y}_{0}^{p}{y}_{1}^{p}+{y}_{1}^{p}{y}_{2}^{p}+{y}_{0}^{p}{y}_{2}^{p}.\) Furthermore, \({D}_{0}=3,{D}_{1}={X}_{2},{D}_{2}={\left({X}_{2}\right)}^{2}-2{X}_{3}{X}_{1}\) and \({E}_{0}=3,{E}_{1}={Y}_{2},{E}_{2}={\left({Y}_{2}\right)}^{2}-2{Y}_{3}{Y}_{1}.\) Using Eq. (B.2) and (B.3), we arrive at
$$\begin{aligned} S_{3} & = \left( {X_{1} } \right)^{3} - 3X_{1} X_{2} + 3X_{3} \\ S_{4} & = \left( {X_{1} } \right)^{4} - 4\left( {X_{1} } \right)^{2} X_{2} + 4X_{1} X_{3} + 2\left( {X_{2} } \right)^{2} \\ S_{5} & = \left( {X_{1} } \right)^{5} - 5\left( {X_{1} } \right)^{3} X_{2} + 5\left( {X_{1} } \right)^{2} X_{3} + 5X_{1} \left( {X_{2} } \right)^{2} - 5X_{2} X_{3} \\ S_{6} & = \left( {X_{1} } \right)^{6} - 6\left( {X_{1} } \right)^{4} X_{2} + 6\left( {X_{1} } \right)^{3} X_{3} + 9\left( {X_{1} X_{2} } \right)^{2} - 12X_{1} X_{2} X_{3} - 2\left( {X_{2} } \right)^{3} + 3\left( {X_{3} } \right)^{2} \\ D_{3} & = \left( {X_{2} } \right)^{3} - 3X_{2} X_{3} X_{1} + 3\left( {X_{3} } \right)^{2} \\ \end{aligned}$$
(B.4)
Some terms of Eq. (15) can be rewritten in terms of variables \({X}_{1},{X}_{2},{X}_{3}\) by use of relations (B.4) as follows
$$\begin{aligned} & x_{0}^{6} + x_{1}^{6} + x_{2}^{6} + x_{0}^{5} \left( {x_{1} + x_{2} } \right) + x_{1}^{5} \left( {x_{0} + x_{2} } \right) + x_{2}^{5} \left( { x_{0} + x_{1} } \right) \\ & \qquad + x_{0} x_{1} x_{2} \left( {x_{0} + x_{1} + x_{2} } \right)\left( {x_{0}^{2} + x_{1}^{2} + x_{2}^{2} } \right) + \left( {x_{0}^{2} + x_{1}^{2} } \right)x_{0}^{2} x_{1}^{2} \\ & \qquad + x_{0}^{2} x_{2}^{2} \left( { x_{0}^{2} + x_{2}^{2} } \right) + x_{1}^{2} x_{2}^{2} \left( {x_{1}^{2} + x_{2}^{2} } \right) + x_{0}^{3} x_{1}^{3} + x_{1}^{3} x_{2}^{3} + x_{0}^{3} x_{2}^{3} + x_{0}^{2} x_{1}^{2} x_{2}^{2} \\ &\quad = X_{1} S_{5} + X_{3} X_{1} S_{2} + S_{2} D_{2} + D_{3} - 2 \left( {X_{3} } \right)^{2} \\ &\quad = X_{1}^{6} - 5X_{1}^{4} X_{2} + 4X_{1}^{3} X_{3} + 6X_{1}^{2} X_{2}^{2} - 6X_{1} X_{2} X_{3} - X_{2}^{3} + X_{3}^{2} , \\ & \qquad + x_{0}^{2} x_{2}^{2} \left( { x_{0}^{2} + x_{2}^{2} } \right) + x_{1}^{2} x_{2}^{2} \left( {x_{1}^{2} + x_{2}^{2} } \right) + x_{0}^{3} x_{1}^{3} + x_{1}^{3} x_{2}^{3} + x_{0}^{3} x_{2}^{3} + x_{0}^{2} x_{1}^{2} x_{2}^{2} \\ &\quad = X_{1} S_{5} + X_{3} X_{1} S_{2} + S_{2} D_{2} + D_{3} - 2 \left( {X_{3} } \right)^{2} \\ &\quad = X_{1}^{6} - 5X_{1}^{4} X_{2} + 4X_{1}^{3} X_{3} + 6X_{1}^{2} X_{2}^{2} - 6X_{1} X_{2} X_{3} - X_{2}^{3} + X_{3}^{2} , \\ & x_{0}^{5} + x_{1}^{5} + x_{2}^{5} + x_{0}^{4} \left( {x_{1} + x_{2} } \right) + x_{1}^{4} \left( { x_{0} + x_{2} } \right) + x_{2}^{4} \left( { x_{0} + x_{1} } \right) \\ & \qquad + x_{0}^{3} \left( {x_{1}^{2} + x_{1} x_{2} + x_{2}^{2} } \right) + x_{1}^{3} \left( { x_{0}^{2} + x_{0} x_{2} + x_{2}^{2} } \right) + x_{2}^{3} \left( {x_{0}^{2} + x_{0} x_{1} + x_{1}^{2} } \right) \\ & \qquad + x_{0} x_{1} x_{2} \left( {x_{1} x_{2} + x_{0} x_{2} + x_{0} x_{1} } \right) = X_{1} S_{4} + X_{3} S_{2} + S_{2} S_{3} - S_{5} + X_{3} D_{1} \\ &\quad = X_{1}^{5} - 4X_{1}^{3} X_{2} + 3X_{3} X_{1}^{2} + 3X_{1} X_{2}^{2} - 2 X_{3} X_{2} , \\ & x_{0}^{4} + x_{1}^{4} + x_{2}^{4} + x_{0}^{3} \left( {x_{1} + x_{2} } \right) + x_{1}^{3} \left( { x_{0} + x_{2} } \right) + x_{2}^{3} \left( {x_{0} + x_{1} } \right) \\ & \qquad + x_{0} x_{1} x_{2} \left( {x_{0} + x_{1} + x_{2} } \right) + x_{0}^{2} x_{1}^{2} + x_{1}^{2} x_{2}^{2} + x_{0}^{2} x_{2}^{2} = X_{1} S_{3} + X_{3} X_{1} + D_{2} \\ &\quad = X_{1}^{4} - 3X_{1}^{2} X_{2} + 2X_{3} X_{1} + X_{2}^{2} , \\ & x_{0}^{3} + x_{1}^{3} + x_{2}^{3} + x_{0}^{2} \left( {x_{1} + x_{2} } \right) + x_{1}^{2} \left( {x_{0} + x_{2} } \right) + x_{2}^{2} \left( { x_{0} + x_{1} } \right) + x_{0} x_{1} x_{2} \\ &\quad = X_{1} S_{2} + X_{3} = \left( {X_{1} } \right)^{3} - 2X_{1} X_{2} + X_{3} \\ \end{aligned}$$
(B.5)
Appendix C
$$\begin{aligned} C_{{30}} & = 2\left( {m_{{0,0}}^{1} } \right)^{3} \left( {m_{{0,0}}^{1} + 1} \right)^{2} , \\ C_{{22}} & = - 3\left( {m_{{0,0}}^{1} } \right)^{2} \left( {m_{{0,0}}^{1} - 1} \right)\left( {m_{{0,0}}^{1} + 1} \right)^{2} , \\ C_{{21}} & = - 6\left( {m_{{0,0}}^{1} } \right)^{2} m_{{1,0}}^{t} \left[ {\left( {m_{{0,0}}^{1} } \right)^{2} ~ + ~4m_{{0,0}}^{1} ~ + ~3} \right], \\ C_{{20}} & = - 9\left( {m_{{0,0}}^{1} } \right)^{2} \left[ {4m_{{0,0}}^{1} \left( {m_{{0,0}}^{1} + 1} \right)m_{{2,0}}^{t} ~ - ~5m_{{0,0}}^{1} \left( {m_{{1,0}}^{t} } \right)^{2} - 3\left( {m_{{1,0}}^{t} } \right)^{2} } \right], \\ C_{{14}} & = 3m_{{0,0}}^{1} \left( {m_{{0,0}}^{1} + 1} \right)^{2} \left( {\left( {m_{{0,0}}^{1} } \right)^{2} + 1} \right), \\ C_{{13}} & = - 12m_{{0,0}}^{1} m_{{1,0}}^{t} [\left( {m_{{0,0}}^{1} } \right)^{3} + ~3\left( {m_{{0,0}}^{1} } \right)^{2} ~ + ~5m_{{0,0}}^{1} + ~3 \\ C_{{12}} & = 6m_{{0,0}}^{1} \left( {m_{{0,0}}^{1} + 1} \right)\left[ {2m_{{0,0}}^{1} \left( {2m_{{0,0}}^{1} - 3} \right)m_{{2,0}}^{t} + 9\left( {m_{{0,0}}^{1} + 3} \right)\left( {m_{{1,0}}^{t} } \right)^{2} } \right], \\ C_{{11}} & = 36m_{{0,0}}^{1} m_{{1,0}}^{t} \left[ {2m_{{0,0}}^{1} \left( {2m_{{0,0}}^{1} + 3} \right)m_{{2,0}}^{t} - 9\left( {m_{{0,0}}^{1} + 1} \right)\left( {m_{{1,0}}^{t} } \right)^{2} } \right], \\ C_{{10}} & = 3m_{{0,0}}^{1} \left\{ { - ~10m_{{4,0}}^{t} \left( {m_{{0,0}}^{1} } \right)^{3} \left( {m_{{0,0}}^{1} + 1} \right) + ~40\left( {m_{{0,0}}^{1} } \right)^{3} m_{{3,0}}^{t} m_{{1,0}}^{t} } \right. \\ & \quad \left. { + 24\left( {m_{{0,0}}^{1} + 3} \right)\left( {m_{{0,0}}^{1} } \right)^{2} \left( {m_{{2,0}}^{t} } \right)^{2} - 108m_{{0,0}}^{1} \left( {m_{{0,0}}^{1} + 1} \right)\left( {m_{{1,0}}^{t} } \right)^{2} m_{{2,0}}^{t} + 27\left( {2m_{{0,0}}^{1} + 3} \right)\left( {m_{{1,0}}^{t} } \right)^{4} } \right\}, \\ C_{{06}} & = - \left[ {\left( {m_{{0,0}}^{1} } \right)^{3} - ~1} \right]\left( {m_{{0,0}}^{1} + ~1} \right)^{2} , \\ C_{{05}} & = 6m_{{1,0}}^{t} \left[ {\left( {m_{{0,0}}^{1} } \right)^{4} + \left( {m_{{0,0}}^{1} } \right)^{3} - ~2\left( {m_{{0,0}}^{1} } \right)^{2} - ~5m_{{0,0}}^{1} - ~3} \right], \\ C_{{04}} & = 18\left( {m_{{1,0}}^{t} } \right)^{2} ~ - 3\left[ {\left( {m_{{0,0}}^{1} } \right)^{3} ~ + ~1} \right]\left[ {3\left( {m_{{1,0}}^{t} } \right)^{2} + 4m_{{0,0}}^{1} m_{{2,0}}^{t} } \right] \\ & - 18\left[ {\left( {m_{{0,0}}^{1} } \right)^{2} - 1} \right]\left[ {m_{{0,0}}^{1} m_{{2,0}}^{t} - 3\left( {m_{{1,0}}^{t} } \right)^{2} } \right] - 12\left( {m_{{0,0}}^{1} + 1} \right)\left[ {2m_{{0,0}}^{1} m_{{2,0}}^{t} - 15\left( {m_{{1,0}}^{t} } \right)^{2} } \right], \\ C_{{03}} & = 4\left( {m_{{0,0}}^{1} + 1} \right)\left[ { - 5m_{{3,0}}^{t} \left( {m_{{0,0}}^{1} } \right)^{3} + 54m_{{0,0}}^{1} m_{{2,0}}^{t} m_{{1,0}}^{t} - 135\left( {m_{{1,0}}^{t} } \right)^{3} } \right] \\ & \quad + 108\left( {m_{{0,0}}^{1} } \right)^{2} \left[ {m_{{0,0}}^{1} m_{{2,0}}^{t} m_{{1,0}}^{t} - \left( {m_{{1,0}}^{t} } \right)^{3} } \right], \\ C_{{02}} & = 3\left\{ {5\left( {m_{{0,0}}^{1} } \right)^{3} \left[ {1 - \left( {m_{{0,0}}^{1} } \right)^{2} } \right]m_{{4,0}}^{t} + 20\left( {m_{{0,0}}^{1} } \right)^{3} \left( {~m_{{0,0}}^{1} + 2} \right)m_{{3,0}}^{t} m_{{1,0}}^{t} } \right.~ \\ & \quad + 12\left( {m_{{0,0}}^{1} } \right)^{2} \left[ {\left( {m_{{0,0}}^{1} } \right)^{2} - 2m_{{0,0}}^{1} + ~3} \right]\left( {m_{{2,0}}^{t} } \right)^{2} - 54m_{{0,0}}^{1} \left[ {~\left( {m_{{0,0}}^{1} } \right)^{2} + 4m_{{0,0}}^{1} + 6} \right]\left( {m_{{1,0}}^{t} } \right)^{2} m_{{2,0}}^{t} \\ & \quad \left. { + 27\left[ {~\left( {m_{{0,0}}^{1} } \right)^{2} + 10m_{{0,0}}^{1} ~ + 15} \right]\left( {m_{{1,0}}^{t} } \right)^{4} } \right\}, \\ C_{{01}} & = 6\left\{ {\left( {m_{{0,0}}^{1} } \right)^{4} \left[ {7\left( {m_{{0,0}}^{1} + 1} \right)m_{{5,0}}^{t} - ~40m_{{3,0}}^{t} m_{{2,0}}^{t} } \right]} \right. \\ & \quad - ~15\left( {m_{{0,0}}^{1} } \right)^{3} \left[ {m_{{4,0}}^{t} m_{{1,0}}^{t} + 2\left( {m_{{1,0}}^{t} } \right)^{2} m_{{3,0}}^{t} } \right] + 36\left( {m_{{0,0}}^{1} } \right)^{2} \left( {m_{{0,0}}^{1} - 3} \right)m_{{1,0}}^{t} \left( {m_{{2,0}}^{t} } \right)^{2} ~ \\ & \quad \left. { + 108m_{{0,0}}^{1} \left( {m_{{0,0}}^{1} + 3} \right)\left( {m_{{1,0}}^{t} } \right)^{3} m_{{2,0}}^{t} ~ - ~81\left( {m_{{0,0}}^{1} + 3} \right)\left( {m_{{1,0}}^{t} } \right)^{5} } \right\}, \\ C_{{00}} & = - 9\left\{ {2~\left( {m_{{0,0}}^{1} } \right)^{4} \left( {7m_{{5,0}}^{t} m_{{1,0}}^{t} - 10m_{{4,0}}^{t} m_{{2,0}}^{t} } \right) + 3\left( {m_{{0,0}}^{1} } \right)^{3} \left[ { - 5m_{{4,0}}^{t} \left( {m_{{1,0}}^{t} } \right)^{2} + 16\left( {m_{{2,0}}^{t} } \right)^{3} } \right]} \right. \\ & \quad \left. { - ~108\left( {m_{{0,0}}^{1} } \right)^{2} \left( {m_{{1,0}}^{t} } \right)^{2} \left( {m_{{2,0}}^{t} } \right)^{2} + ~162m_{{0,0}}^{1} \left( {m_{{1,0}}^{t} } \right)^{4} m_{{2,0}}^{t} - ~81\left( {m_{{1,0}}^{t} } \right)^{6} } \right\}, \\ \end{aligned}$$
$$\begin{aligned} D_{{40}} & = \left( {m_{{0,0}}^{1} } \right)^{4} \left( {m_{{0,0}}^{1} ~ + ~1} \right)^{3} , \\ D_{{32}} & = - 6\left( {m_{{0,0}}^{1} } \right)^{3} \left( {m_{{0,0}}^{1} ~ + ~1} \right)^{2} \left[ {\left( {m_{{0,0}}^{1} } \right)^{2} ~ - 1} \right], \\ D_{{31}} & = - 12\left( {m_{{0,0}}^{1} } \right)^{3} \left( {m_{{0,0}}^{1} + ~1} \right)^{2} \left( {m_{{0,0}}^{1} + 3} \right)m_{{1,0}}^{1} , \\ D_{{30}} & = - 6\left( {m_{{0,0}}^{1} } \right)^{3} \left( {m_{{0,0}}^{1} ~ + ~1} \right)\left[ {4\left( {\left( {m_{{0,0}}^{1} } \right)^{2} + m_{{0,0}}^{1} } \right)m_{{2,0}}^{t} - 3\left( {5m_{{0,0}}^{1} + 3} \right)\left( {m_{{1,0}}^{t} } \right)^{2} } \right], \\ D_{{24}} & = 3\left( {m_{{0,0}}^{1} } \right)^{2} \left( {m_{{0,0}}^{1} + ~1} \right)^{3} \left[ {\left( {m_{{0,0}}^{1} } \right)^{2} ~ - ~8m_{{0,0}}^{1} ~ + ~1} \right], \\ D_{{23}} & = 12\left( {m_{{0,0}}^{1} } \right)^{2} m_{{1,0}}^{t} \left( {m_{{0,0}}^{1} ~ + ~1} \right)^{2} \left[ {7\left( {m_{{0.0}}^{1} } \right)^{2} + ~14m_{{0,0}}^{1} - ~3} \right], \\ D_{{22}} & = - 6\left( {m_{{0,0}}^{1} } \right)^{2} \left( {m_{{0,0}}^{1} ~ + ~1} \right)\left\{ {3\left[ {25\left( {m_{{0,0}}^{1} } \right)^{2} + ~12m_{{0,0}}^{1} - 9} \right]\left( {m_{{1,0}}^{t} } \right)^{2} } \right. \\ & \quad \left. { + 2m_{{0,0}}^{1} \left[ { - ~10\left( {m_{{0,0}}^{1} } \right)^{2} - m_{{0,0}}^{1} + 9} \right]m_{{2,0}}^{t} } \right\}, \\ D_{{21}} & = 36\left( {m_{{0,0}}^{1} } \right)^{2} m_{{1,0}}^{t} \left\{ {3\left[ {\left( {m_{{0,0}}^{1} } \right)^{2} - 6m_{{0,0}}^{1} - 3} \right]\left( {m_{{1,0}}^{t} } \right)^{2} } \right. \\ & \quad \left. { + 2m_{{0.0}}^{1} \left[ {2\left( {m_{{0.0}}^{1} } \right)^{2} + 11m_{{0,0}}^{1} + 9} \right]m_{{2,0}}^{t} } \right\}, \\ D_{{20}} & = 3\left( {m_{{0,0}}^{1} } \right)^{2} \left\{ {9\left[ {6\left( {m_{{0,0}}^{1} } \right)^{2} + 39m_{{0,0}}^{1} + 9} \right]\left( {m_{{1,0}}^{t} } \right)^{4} + 24\left( {m_{{0,0}}^{1} } \right)^{2} [\left( {m_{{0,0}}^{1} } \right)^{2} + 4m_{{0,0}}^{1} } \right. \\ & \quad + 3]\left( {m_{{2,0}}^{t} } \right)^{2} - 36m_{{0,0}}^{1} \left[ {3\left( {m_{{0,0}}^{1} } \right)^{2} + 14m_{{0,0}}^{1} + 9} \right]\left( {m_{{1,0}}^{t} } \right)^{2} m_{{2,0}}^{t} - 10\left( {m_{{0,0}}^{1} } \right)^{3} \left( {m_{{0,0}}^{1} + 1} \right)^{2} m_{{4,0}}^{t} \\ & \quad \left. { + 40\left( {m_{{0,0}}^{1} } \right)^{3} \left( {m_{{0,0}}^{1} + 1} \right)m_{{3,0}}^{t} m_{{1,0}}^{t} } \right\}, \\ D_{{16}} & = - 2m_{{0,0}}^{1} \left( {m_{{0,0}}^{1} + 1} \right)^{3} \left( {m_{{0,0}}^{1} - 1} \right)\left[ {\left( {m_{{0,0}}^{1} } \right)^{2} - ~5m_{{0,0}}^{1} ~ + ~1} \right], \\ D_{{15}} & = - 12m_{{0,0}}^{1} m_{{1,0}}^{t} \left( {m_{{00}}^{1} + 1} \right)^{2} \left[ {3\left( {m_{{0,0}}^{1} } \right)^{3} + 2\left( {m_{{0,0}}^{1} } \right)^{2} - 12m_{{0,0}}^{1} ~ + ~3} \right], \\ D_{{14}} & = - 6m_{{00}}^{1} \left( {m_{{00}}^{1} ~ + ~1} \right)\left\{ {3\left[ { - ~7\left( {m_{{0,0}}^{1} } \right)^{3} + 32\left( {m_{{0,0}}^{1} } \right)^{2} + 30m_{{0,0}}^{1} - 15} \right]\left( {m_{{1,0}}^{t} } \right)^{2} } \right. \\ & \quad \left. { + 2m_{{0,0}}^{1} \left[ {2\left( {m_{{0,0}}^{1} } \right)^{3} - 21\left( {m_{{0,0}}^{1} } \right)^{2} - 20m_{{0,0}}^{1} + 3} \right]m_{{2,0}}^{t} } \right\}, \\ D_{{13}} & = 8m_{{0,0}}^{1} \left\{ {10\left( {m_{{0,0}}^{1} } \right)^{3} \left( {m_{{0,0}}^{1} + 1} \right)^{2} m_{{3,0}}^{t} + 27\left[ {5\left( {m_{{0,0}}^{1} } \right)^{3} + 12\left( {m_{{0,0}}^{1} } \right)^{2} - 5} \right]\left( {m_{{1,0}}^{t} } \right)^{3} } \right. \\ & \quad \left. { - 18m_{{0,0}}^{1} \left[ {9\left( {m_{{0,0}}^{1} } \right)^{3} + 22\left( {m_{{0,0}}^{1} } \right)^{2} + 10m_{{0,0}}^{1} - 3} \right]m_{{2,0}}^{t} m_{{1,0}}^{t} } \right\}, \\ D_{{12}} & = - 6m_{{0,0}}^{1} \left\{ {25\left( {m_{{0,0}}^{1} } \right)^{3} \left( {m_{{0,0}}^{1} + 1} \right)\left[ {1 - \left( {m_{{0,0}}^{1} } \right)^{2} } \right]m_{{4,0}}^{t} } \right. \\ & \quad + 20\left( {m_{{0,0}}^{1} } \right)^{3} \left( {5m_{{0,0}}^{1} + 1} \right)\left( {m_{{0,0}}^{1} + 1} \right)m_{{3,0}}^{t} m_{{1,0}}^{t} \\ & \quad + 12\left( {m_{{0,0}}^{1} } \right)^{2} \left[ {11\left( {m_{{0,0}}^{1} } \right)^{2} + 5\left( {m_{{0,0}}^{1} } \right)^{3} - 3m_{{0,0}}^{1} - 9} \right]\left( {m_{{2,0}}^{t} } \right)^{2} \\ & \quad + 18m_{{0,0}}^{1} \left[ { - ~53\left( {m_{{0,0}}^{1} } \right)^{2} - 15\left( {m_{{0,0}}^{1} } \right)^{3} - 18m_{{0,0}}^{1} + 18} \right]\left( {m_{{1,0}}^{t} } \right)^{2} m_{{2,0}}^{t} \\ & \quad \left. { + ~27\left[ {5\left( {m_{{0,0}}^{1} } \right)^{3} + 17\left( {m_{{0,0}}^{1} } \right)^{2} - 15m_{{0,0}}^{1} - ~15} \right]\left( {m_{{1,0}}^{t} } \right)^{4} } \right\}, \\ \end{aligned}$$
$$\begin{aligned} D_{{11}} & = - 36m_{{0,0}}^{1} m_{{1,0}}^{t} \{ 5\left( {m_{{0,0}}^{1} } \right)^{3} \left[ {\left( {m_{{0,0}}^{1} } \right)^{2} - 4m_{{0,0}}^{1} - 5} \right]m_{{4,0}}^{t} + 20\left( {m_{{0,0}}^{1} } \right)^{3} \left( {2 - m_{{0,0}}^{1} } \right)m_{{3,0}}^{t} m_{{1,0}}^{t} \\ & \quad + 12\left( {m_{{0,0}}^{1} } \right)^{2} \left[ { - \left( {m_{{0,0}}^{1} } \right)^{2} + 6m_{{0,0}}^{1} + 9} \right]\left( {m_{{2,0}}^{t} } \right)^{2} + 18m_{{0,0}}^{1} \left[ {3\left( {m_{{0,0}}^{1} } \right)^{2} - 8m_{{0,0}}^{1} - 6} \right]\left( {m_{{1,0}}^{t} } \right)^{2} m_{{2,0}}^{t} \\ & \quad + 9\left[ { - 3\left( {m_{{0,0}}^{1} } \right)^{2} + 18m_{{0,0}}^{1} + 9} \right]\left( {m_{{1,0}}^{t} } \right)^{4} \} , \\ D_{{10}} & = 18m_{{0,0}}^{1} \left\{ {20\left( {m_{{0,0}}^{1} } \right)^{4} \left[ {\left( {m_{{0,0}}^{1} + 1} \right)m_{{4,0}}^{t} m_{{2,0}}^{t} - ~4m_{{1,0}}^{t} m_{{2,0}}^{t} m_{{3,0}}^{t} } \right]} \right. \\ & \quad + ~324\left( {m_{{0,0}}^{1} } \right)^{2} \left( {m_{{0,0}}^{1} + 1} \right)\left( {m_{{1,0}}^{t} } \right)^{2} \left( {m_{{2,0}}^{t} } \right)^{2} \\ & \quad + 3\left( {m_{{0,0}}^{1} } \right)^{3} \left[ {60m_{{3,0}}^{t} \left( {m_{{1,0}}^{t} } \right)^{3} - 16\left( {m_{{0,0}}^{1} + 1} \right)\left( {m_{{2,0}}^{t} } \right)^{3} - 5\left( {3m_{{0,0}}^{1} + 5} \right)m_{{4,0}}^{t} \left( {m_{{1,0}}^{t} } \right)^{2} } \right] \\ & \quad \left. { - 54m_{{0,0}}^{1} \left( {11m_{{0,0}}^{1} + 3} \right)\left( {m_{{1,0}}^{t} } \right)^{4} m_{{2,0}}^{t} ~ + 81\left( {3m_{{0,0}}^{1} + 1} \right)\left( {m_{{1,0}}^{t} } \right)^{6} } \right\}, \\ D_{{08}} & = \left( {m_{{0,0}}^{1} ~ + ~1} \right)^{3} \left[ {\left( {m_{{0,0}}^{1} } \right)^{2} - m_{{0,0}}^{1} ~ + ~1} \right]^{2} , \\ D_{{07}} & = - 24m_{{1,0}}^{t} \left( {m_{{0,0}}^{1} ~ + ~1} \right)^{2} \left( {\left( {m_{{0,0}}^{1} } \right)^{2} - m_{{0,0}}^{1} + ~1} \right), \\ D_{{06}} & = 12\left( {m_{{0,0}}^{1} + ~1} \right)\left\{ {\left[ {9\left( {m_{{0,0}}^{1} } \right)^{3} + ~21} \right]\left( {m_{{1,0}}^{t} } \right)^{2} + m_{{0,0}}^{1} \left[ { - ~5\left( {m_{{0,0}}^{1} } \right)^{3} + 4m_{{0,0}}^{1} - 1} \right]m_{{2,0}}^{t} } \right\}, \\ D_{{05}} & = 72m_{{1,0}}^{t} \left( {m_{{0,0}}^{1} + ~1} \right)\{ m_{{0,0}}^{1} \left[ {4\left( {m_{{0,0}}^{1} } \right)^{3} + \left( {m_{{0,0}}^{1} } \right)^{2} - 9m_{{0,0}}^{1} + 3} \right]m_{{2,0}}^{t} \\ & \quad \left. { - 3\left[ {\left( {m_{{0,0}}^{1} } \right)^{3} + 7} \right]\left( {m_{{1,0}}^{t} } \right)^{2} } \right\}, \\ D_{{04}} & = \left\{ {120\left( {m_{{0,0}}^{1} } \right)^{3} \left[ {\left( {m_{{0,0}}^{1} } \right)^{3} + 1} \right]m_{{3,0}}^{t} m_{{1,0}}^{t} - ~30\left( {m_{{0,0}}^{1} } \right)^{3} \left( {m_{{0,0}}^{1} + 1} \right)\left[ {\left( {m_{{0,0}}^{1} } \right)^{3} + 1} \right]m_{{4,0}}^{t} } \right. \\ & \quad + 162\left( {m_{{0,0}}^{1} + 1} \right)\left[ {\left( {m_{{0,0}}^{1} } \right)^{3} + 35} \right]\left( {m_{{1,0}}^{t} } \right)^{3} + 36\left( {m_{{0,0}}^{1} } \right)^{2} [2\left( {m_{{0,0}}^{1} } \right)^{4} ~ + 2\left( {m_{{0,0}}^{1} } \right)^{3} - 20\left( {m_{{0,0}}^{1} } \right)^{2} \\ & \quad - 17m_{{0,0}}^{1} + 3\left] {\left( {m_{{2,0}}^{t} } \right)^{2} + 108m_{{0,0}}^{1} } \right[24\left( {m_{{0,0}}^{1} } \right)^{2} - 5\left( {m_{{0,0}}^{1} } \right)^{3} - 15 + ~15m_{{0,0}}^{1} \\ & \quad \left. { - 3\left( {m_{{0,0}}^{1} } \right)^{4} ]\left( {m_{{1,0}}^{t} } \right)^{2} m_{{2,0}}^{t} } \right\}, \\ D_{{03}} & = 24\left\{ { - 20\left( {m_{{0,0}}^{1} } \right)^{5} m_{{3,0}}^{t} m_{{2,0}}^{t} - 20\left( {m_{{0,0}}^{1} } \right)^{4} m_{{3,0}}^{t} m_{{2,0}}^{t} - 60\left( {m_{{0,0}}^{1} } \right)^{3} m_{{3,0}}^{t} \left( {m_{{1,0}}^{t} } \right)^{2} } \right. \\ & \quad + 15\left( {m_{{0,0}}^{1} } \right)^{3} \left( {m_{{0,0}}^{1} + 1} \right)m_{{4,0}}^{t} m_{{1,0}}^{t} + 18\left( {m_{{0,0}}^{1} } \right)^{2} \left[ {10\left( {m_{{0,0}}^{1} } \right)^{2} + 10m_{{0,0}}^{1} - 3} \right]m_{{1,0}}^{t} \left( {m_{{2,0}}^{t} } \right)^{2} \\ & \quad \left. { + 54m_{{0,0}}^{1} \left[ { - 4\left( {m_{{0,0}}^{1} } \right)^{2} + 5} \right]\left( {m_{{1,0}}^{t} } \right)^{3} m_{{2,0}}^{t} - 567\left( {m_{{0,0}}^{1} + 1} \right)\left( {m_{{1,0}}^{t} } \right)^{5} } \right\}, \\ D_{{02}} & = 4\left\{ {\left( {m_{{0,0}}^{1} + 1} \right)[5103\left( {m_{{1,0}}^{t} } \right)^{6} - 3645m_{{0,0}}^{1} \left( {m_{{1,0}}^{t} } \right)^{4} m_{{2,0}}^{t} - 405\left( {m_{{0,0}}^{1} } \right)^{3} m_{{4,0}}^{t} \left( {m_{{1,0}}^{t} } \right)^{2} } \right. \\ & \quad \left. { + 225\left( {m_{{0,0}}^{1} } \right)^{4} m_{{4,0}}^{t} m_{{2,0}}^{t} ~ - 28\left( {m_{{0,0}}^{1} } \right)^{5} \left( {m_{{0,0}}^{1} + 1} \right)m_{{6,0}}^{t} + 100\left( {m_{{0,0}}^{1} } \right)^{5} \left( {m_{{3,0}}^{t} } \right)^{2} } \right] \\ & \quad + 972\left( {m_{{0,0}}^{1} } \right)^{3} \left( {m_{{1,0}}^{t} } \right)^{4} m_{{2,0}}^{t} + 1620\left( {m_{{0,0}}^{1} } \right)^{3} \left( {m_{{1,0}}^{t} } \right)^{3} m_{{3,0}}^{t} \\ & \quad + 162\left( {m_{{0,0}}^{1} } \right)^{2} \left[ {~ - 10\left( {m_{{0,0}}^{1} } \right)^{2} - 9m_{{0,0}}^{1} + 9} \right]\left( {m_{{1,0}}^{t} m_{{2,0}}^{t} } \right)^{2} \\ & \quad \left. { + 108\left( {m_{{0,0}}^{1} } \right)^{3} \left[ {2\left( {m_{{0,0}}^{1} } \right)^{2} - m_{{0,0}}^{1} - 3} \right]\left( {m_{{2,0}}^{t} } \right)^{3} + 180\left( {m_{{0,0}}^{1} } \right)^{4} \left( {1 + 2m_{{0,0}}^{1} } \right)m_{{1,0}}^{t} m_{{2,0}}^{t} m_{{3,0}}^{t} } \right\}, \\ \end{aligned}$$
$$\begin{aligned} D_{01} & = 24m_{1,0}^{t} \left\{ {\left( {m_{0,0}^{1} + 1} \right)\left[ {28\left( {m_{0,0}^{1} } \right)^{5} m_{6,0}^{t} - 729\left( {m_{1,0}^{t} } \right)^{6} + 135\left( {m_{0,0}^{1} } \right)^{3} m_{4,0}^{t} \left( {m_{1,0}^{t} } \right)^{2} } \right]} \right. \\ & \quad - 162\left( {4m_{0,0}^{1} + 3} \right)\left( {m_{0,0}^{1} } \right)^{2} \left( {m_{1,0}^{t} m_{2,0}^{t} } \right)^{2} - 45\left( {m_{0,0}^{1} } \right)^{4} \left( { 4m_{0,0}^{1} + 5} \right)m_{4,0}^{t} m_{2,0}^{t} \\ & \quad + 108\left( {2m_{0,0}^{1} + 3} \right)\left( {m_{0,0}^{1} } \right)^{3} \left( {m_{2,0}^{t} } \right)^{3} + 729m_{0,0}^{1} \left( {2m_{0,0}^{1} + 1} \right)\left( {m_{1,0}^{t} } \right)^{4} m_{2,0}^{t} \\ & \quad \left. { - 100\left( {m_{0,0}^{1} } \right)^{5} \left( {m_{3,0}^{t} } \right)^{2} + 360\left( {m_{0,0}^{1} } \right)^{4} m_{1,0}^{t} m_{2,0}^{t} m_{3,0}^{t} - 540\left( {m_{0,0}^{1} } \right)^{3} \left( {m_{1,0}^{t} } \right)^{3} m_{3,0}^{t} } \right\}, \\ D_{00} & = 9\left\{ {m_{0,0}^{1} [ - 27\left( {m_{1,0}^{t} } \right)^{4} + 5\left( {m_{0,0}^{1} } \right)^{3} m_{4,0}^{t} - 4\left( {m_{0,0}^{1} } \right)^{2} \left[ {3\left( {m_{2,0}^{t} } \right)^{2} + 5m_{1,0}^{t} m_{3,0}^{t} } \right]} \right. \\ & \quad + 54m_{0,0}^{1} \left( {m_{1,0}^{t} } \right)^{2} m_{2,0}^{t} ]^{2} + [5\left( {m_{0,0}^{1} } \right)^{3} m_{4,0}^{t} - 12\left( {m_{0,0}^{1} } \right)^{2} \left( {m_{2,0}^{t} } \right)^{2} + 54m_{0,0}^{1} \left( {m_{1,0}^{t} } \right)^{2} m_{2,0}^{t} \\ & \quad - 27\left( {m_{1,0}^{t} } \right)^{4} ]\left[ {5\left( {m_{0,0}^{1} } \right)^{3} m_{4,0}^{t} - 12\left( {m_{0,0}^{1} } \right)^{2} \left( {m_{2,0}^{t} } \right)^{2} + 126m_{0,0}^{1} \left( {m_{1,0}^{t} } \right)^{2} m_{2,0}^{t} - 27\left( {m_{1,0}^{t} } \right)^{4} } \right] \\ & \quad + 16m_{0,0}^{1} \left( {m_{1,0}^{t} } \right)^{2} \{ 27m_{2,0}^{t} \left[ {2\left( {m_{00}^{1} } \right)^{2} \left( {m_{2,0}^{t} } \right)^{2} - 15m_{0,0}^{1} \left( {m_{1,0}^{t} } \right)^{2} m_{2,0}^{t} + 9\left( {m_{1,0}^{t} } \right)^{4} } \right] \\ & \quad \left. { - 7\left( {m_{0,0}^{1} } \right)^{4} m_{6,0}^{t} } \right\} \\ \end{aligned}$$
(C.1)
