Appendix 1
From Eq. 20, the radial Racah polynomials \( {\tilde{u}}_n^{\left(\alpha, \beta \right)}\left( r, a, b\right) \)can be expressed as a series of decreasing power of r as follows:
$$ \left(\begin{array}{c}\hfill {\tilde{u}}_0^{\left(\alpha, \beta \right)}\left( r, a, b\right)\hfill \\ {}\hfill {\tilde{u}}_1^{\left(\alpha, \beta \right)}\left( r, a, b\right)\hfill \\ {}\hfill \begin{array}{c}\hfill \vdots \hfill \\ {}\hfill {\tilde{u}}_n^{\left(\alpha, \beta \right)}\left( r, a, b\right)\hfill \end{array}\hfill \end{array}\right)=\left(\begin{array}{c}\hfill \begin{array}{c}\hfill {B}_{00}\kern0.5em \begin{array}{cc}\hfill \hfill & \hfill \hfill \end{array}\hfill \\ {}\hfill \begin{array}{cc}\hfill {B}_{10}\hfill & \hfill {B}_{11}\hfill \end{array}\begin{array}{cc}\hfill \hfill & \hfill \hfill \end{array}\hfill \\ {}\hfill \kern0.48em \begin{array}{cc}\hfill \vdots \hfill & \hfill \hfill \end{array}\begin{array}{cc}\hfill \vdots \hfill & \hfill \ddots \hfill \end{array}\hfill \end{array}\hfill \\ {}\hfill \begin{array}{cc}\hfill {B}_{n0}\hfill & \hfill {B}_{n1}\hfill \end{array}\begin{array}{cc}\hfill \cdots \hfill & \hfill {B}_{n n}\hfill \end{array}\hfill \end{array}\right)\left(\begin{array}{c}\hfill {r}^0\hfill \\ {}\hfill {r}^1\hfill \\ {}\hfill \begin{array}{c}\hfill \vdots \hfill \\ {}\hfill {r}^n\hfill \end{array}\hfill \end{array}\right) $$
(41)
$$ \begin{array}{l}\left(\begin{array}{c}\hfill {r}^0\hfill \\ {}\hfill {r}^1\hfill \\ {}\hfill \begin{array}{c}\hfill \vdots \hfill \\ {}\hfill {r}^n\hfill \end{array}\hfill \end{array}\right)={\left(\begin{array}{c}\hfill \begin{array}{c}\hfill \begin{array}{cc}\hfill {B}_{00}\hfill & \hfill \hfill \end{array}\begin{array}{cc}\hfill \hfill & \hfill \hfill \end{array}\hfill \\ {}\hfill \begin{array}{cc}\hfill {B}_{10}\hfill & \hfill {B}_{11}\hfill \end{array}\begin{array}{cc}\hfill \hfill & \hfill \hfill \end{array}\hfill \\ {}\hfill \kern0.48em \begin{array}{cc}\hfill \vdots \hfill & \hfill \hfill \end{array}\begin{array}{cc}\hfill \vdots \hfill & \hfill \ddots \hfill \end{array}\hfill \end{array}\hfill \\ {}\hfill \begin{array}{cc}\hfill {B}_{n0}\hfill & \hfill {B}_{n1}\hfill \end{array}\begin{array}{cc}\hfill \cdots \hfill & \hfill {B}_{n n}\hfill \end{array}\hfill \end{array}\right)}^{-1}\left(\begin{array}{c}\hfill {\tilde{u}}_0^{\left(\alpha, \beta \right)}\left( r, a, b\right)\hfill \\ {}\hfill {\tilde{u}}_1^{\left(\alpha, \beta \right)}\left( r, a, b\right)\hfill \\ {}\hfill \begin{array}{c}\hfill \vdots \hfill \\ {}\hfill {\tilde{u}}_n^{\left(\alpha, \beta \right)}\left( r, a, b\right)\hfill \end{array}\hfill \end{array}\right)\\ {}\ \begin{array}{cc}\hfill \hfill & \hfill \hfill \end{array}=\left(\begin{array}{c}\hfill \begin{array}{c}\hfill \begin{array}{cc}\hfill {D}_{00}\hfill & \hfill \hfill \end{array}\begin{array}{cc}\hfill \hfill & \hfill \hfill \end{array}\hfill \\ {}\hfill \begin{array}{cc}\hfill {D}_{10}\hfill & \hfill {D}_{11}\hfill \end{array}\begin{array}{cc}\hfill \hfill & \hfill \hfill \end{array}\hfill \\ {}\hfill \kern0.48em \begin{array}{cc}\hfill \vdots \hfill & \hfill \hfill \end{array}\begin{array}{cc}\hfill \vdots \hfill & \hfill \ddots \hfill \end{array}\hfill \end{array}\hfill \\ {}\hfill \begin{array}{cc}\hfill {D}_{n0}\hfill & \hfill {D}_{n1}\hfill \end{array}\begin{array}{cc}\hfill \cdots \hfill & \hfill {D}_{n n}\hfill \end{array}\hfill \end{array}\right)\left(\begin{array}{c}\hfill {\tilde{u}}_0^{\left(\alpha, \beta \right)}\left( r, a, b\right)\hfill \\ {}\hfill {\tilde{u}}_1^{\left(\alpha, \beta \right)}\left( r, a, b\right)\hfill \\ {}\hfill \begin{array}{c}\hfill \vdots \hfill \\ {}\hfill {\tilde{u}}_n^{\left(\alpha, \beta \right)}\left( r, a, b\right)\hfill \end{array}\hfill \end{array}\right)\end{array} $$
(42)
From Eq. 20, the radial Racah polynomials \( {\tilde{u}}_n^{\left(\alpha, \beta \right)}\left( r, a, b\right) \) can also be expressed as a series of decreasing power of xr as follows:
$$ \begin{array}{l}\left(\begin{array}{c}\hfill {\tilde{u}}_0^{\left(\alpha, \beta \right)}\left( r, a, b\right)\hfill \\ {}\hfill {\tilde{u}}_1^{\left(\alpha, \beta \right)}\left( r, a, b\right)\hfill \\ {}\hfill \begin{array}{c}\hfill \vdots \hfill \\ {}\hfill {\tilde{u}}_n^{\left(\alpha, \beta \right)}\left( r, a, b\right)\hfill \end{array}\hfill \end{array}\right)=\left(\begin{array}{c}\hfill \begin{array}{c}\hfill \begin{array}{cc}\hfill {B}_{00}\hfill & \hfill \hfill \end{array}\begin{array}{cc}\hfill \hfill & \hfill \hfill \end{array}\hfill \\ {}\hfill \begin{array}{cc}\hfill {B}_{10}\hfill & \hfill {B}_{11}\hfill \end{array}\begin{array}{cc}\hfill \hfill & \hfill \hfill \end{array}\hfill \\ {}\hfill \kern0.48em \begin{array}{cc}\hfill \vdots \hfill & \hfill \hfill \end{array}\begin{array}{cc}\hfill \vdots \hfill & \hfill \ddots \hfill \end{array}\hfill \end{array}\hfill \\ {}\hfill \begin{array}{cc}\hfill {B}_{n0}\hfill & \hfill {B}_{n1}\hfill \end{array}\begin{array}{cc}\hfill \cdots \hfill & \hfill {B}_{n n}\hfill \end{array}\hfill \end{array}\right)\left(\begin{array}{c}\hfill {(xr)}^0\hfill \\ {}\hfill {r}^1\hfill \\ {}\hfill \begin{array}{c}\hfill \vdots \hfill \\ {}\hfill {r}^n\hfill \end{array}\hfill \end{array}\right)\\ {}\begin{array}{cc}\hfill \begin{array}{cccc}\hfill \hfill & \hfill \hfill & \hfill \hfill & \hfill \hfill \end{array}\hfill & \hfill \hfill \end{array}=\left(\begin{array}{c}\hfill \begin{array}{c}\hfill \begin{array}{cc}\hfill {B}_{00}\hfill & \hfill \hfill \end{array}\begin{array}{cc}\hfill \hfill & \hfill \hfill \end{array}\hfill \\ {}\hfill \begin{array}{cc}\hfill {B}_{10}\hfill & \hfill {B}_{11}\hfill \end{array}\begin{array}{cc}\hfill \hfill & \hfill \hfill \end{array}\hfill \\ {}\hfill \kern0.48em \begin{array}{cc}\hfill \vdots \hfill & \hfill \hfill \end{array}\begin{array}{cc}\hfill \vdots \hfill & \hfill \ddots \hfill \end{array}\hfill \end{array}\hfill \\ {}\hfill \begin{array}{cc}\hfill {B}_{n0}\hfill & \hfill {B}_{n1}\hfill \end{array}\begin{array}{cc}\hfill \cdots \hfill & \hfill {B}_{n n}\hfill \end{array}\hfill \end{array}\right)\left(\begin{array}{c}\hfill \begin{array}{c}\hfill \begin{array}{cc}\hfill {x}^0\hfill & \hfill \hfill \end{array}\begin{array}{cc}\hfill \hfill & \hfill \hfill \end{array}\hfill \\ {}\hfill \begin{array}{cc}\hfill \hfill & \hfill {x}^1\hfill \end{array}\begin{array}{cc}\hfill \hfill & \hfill \hfill \end{array}\hfill \\ {}\hfill \begin{array}{cc}\hfill \hfill & \hfill \hfill \end{array}\begin{array}{cc}\hfill \hfill & \hfill \ddots \hfill \end{array}\hfill \end{array}\hfill \\ {}\hfill \begin{array}{cc}\hfill \hfill & \hfill \hfill \end{array}\begin{array}{cc}\hfill \hfill & \hfill {x}^n\hfill \end{array}\hfill \end{array}\right)\left(\begin{array}{c}\hfill \begin{array}{c}\hfill \begin{array}{cc}\hfill {D}_{00}\hfill & \hfill \hfill \end{array}\begin{array}{cc}\hfill \hfill & \hfill \hfill \end{array}\hfill \\ {}\hfill \begin{array}{cc}\hfill {D}_{10}\hfill & \hfill {D}_{11}\hfill \end{array}\begin{array}{cc}\hfill \hfill & \hfill \hfill \end{array}\hfill \\ {}\hfill \kern0.48em \begin{array}{cc}\hfill \vdots \hfill & \hfill \hfill \end{array}\begin{array}{cc}\hfill \vdots \hfill & \hfill \ddots \hfill \end{array}\hfill \end{array}\hfill \\ {}\hfill \begin{array}{cc}\hfill {D}_{n0}\hfill & \hfill {D}_{n1}\hfill \end{array}\begin{array}{cc}\hfill \cdots \hfill & \hfill {D}_{n n}\hfill \end{array}\hfill \end{array}\right)\left(\begin{array}{c}\hfill {\tilde{u}}_0^{\left(\alpha, \beta \right)}\left( r, a, b\right)\hfill \\ {}\hfill {\tilde{u}}_1^{\left(\alpha, \beta \right)}\left( r, a, b\right)\hfill \\ {}\hfill \begin{array}{c}\hfill \vdots \hfill \\ {}\hfill {\tilde{u}}_n^{\left(\alpha, \beta \right)}\left( r, a, b\right)\hfill \end{array}\hfill \end{array}\right)\\ {}\begin{array}{cc}\hfill \begin{array}{cccc}\hfill \hfill & \hfill \hfill & \hfill \hfill & \hfill \hfill \end{array}\hfill & \hfill \hfill \end{array}=\sum_{k=0}^n{\tilde{u}}_k^{\left(\alpha, \beta \right)}\left( r, a, b\right)\sum_{i= k}^n{x}^i{B}_{n i}{D}_{i k}\end{array} $$
(43)
Appendix 2
We can rewrite Eq. 32 in matrix from as
$$ \left(\begin{array}{c}\hfill {I}_{0 nml}^{sr}\hfill \\ {}\hfill {I}_{1 nml}^{sr}\hfill \\ {}\hfill \begin{array}{c}\hfill \begin{array}{c}\hfill \vdots \hfill \\ {}\hfill \begin{array}{c}\hfill \vdots \hfill \\ {}\hfill \vdots \hfill \end{array}\hfill \end{array}\hfill \\ {}\hfill {I}_{k nml}^{sr}\hfill \end{array}\hfill \end{array}\right)={e}^{jn \arg \left({R}_{0100}^{sr}\right)}{e}^{jm \arg \left({R}_{0010}^{sr}\right)}{e}^{jl \arg \left({R}_{0001}^{sr}\right)}\left(\begin{array}{c}\hfill \begin{array}{ccc}\hfill {\left({R}_{0000}^{sr}\right)}^{-1}\hfill & \hfill \hfill & \hfill \hfill \\ {}\hfill \hfill & \hfill {\left({R}_{0000}^{sr}\right)}^{-2}\hfill & \hfill \hfill \\ {}\hfill \vdots \hfill & \hfill \hfill & \hfill {\left({R}_{0000}^{sr}\right)}^{-\left( k+1\right)}\hfill \end{array}\hfill \\ {}\hfill \begin{array}{ccc}\hfill \hfill & \hfill \hfill & \hfill \hfill \end{array}\hfill \end{array}\right)\left(\begin{array}{c}\hfill \begin{array}{ccc}\hfill {B}_{00}\hfill & \hfill \hfill & \hfill \hfill \\ {}\hfill {B}_{10}\hfill & \hfill {B}_{11}\hfill & \hfill \hfill \\ {}\hfill \vdots \hfill & \hfill \ddots \hfill & \hfill \hfill \end{array}\hfill \\ {}\hfill \begin{array}{ccc}\hfill {B}_{k0}\hfill & \hfill {B}_{k1}\hfill & \hfill {B}_{k k}\hfill \end{array}\hfill \end{array}\right)\left(\begin{array}{c}\hfill \begin{array}{ccc}\hfill {D}_{00}\hfill & \hfill \hfill & \hfill \hfill \\ {}\hfill {D}_{10}\hfill & \hfill {D}_{11}\hfill & \hfill \hfill \\ {}\hfill \vdots \hfill & \hfill \ddots \hfill & \hfill \hfill \end{array}\hfill \\ {}\hfill \begin{array}{ccc}\hfill {D}_{k0}\hfill & \hfill \hfill & \hfill {D}_{k k}\hfill \end{array}\hfill \end{array}\right)\left(\begin{array}{c}\hfill {R}_{0 nml}^{sr}\hfill \\ {}\hfill {R}_{1 nml}^{sr}\hfill \\ {}\hfill \begin{array}{c}\hfill \begin{array}{c}\hfill \vdots \hfill \\ {}\hfill \begin{array}{c}\hfill \vdots \hfill \\ {}\hfill \vdots \hfill \end{array}\hfill \end{array}\hfill \\ {}\hfill {R}_{k nml}^{sr}\hfill \end{array}\hfill \end{array}\right) $$
(44)
From Eq. 32 we can also get
$$ \arg \left({R}_{0100}^{sr}\right)= \arg \left({R}_{0100}\right)+{\theta}^{\hbox{'}}, \arg \left({R}_{0010}^{sr}\right)= \arg \left({R}_{0010}\right)+{\varphi^{\hbox{'}}}_{,}, \arg \left({SR}_{0001}^{sr}\right)= \arg \left({R}_{0001}\right)+{\psi}^{\hbox{'}},\mathrm{and}\kern0.37em {R}_{0000}^{sr}={\lambda}^2{R}_{0100} $$
(45)
Similarly Eq. 31 can also be written in the matrix form as
$$ \left(\begin{array}{c}\hfill {R}_{0 nml}^{sr}\hfill \\ {}\hfill {R}_{1 nml}^{sr}\hfill \\ {}\hfill \begin{array}{c}\hfill \begin{array}{c}\hfill \vdots \hfill \\ {}\hfill \begin{array}{c}\hfill \vdots \hfill \\ {}\hfill \vdots \hfill \end{array}\hfill \end{array}\hfill \\ {}\hfill {R}_{k nml}^{sr}\hfill \end{array}\hfill \end{array}\right)={e}^{{jn\theta}^{\hbox{'}}}{e}^{{jm\varphi}^{\hbox{'}}}{e}^{{jl\psi}^{\hbox{'}}}\left(\begin{array}{c}\hfill \begin{array}{ccc}\hfill {B}_{00}\hfill & \hfill \hfill & \hfill \hfill \\ {}\hfill {B}_{10}\hfill & \hfill {B}_{11}\hfill & \hfill \hfill \\ {}\hfill \vdots \hfill & \hfill \ddots \hfill & \hfill \hfill \end{array}\hfill \\ {}\hfill \begin{array}{ccc}\hfill {B}_{k0}\hfill & \hfill {B}_{k1}\hfill & \hfill {B}_{k k}\hfill \end{array}\hfill \end{array}\right)\left(\begin{array}{c}\hfill \begin{array}{ccc}\hfill {D}_{00}\hfill & \hfill \hfill & \hfill \hfill \\ {}\hfill {D}_{10}\hfill & \hfill {D}_{11}\hfill & \hfill \hfill \\ {}\hfill \vdots \hfill & \hfill \ddots \hfill & \hfill \hfill \end{array}\hfill \\ {}\hfill \begin{array}{ccc}\hfill {D}_{k0}\hfill & \hfill \hfill & \hfill {D}_{k k}\hfill \end{array}\hfill \end{array}\right)\left(\begin{array}{c}\hfill {R}_{0 nml}\hfill \\ {}\hfill {R}_{1 nml}\hfill \\ {}\hfill \begin{array}{c}\hfill \begin{array}{c}\hfill \vdots \hfill \\ {}\hfill \begin{array}{c}\hfill \vdots \hfill \\ {}\hfill \vdots \hfill \end{array}\hfill \end{array}\hfill \\ {}\hfill {R}_{k nml}\hfill \end{array}\hfill \end{array}\right) $$
(46)
By substituting Eqs. 46 and 45 into Eq. 44, we get
$$ \begin{array}{l}\left(\begin{array}{c}\hfill {I}_{0 nml}^{sr}\hfill \\ {}\hfill {I}_{1 nml}^{sr}\hfill \\ {}\hfill \begin{array}{c}\hfill \begin{array}{c}\hfill \vdots \hfill \\ {}\hfill \begin{array}{c}\hfill \vdots \hfill \\ {}\hfill \vdots \hfill \end{array}\hfill \end{array}\hfill \\ {}\hfill {I}_{k nml}^{sr}\hfill \end{array}\hfill \end{array}\right)={e}^{jn \arg \left({R}_{0100}\right)}{e}^{jm \arg \left({R}_{0010}\right)}{e}^{jl \arg \left({R}_{0001}\right)}\left(\begin{array}{c}\hfill \begin{array}{ccc}\hfill {B}_{00}\hfill & \hfill \hfill & \hfill \hfill \\ {}\hfill {B}_{10}\hfill & \hfill {B}_{11}\hfill & \hfill \hfill \\ {}\hfill \vdots \hfill & \hfill \ddots \hfill & \hfill \hfill \end{array}\hfill \\ {}\hfill \begin{array}{ccc}\hfill {B}_{k0}\hfill & \hfill {B}_{k1}\hfill & \hfill {B}_{k k}\hfill \end{array}\hfill \end{array}\right)\left(\begin{array}{c}\hfill \begin{array}{ccc}\hfill {R_{0000}}^{-1}\hfill & \hfill \hfill & \hfill \hfill \\ {}\hfill \hfill & \hfill {R_{0000}}^{-2}\hfill & \hfill \hfill \\ {}\hfill \vdots \hfill & \hfill \hfill & \hfill {R_{0000}}^{-\left( k+1\right)}\hfill \end{array}\hfill \\ {}\hfill \begin{array}{ccc}\hfill \hfill & \hfill \hfill & \hfill \hfill \end{array}\hfill \end{array}\right)\\ {}\times \left(\begin{array}{c}\hfill \begin{array}{ccc}\hfill {x}^{-1}\hfill & \hfill \hfill & \hfill \hfill \\ {}\hfill \hfill & \hfill {x}^{-2}\hfill & \hfill \hfill \\ {}\hfill \vdots \hfill & \hfill \hfill & \hfill \hfill \end{array}\hfill \\ {}\hfill \begin{array}{ccc}\hfill \hfill & \hfill \hfill & \hfill {x}^{-\left( k+1\right)}\hfill \end{array}\hfill \end{array}\right)\left(\begin{array}{c}\hfill \begin{array}{ccc}\hfill {D}_{00}\hfill & \hfill \hfill & \hfill \hfill \\ {}\hfill {D}_{10}\hfill & \hfill {D}_{11}\hfill & \hfill \hfill \\ {}\hfill \vdots \hfill & \hfill \ddots \hfill & \hfill \hfill \end{array}\hfill \\ {}\hfill \begin{array}{ccc}\hfill {D}_{k0}\hfill & \hfill \hfill & \hfill {D}_{k k}\hfill \end{array}\hfill \end{array}\right)\ \left(\begin{array}{c}\hfill \begin{array}{ccc}\hfill {B}_{00}\hfill & \hfill \hfill & \hfill \hfill \\ {}\hfill {B}_{10}\hfill & \hfill {B}_{11}\hfill & \hfill \hfill \\ {}\hfill \vdots \hfill & \hfill \ddots \hfill & \hfill \hfill \end{array}\hfill \\ {}\hfill \begin{array}{ccc}\hfill {B}_{k0}\hfill & \hfill {B}_{k1}\hfill & \hfill {B}_{k k}\hfill \end{array}\hfill \end{array}\right)\left(\begin{array}{c}\hfill \begin{array}{ccc}\hfill {x}^1\hfill & \hfill \hfill & \hfill \hfill \\ {}\hfill \hfill & \hfill {x}^2\hfill & \hfill \hfill \\ {}\hfill \vdots \hfill & \hfill \hfill & \hfill \hfill \end{array}\hfill \\ {}\hfill \begin{array}{ccc}\hfill \hfill & \hfill \hfill & \hfill {x}^{k+1}\hfill \end{array}\hfill \end{array}\right)\ \left(\begin{array}{c}\hfill \begin{array}{ccc}\hfill {D}_{00}\hfill & \hfill \hfill & \hfill \hfill \\ {}\hfill {D}_{10}\hfill & \hfill {D}_{11}\hfill & \hfill \hfill \\ {}\hfill \vdots \hfill & \hfill \ddots \hfill & \hfill \hfill \end{array}\hfill \\ {}\hfill \begin{array}{ccc}\hfill {D}_{k0}\hfill & \hfill \hfill & \hfill {D}_{k k}\hfill \end{array}\hfill \end{array}\right)\left(\begin{array}{c}\hfill {R}_{0 nml}\hfill \\ {}\hfill {R}_{1 nml}^{sr}\hfill \\ {}\hfill \begin{array}{c}\hfill \begin{array}{c}\hfill \vdots \hfill \\ {}\hfill \begin{array}{c}\hfill \vdots \hfill \\ {}\hfill \vdots \hfill \end{array}\hfill \end{array}\hfill \\ {}\hfill {R}_{k nml}\hfill \end{array}\hfill \end{array}\right)\end{array} $$
(47)
Science
$$ \left(\begin{array}{c}\hfill \begin{array}{ccc}\hfill {B}_{00}\hfill & \hfill \hfill & \hfill \hfill \\ {}\hfill {B}_{10}\hfill & \hfill {B}_{11}\hfill & \hfill \hfill \\ {}\hfill \vdots \hfill & \hfill \ddots \hfill & \hfill \hfill \end{array}\hfill \\ {}\hfill \begin{array}{ccc}\hfill {B}_{k0}\hfill & \hfill {B}_{k1}\hfill & \hfill {B}_{k k}\hfill \end{array}\hfill \end{array}\right)\left(\begin{array}{c}\hfill \begin{array}{ccc}\hfill {D}_{00}\hfill & \hfill \hfill & \hfill \hfill \\ {}\hfill {D}_{10}\hfill & \hfill {D}_{11}\hfill & \hfill \hfill \\ {}\hfill \vdots \hfill & \hfill \ddots \hfill & \hfill \hfill \end{array}\hfill \\ {}\hfill \begin{array}{ccc}\hfill {D}_{k0}\hfill & \hfill \hfill & \hfill {D}_{k k}\hfill \end{array}\hfill \end{array}\right)=\left(\begin{array}{c}\hfill \begin{array}{ccc}\hfill 1\hfill & \hfill \hfill & \hfill \hfill \\ {}\hfill \hfill & \hfill 1\hfill & \hfill \hfill \\ {}\hfill \vdots \hfill & \hfill \ddots \hfill & \hfill \hfill \end{array}\hfill \\ {}\hfill \begin{array}{ccc}\hfill \hfill & \hfill \hfill & \hfill 1\hfill \end{array}\hfill \end{array}\right)\left(\begin{array}{c}\hfill \begin{array}{ccc}\hfill {x}^{-1}\hfill & \hfill \hfill & \hfill \hfill \\ {}\hfill \hfill & \hfill {x}^{-2}\hfill & \hfill \hfill \\ {}\hfill \vdots \hfill & \hfill \hfill & \hfill \hfill \end{array}\hfill \\ {}\hfill \begin{array}{ccc}\hfill \hfill & \hfill \hfill & \hfill {x}^{-\left( k+1\right)}\hfill \end{array}\hfill \end{array}\right)\left(\begin{array}{c}\hfill \begin{array}{ccc}\hfill {x}^1\hfill & \hfill \hfill & \hfill \hfill \\ {}\hfill \hfill & \hfill {x}^2\hfill & \hfill \hfill \\ {}\hfill \vdots \hfill & \hfill \hfill & \hfill \hfill \end{array}\hfill \\ {}\hfill \begin{array}{ccc}\hfill \hfill & \hfill \hfill & \hfill {x}^{k+1}\hfill \end{array}\hfill \end{array}\right)=\left(\begin{array}{c}\hfill \begin{array}{ccc}\hfill 1\hfill & \hfill \hfill & \hfill \hfill \\ {}\hfill \hfill & \hfill 1\hfill & \hfill \hfill \\ {}\hfill \vdots \hfill & \hfill \ddots \hfill & \hfill \hfill \end{array}\hfill \\ {}\hfill \begin{array}{ccc}\hfill \hfill & \hfill \hfill & \hfill 1\hfill \end{array}\hfill \end{array}\right) $$
Eq. 47 can be rewritten as
$$ \begin{array}{l}\left(\begin{array}{c}\hfill {I}_{0 nml}^{sr}\hfill \\ {}\hfill {I}_{1 nml}^{sr}\hfill \\ {}\hfill \begin{array}{c}\hfill \begin{array}{c}\hfill \vdots \hfill \\ {}\hfill \begin{array}{c}\hfill \vdots \hfill \\ {}\hfill \vdots \hfill \end{array}\hfill \end{array}\hfill \\ {}\hfill {I}_{k nml}^{sr}\hfill \end{array}\hfill \end{array}\right)={e}^{jn \arg \left({R}_{0100}\right)}{e}^{jm \arg \left({R}_{0010}\right)}{e}^{jl \arg \left({R}_{0001}\right)}\left(\begin{array}{c}\hfill \begin{array}{ccc}\hfill {B}_{00}\hfill & \hfill \hfill & \hfill \hfill \\ {}\hfill {B}_{10}\hfill & \hfill {B}_{11}\hfill & \hfill \hfill \\ {}\hfill \vdots \hfill & \hfill \ddots \hfill & \hfill \hfill \end{array}\hfill \\ {}\hfill \begin{array}{ccc}\hfill {B}_{k0}\hfill & \hfill {B}_{k1}\hfill & \hfill {B}_{k k}\hfill \end{array}\hfill \end{array}\right)\left(\begin{array}{c}\hfill \begin{array}{ccc}\hfill {R_{0000}}^{-1}\hfill & \hfill \hfill & \hfill \hfill \\ {}\hfill \hfill & \hfill {R_{0000}}^{-2}\hfill & \hfill \hfill \\ {}\hfill \vdots \hfill & \hfill \hfill & \hfill {R_{0000}}^{-\left( k+1\right)}\hfill \end{array}\hfill \\ {}\hfill \begin{array}{ccc}\hfill \hfill & \hfill \hfill & \hfill \hfill \end{array}\hfill \end{array}\right)\left(\begin{array}{c}\hfill \begin{array}{ccc}\hfill {D}_{00}\hfill & \hfill \hfill & \hfill \hfill \\ {}\hfill {D}_{10}\hfill & \hfill {D}_{11}\hfill & \hfill \hfill \\ {}\hfill \vdots \hfill & \hfill \ddots \hfill & \hfill \hfill \end{array}\hfill \\ {}\hfill \begin{array}{ccc}\hfill {D}_{k0}\hfill & \hfill \hfill & \hfill {D}_{k k}\hfill \end{array}\hfill \end{array}\right)\left(\begin{array}{c}\hfill {R}_{0 nml}\hfill \\ {}\hfill {R}_{1 nml}^{sr}\hfill \\ {}\hfill \begin{array}{c}\hfill \begin{array}{c}\hfill \vdots \hfill \\ {}\hfill \begin{array}{c}\hfill \vdots \hfill \\ {}\hfill \vdots \hfill \end{array}\hfill \end{array}\hfill \\ {}\hfill {R}_{k nml}\hfill \end{array}\hfill \end{array}\right)\\ {}\ \begin{array}{ccc}\hfill \hfill & \hfill \hfill & \hfill =\left(\begin{array}{c}\hfill {I}_{0 nml}\hfill \\ {}\hfill {I}_{1 nml}\hfill \\ {}\hfill \begin{array}{c}\hfill \begin{array}{c}\hfill \vdots \hfill \\ {}\hfill \begin{array}{c}\hfill \vdots \hfill \\ {}\hfill \vdots \hfill \end{array}\hfill \end{array}\hfill \\ {}\hfill {I}_{k nml}\hfill \end{array}\hfill \end{array}\right)\hfill \end{array}\end{array} $$
