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For $\alpha=\frac{d}{2}$, the desired probabilities ($\frac{29}{64}, \frac{8}{33},\ldots$) are yielded by \begin{equation} \label{Hou1} P(\alpha) =\Sigma_{i=0}^\infty f(\alpha+i), \end{equation} where \begin{equation} \label{Hou2} f(\alpha) = P(\alpha)-P(\alpha +1) = \frac{ q(\alpha) 2^{-4 \alpha -6} \Gamma{(3 \alpha +\frac{5}{2})} \Gamma{(5 \alpha +2})}{6 \Gamma{(\alpha +1)} \Gamma{(2 \alpha +3)} \Gamma{(5 \alpha +\frac{13}{2})}}, \end{equation} and \begin{equation} \label{Hou3} q(\alpha) = 185000 \alpha ^5+779750 \alpha ^4+1289125 \alpha ^3+1042015 \alpha ^2+410694 \alpha +63000 = \end{equation} \begin{equation} \alpha \bigg(5 \alpha \Big(25 \alpha \big(2 \alpha (740 \alpha +3119)+10313\big)+208403\Big)+410694\bigg)+63000. \end{equation} (Qing-Hu Hou helped in the derivation of this formula, using "Zeilberger's algorithm" ["creative telescoping"] ConciseFormula. An equivalent formula--also to be presented here--was given by C. F. Dunkl in App. D of MasterLovasAndai .)

However, for $n$ or $m$ greater than 2, no analogous formulas are yet available.

For $\alpha=\frac{d}{2}$, the desired probabilities ($\frac{29}{64}, \frac{8}{33},\ldots$) are yielded by \begin{equation} \label{Hou1} P(\alpha) =\Sigma_{i=0}^\infty f(\alpha+i), \end{equation} where \begin{equation} \label{Hou2} f(\alpha) = P(\alpha)-P(\alpha +1) = \frac{ q(\alpha) 2^{-4 \alpha -6} \Gamma{(3 \alpha +\frac{5}{2})} \Gamma{(5 \alpha +2})}{6 \Gamma{(\alpha +1)} \Gamma{(2 \alpha +3)} \Gamma{(5 \alpha +\frac{13}{2})}}, \end{equation} and \begin{equation} \label{Hou3} q(\alpha) = 185000 \alpha ^5+779750 \alpha ^4+1289125 \alpha ^3+1042015 \alpha ^2+410694 \alpha +63000 = \end{equation} \begin{equation} \alpha \bigg(5 \alpha \Big(25 \alpha \big(2 \alpha (740 \alpha +3119)+10313\big)+208403\Big)+410694\bigg)+63000. \end{equation} (Qing-Hu Hou helped in the derivation of this formula, using "Zeilberger's algorithm" ["creative telescoping"] ConciseFormula.)

An equivalent formula--now employing $d$ rather than $\alpha=\frac{d}{2}$--was given by C. F. Dunkl in App. D of MasterLovasAndai \begin{equation} \mathcal{P}(d) =3456^{d}\frac{\left( \frac{1}{2}\right) _{d/2}% ^{3}\left( \frac{7}{6}\right) _{d/2}^{2}\left( \frac{5}{6}\right) _{d/2}^{2}\left( 2d\right) !}{\left( \frac{d}{2}\right) !\left( 3\right) _{5d}}\sum_{i\geq0,j\geq0}^{i+j\leq d/2}\frac{\left( -\frac{d}{2}\right) _{i+j}\left( \frac{d}{2}\right) _{j}\left( d\right) _{j}\left( 2+3d\right) _{i}\left( 1+d\right) _{i}}{\left( 2+\frac{5d}{2}\right) _{i+j}\left( 1+\frac{d}{2}\right) _{j}i!j!\left( -2d\right) _{i}}. \end{equation}

However, for dimensions $n$ or $m$ greater than 2, no analogous formulas are yet available.

For $\alpha=\frac{d}{2}$, the desired probabilities ($\frac{29}{64}, \frac{8}{33},\ldots$) are yielded by \begin{equation} \label{Hou1} P(\alpha) =\Sigma_{i=0}^\infty f(\alpha+i), \end{equation} where \begin{equation} \label{Hou2} f(\alpha) = P(\alpha)-P(\alpha +1) = \frac{ q(\alpha) 2^{-4 \alpha -6} \Gamma{(3 \alpha +\frac{5}{2})} \Gamma{(5 \alpha +2})}{6 \Gamma{(\alpha +1)} \Gamma{(2 \alpha +3)} \Gamma{(5 \alpha +\frac{13}{2})}}, \end{equation} and \begin{equation} \label{Hou3} q(\alpha) = 185000 \alpha ^5+779750 \alpha ^4+1289125 \alpha ^3+1042015 \alpha ^2+410694 \alpha +63000 = \end{equation} \begin{equation} \alpha \bigg(5 \alpha \Big(25 \alpha \big(2 \alpha (740 \alpha +3119)+10313\big)+208403\Big)+410694\bigg)+63000. \end{equation} (Qing-Hu Hou helped in the derivation of this formula, using "Zeilberger's algorithm" ["creative telescoping"] ConciseFormula. An equivalent formula was given by C. F. Dunkl in App. D of MasterLovasAndai .)

For $\alpha=\frac{d}{2}$, the desired probabilities ($\frac{29}{64}, \frac{8}{33},\ldots$) are yielded by \begin{equation} \label{Hou1} P(\alpha) =\Sigma_{i=0}^\infty f(\alpha+i), \end{equation} where \begin{equation} \label{Hou2} f(\alpha) = P(\alpha)-P(\alpha +1) = \frac{ q(\alpha) 2^{-4 \alpha -6} \Gamma{(3 \alpha +\frac{5}{2})} \Gamma{(5 \alpha +2})}{6 \Gamma{(\alpha +1)} \Gamma{(2 \alpha +3)} \Gamma{(5 \alpha +\frac{13}{2})}}, \end{equation} and \begin{equation} \label{Hou3} q(\alpha) = 185000 \alpha ^5+779750 \alpha ^4+1289125 \alpha ^3+1042015 \alpha ^2+410694 \alpha +63000 = \end{equation} \begin{equation} \alpha \bigg(5 \alpha \Big(25 \alpha \big(2 \alpha (740 \alpha +3119)+10313\big)+208403\Big)+410694\bigg)+63000. \end{equation} (Qing-Hu Hou helped in the derivation of this formula, using "Zeilberger's algorithm" ["creative telescoping"] ConciseFormula. An equivalent formula--also to be presented here--was given by C. F. Dunkl in App. D of MasterLovasAndai .)

Slater was able to construct--though yet without formalized proof--the much simpler \begin{equation} \label{BasicFormula2} \tilde{\chi}_2 (\varepsilon ) = \frac{1}{3} \varepsilon^2 (4-\varepsilon^2) \end{equation} leading to the two-qubit separability probability of $\frac{8}{33}$. (Also, in this paper, counterparts were given for quaternionic [$\tilde{\chi}_4 (\varepsilon ) = \frac{1}{35} \varepsilon^4 (84-64\varepsilon^2+15 \varepsilon^4)$ yielding $\frac{26}{323}$],...density matrices.)

Slater was able to construct--though yet without formalized proof--the much simpler \begin{equation} \label{BasicFormula2} \tilde{\chi}_2 (\varepsilon ) = \frac{1}{3} \varepsilon^2 (4-\varepsilon^2) \end{equation} leading to the two-qubit separability probability of $\frac{8}{33}$.

Also, in this paper, counterparts were given for quaternionic [$\tilde{\chi}_4 (\varepsilon ) = \frac{1}{35} \varepsilon^4 (84-64\varepsilon^2+15 \varepsilon^4)$ yielding $\frac{26}{323}$],...density matrices.

Then, these three formulas were incorporated into a "Master Lovas-Andai" formula--the index $d$ being a form of "Dyson-index" of random matrix theory- \begin{equation} \tilde{\chi}_d (\varepsilon ) = \frac{\varepsilon^d \Gamma(d+1)^3 \, _3\tilde{F}_2\left(-\frac{d}{2},\frac{d}{2},d;\frac{d}{2}+1,\frac{3 d}{2}+1;\varepsilon^2\right)}{\Gamma(\frac{d}{2}+1)^2}, \end{equation} where the regularized hypergeometric function is indicated.

For $\alpha=\frac{d}{2}$, the desired probabilities ($\frac{29}{64}, \frac{8}{33},\ldots$) are yielded by \begin{equation} \label{Hou1} P(\alpha) =\Sigma_{i=0}^\infty f(\alpha+i), \end{equation} where \begin{equation} \label{Hou2} f(\alpha) = P(\alpha)-P(\alpha +1) = \frac{ q(\alpha) 2^{-4 \alpha -6} \Gamma{(3 \alpha +\frac{5}{2})} \Gamma{(5 \alpha +2})}{6 \Gamma{(\alpha +1)} \Gamma{(2 \alpha +3)} \Gamma{(5 \alpha +\frac{13}{2})}}, \end{equation} and \begin{equation} \label{Hou3} q(\alpha) = 185000 \alpha ^5+779750 \alpha ^4+1289125 \alpha ^3+1042015 \alpha ^2+410694 \alpha +63000 = \end{equation} \begin{equation} \alpha \bigg(5 \alpha \Big(25 \alpha \big(2 \alpha (740 \alpha +3119)+10313\big)+208403\Big)+410694\bigg)+63000. \end{equation} (Qing-Hu Hou helped in the derivation of this formula, using "Zeilberger's algorithm" ["creative telescoping"] ConciseFormula. An equivalent formula was given by C. F. Dunkl in App. D of MasterLovasAndai .)