I have a set of individual formulas ($a=1/2, 1, 3/2,\ldots,6$), each itself a function of an integer variable $k$, of increasing complexity. I would like to find a "master" formula (conjecturally of a hypergeometric nature), itself a function of both $a$ and $k$, encompassing these individual cases. I will now list all the results I currently have. More may be added, but it is becoming quite challenging to do so. (A Mathematica notebook is available containing the formulas listed below. The problem pertains to the issue of the entanglement of two quantum bits, where $a$ serves as a "Dyson-index-like" parameter of random matrix theory.)
($a=\frac{1}{2}$) $\dfrac{\Gamma \left(2 k+\dfrac{9}{2}\right)}{\sqrt{\pi } \Gamma (2 k+5)}$
($a=1$) $\dfrac{4^{k+3} \Gamma \left(k+\dfrac{7}{2}\right)^2 \Gamma \left(k+\dfrac{9}{2}\right)}{\pi \Gamma (k+5) \Gamma \left(2 k+\dfrac{13}{2}\right)}$
($a=\frac{3}{2}$) $\dfrac{(6 k+31) \Gamma \left(2 k+\dfrac{19}{2}\right)}{4 \sqrt{\pi } (k+5) (k+6) \Gamma (2 k+9)}$
($a=2$) $\dfrac{4^{k+6} (k+6) \Gamma \left(k+\dfrac{11}{2}\right) \Gamma \left(k+\frac{13}{2}\right) \Gamma \left(k+\frac{15}{2}\right)}{\pi \Gamma (k+9) \Gamma \left(2 k+\frac{23}{2}\right)}$
($a=\frac{5}{2}$) $\frac{\left(20 k^3+460 k^2+3547 k+9173\right) \Gamma \left(2 k+\frac{29}{2}\right)}{4 \sqrt{\pi } (k+8) (k+9) (k+10) \Gamma (2 k+15)}$
($a=3$) $\dfrac{4^{k+8} \left(3 k^3+77 k^2+668 k+1944\right) \Gamma \left(k+\dfrac{15}{2}\right) \Gamma \left(k+\dfrac{17}{2}\right) \Gamma \left(k+d\frac{21}{2}\right)}{\pi \Gamma (k+13) \Gamma \left(2 k+\dfrac{33}{2}\right)}$
($a=\frac{7}{2}$) $\dfrac{(2 k+19) \left(56 k^4+2380 k^3+38458 k^2+279845 k+772746\right) \Gamma \left(2 k+\dfrac{39}{2}\right)}{8 \sqrt{\pi } (k+11) (k+12) (k+13) (k+14) \Gamma (2 k+21)}$
($a=4$) $\dfrac{2^{2 k+23} (k+11) \left(k^3+34 k^2+402 k+1608\right) \Gamma \left(k+\frac{19}{2}\right) \Gamma \left(k+\frac{23}{2}\right) \Gamma \left(k+\frac{27}{2}\right)}{\pi \Gamma (k+17) \Gamma \left(2 k+\frac{43}{2}\right)}$
($a=5$) $\dfrac{4^{k+13} \left(5 k^6+415 k^5+14613 k^4+278177 k^3+3006982 k^2+17437488 k+42253920\right) \Gamma \left(k+\frac{23}{2}\right) \Gamma \left(k+\frac{27}{2}\right) \Gamma \left(k+\frac{33}{2}\right)}{\pi \Gamma (k+21) \Gamma \left(2 k+\frac{53}{2}\right)}$
($a=6$) $\frac{4^{k+16} (k+16) \left(3 k^6+299 k^5+12795 k^4+298261 k^3+3965226 k^2+28334952 k+84618864\right) \Gamma \left(k+\dfrac{27}{2}\right) \Gamma \left(k+\dfrac{33}{2}\right) \Gamma \left(k+\dfrac{39}{2}\right)}{\pi \Gamma (k+25) \Gamma \left(2 k+\dfrac{63}{2}\right)}$
We also have for $a=\frac{1}{4}$, the result $2^{-2 k-\frac{19}{4}} \Gamma \left(2 k+\frac{13}{4}\right) \, _3\tilde{F}_2\left(1,k+\frac{13}{8},k+\frac{17}{8};k+\frac{19}{8},k+\frac{23}{8};1 \right)$ and for $a=-\frac{1}{4}$, the result $-2^{-2 k-\frac{9}{4}} \Gamma \left(2 k+\frac{3}{4}\right) \, _3\tilde{F}_2\left(1,k+\frac{3}{8},k+\frac{7}{8};k+\frac{9}{8},k+\frac{13}{8};1\right)$.
Finally, for $a=-\frac{1}{2}$, we simply have $-\frac{\Gamma \left(2 k-\frac{1}{2}\right)}{\sqrt{\pi } \Gamma (2 k)}$. The last two formulas for negative values of $a$ have certain issues/breakdowns (too complicated to try to explain at this point) with negative values of $k$. Also, many of the individual formulas listed above have integer or half-integer roots (although I have yet to complete an inventory in this regard).