Q:

Find a generalized hypergeometric-based function yielding certain ratios of fifth-degree polynomials

Find a (presumably, generalized hypergeometric-based function $f(n,a,k)$), yielding for $n=1, a=\frac{1}{2}$,the rational function (ratio of fifth-degree polynomials)
\begin{equation}
f(1,\frac{1}{2},k)=\frac{64 k^5 +128 k^4 -340 k^3 -1032 k^2 -1099 k -384}{k (2 k-1) (2 k+5) (4 k-1) (4 k+1)}
\end{equation}
and for $n=1, a=1$,
\begin{equation}
f(1,1,k)=\frac{8 k^5+36 k^4 -82 k^3 -681 k^2 -1366 k -885}{128 (k+2) (k+3) (k+4) (4 k+5) (4 k+7)}
\end{equation}
These are, respectively, eqs. (25) and (8) in arXiv:1207.1297v2, "Bures and Hilbert-Schmidt $2 \times 2$ Determinantal Moments". Presumably, for general $n$, we have ratios of $5 n$-degree polynomials in $k$. (These ratios pertain to the Bures case. In the Hilbert-Schmidt counterpart, general formulas--incorporating a $_5F_4$ hypergeometric function--yielding ratios of $3 n$-degree polynomials have previously been found, though not yet rigorously demonstrated arXiv:1109.2560, sec. D.6.)

A:

Are there any other conditions this function must satisfy?
Otherwise, there is a trivial solution:
$$\frac{(2-2a) \left(64 k^5+128 k^4-340 k^3-1032 k^2-1099 k-384\right)}{k (2 k-1) (2 k+5) (4 k-1) (4 k+1)}+\frac{(2 a-1) \left(8 k^5+36 k^4-82  k^3-681 k^2-1366 k-885\right)}{128 (k+2) (k+3) (k+4) (4 k+5) (4 k+7)}$$

