This is a follow-up to the first comment (by Nemo) to the posting Compute the two-fold partial integral, where the three-fold full integral is known . (I also just asked this as a comment to that question itself--but suspected that it would remain rather obscure in that form.)

After a simplifying reparameterization (transforming $\beta$ into $\frac{b-1}{3}$) suggested by Matt F., the originally-posed problem took the form, \begin{equation} \int_{p=0}^1 \int_{q=0}^{1-p} (\mu p q (1-p-q))^b (\mu^2 q +p)^{-b-1} dq dp, \end{equation} which has been employed since.

Answers to that question have, in fact, been posted by Martin Rubey and by me, but the possible one that was apparently raised by Nemo--in terms of the original $\beta, p_{11,22}$ (rather than $b,p,q$) parameterization--has not so far been fully detailed. (In his comment, Nemo wrote "Did you try calculating Mellin transform of π(π,π½) wrt to π (the double integral over $p_{11,22}$ can be calculated using Dirichlet's Beta integral) and then recover π(π,π½) by inverse Mellin transform (which reduces to Barnes integral in this case? I did some calculations and if not mistaken this will lead to representation of π(π,π½) as Gauss hypergeometric function in variable π".)

Nemo also later commented "MartinRubey--my observation above about this function having a Gauss hypergeometric form is consistent with this logarithmic terms because 2 parameters of this hypergeometric function coincide", but this has not yet been expanded into an answer after a request of Rubey to do so.

So, to reiterate, I would like to explicitly know the presumed representation of π(π,π½) as Gauss hypergeometric function in variable π, as this might be helpful in the pursuit of the program to construct "separability functions" presented in https://arxiv.org/abs/1701.01973.

In fact, we have been able to find, as already detailed in the earlier answers, a Gauss hypergeometric function expression for the linear (in $\log{\mu}$) term, \begin{equation} w(b,\mu)=\frac{\sqrt{\pi } 4^{-b} \mu^b \left(\mu^2-1\right)^{-2 b-1} \Gamma (b+1) \, _2F_1\left(-b,-b;1;\mu^2\right)}{\Gamma \left(b+\frac{3}{2}\right)}, \end{equation} but not yet for the ``constant term'' $v(b,\mu)$ of the complete functional expression \begin{equation} v(b,\mu) + w(b,\mu) \log(\mu)= \end{equation} \begin{equation} \frac{1}{\Gamma \left(b+\frac{3}{2}\right)} \sqrt{\pi } 4^{-b} \mu^b \left(\mu^2-1\right)^{-2 b-1} \Gamma (b+1) \left(\log (\mu) \sum _{k=0}^b \mu ^{2 k} \binom{b}{k}^2-\left(\mu^2-1\right) \sum _{k=1}^b \mu^{2 k-2} \sum _{i=0}^{k-1} \binom{b}{i}^2 (\psi ^{(0)}(b-i+1)-\psi ^{(0)}(i+1))\right), \end{equation} where, \begin{equation} \sum _{k=0}^b \mu ^{2 k} \binom{b}{k}^2=\, _2F_1\left(-b,-b;1;\mu ^2\right). \end{equation}