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In a quantum-information-theoretic context, I've encountered the problem of integrating over $r \in [0,1]$, the function

\begin{equation} r^{2 d-1} \, _2F_1\left(-\frac{d}{2},\frac{d}{2};\frac{d+2}{2};r^2\right) \left(1-\varepsilon ^2 r^2\right)^{d/2}, \end{equation}

where the parameters $d \geq 1$ and $0<\varepsilon<1$. Mathematica does not appear to directly succeed with this problem. I have tried using certain Pfaff transformations in this regard.

For fixed even values of $d$, the integration yields a polynomial of degree $ d$ in $\varepsilon$, while for odd values of $d$, the integration, quite differently, yields results involving inverse hyperbolic and polylogarithmic functions of $\varepsilon$.

The question pertains to the problem of constructing the functions $\tilde{\chi}_d (\varepsilon)$, raised in https://arxiv.org/abs/1610.01410 and further studied in https://arxiv.org/abs/1701.01973, concerning certain conjectures as to the (rational) values of generalized two-qubit Hilbert-Schmidt separability probabilities''. ($\varepsilon$ is thesingular value ratio'' and $d$ is a ``Dyson-index-like'' parameter, associated with random matrix theory.)

If we set $\varepsilon=0$, then the requested integration yields \begin{equation} \frac{\, _3F_2\left(-\frac{d}{2},\frac{d}{2},d;1+\frac{2}{d},d+1;1\right)}{2 d}, \end{equation} while if we set $\varepsilon=1$, then we obtain \begin{equation} \frac{\Gamma \left(\frac{d}{2}+1\right) \Gamma (d) \, _3F_2\left(-\frac{d}{2},\frac{d}{2},d;1+\frac{2}{d},\frac{3 d}{2}+1;1\right)}{2 \Gamma \left(\frac{3 d}{2}+1\right)}. \end{equation} So, we want a function $f(\varepsilon,d)$ that, in some sense, interpolates between these two endpoints, yielding those polynomials (as indicated by Carlo Beenakker) that result from the integration for even $d$, such as \begin{equation} f(\varepsilon,4)=\frac{13 \varepsilon ^4}{672}-\frac{31 \varepsilon ^2}{630}+\frac{1}{30}. \end{equation}

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    $\begingroup$ for even $d$ the hypergeometric function is just a polynomial and the integral can be evaluated in closed form. $\endgroup$
    – Carlo Beenakker
    Commented Aug 19, 2017 at 11:26
  • $\begingroup$ Thanks, Carlo Beenakker. The integral can apparently be evaluated in closed form for any specific $d>0$, even or odd--being much simpler (as noted) in the even case. But, I was wondering if the integral could be evaluated/expressed in some form as both a function of the parameters $d$ and $\varepsilon$. This might allow me to pursue some further analyses of potential interest. $\endgroup$
    – Paul B. Slater
    Commented Aug 19, 2017 at 13:23
  • $\begingroup$ Dr. Slater, I was reading your paper "Master Lovas-Andai and Equivalent Formulas Verifying the $\frac{8}{33}$ Two-Qubit Hilbert-Schmidt Separability Probability and Companion Rational-Valued Conjectures" and noted a link to here. Along with our websites math.SE we also have a Quantum Computing site which I noticed you are not a member of, and I invite you to join. YT $\endgroup$
    – Rob
    Commented Jun 28, 2018 at 6:58

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An integration-by-parts yielded the result \begin{equation} -\frac{\Gamma \left(\frac{d}{2}+1\right)^2 \Gamma (d) \left(\Gamma (d+1)^2 \, _3\tilde{F}_2\left(-\frac{d}{2},\frac{d}{2},d;\frac{d}{2}+1,\frac{3 d}{2}+1;\varepsilon ^2\right)-2 \, _2F_1\left(-\frac{d}{2},\frac{d}{2};\frac{d+2}{2};\varepsilon ^2\right)\right)}{2 \Gamma (d+1)^2}. \end{equation}

Added together with a set of auxiliary results, this leads to a formula for the "Lovas-Andai function" (https://arxiv.org/abs/1610.01410) \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 \left(\frac{d}{2}+1\right)^2}, \end{equation} where the tilde is used to denote the regularized hypergeometric function.

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