0
$\begingroup$

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 𝑓(𝜇,𝛽) wrt to 𝜇 (the double integral over $p_{11,22}$ can be calculated using Dirichlet's Beta integral) and then recover 𝑓(𝜇,𝛽) 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 𝑓(𝜇,𝛽) as Gauss hypergeometric function in variable 𝜇".)

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 𝑓(𝜇,𝛽) as Gauss hypergeometric function in variable 𝜇, 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}

$\endgroup$
1
$\begingroup$

In trying to implement the strategy of Nemo, I performed the integration of \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} for $b =1,\ldots,5$, then requested Mathematica to take the Mellin transform with respect to $\mu$ of the results.

For $b=1$, I obtained \begin{equation} \frac{1}{3} \left(\mathcal{M}_\mu\left[\frac{\mu}{\left(\mu^2-1\right)^3}\right](s)+\mathcal{M}_\mu\left[\frac{\mu \log (\mu)}{\left(\mu^2-1\right)^3}\right](s)-\mathcal{M}_\mu\left[\frac{\mu^3}{\left(\mu^2-1\right)^ 3}\right](s)+\mathcal{M}_\mu\left[\frac{\mu^3 \log (\mu)}{\left(\mu^2-1\right)^3}\right](s)\right), \end{equation} for $b=2$, \begin{equation} \frac{1}{30} \left(3 \mathcal{M}_\mu\left[\frac{\mu^2}{\left(\mu^2-1\right)^5}\right](s)-3 \mathcal{M}_\mu\left[\frac{\mu^6}{\left(\mu^2-1\right)^5}\right](s)-\frac{\pi (s-2) s \left(e^{i \pi s} \left(\pi s^2-2 (\pi -2 i) s-4 i\right)-4 i (s-1)\right)}{32 \left(-1+e^{i \pi s}\right)^2}\right), \end{equation} for $b=3, \begin{equation} \frac{\pi (s-3) (s-1) (s+1) \left(2 i (3 (s-2) s-1)+e^{i \pi s} (6 i (s-2) s+\pi (s-3) (s-1) (s+1)-2 i)\right)}{161280 \left(1+e^{i \pi s}\right)^2} + \end{equation} \begin{equation} -\frac {11} {420} \mathcal {M} _\mu [\frac {\mu^9} { (\mu^2 - 1 )^7} ] (s) - \frac {9} {140} \mathcal {M} _\mu [\frac {\mu^7} { (\mu^2 - 1 )^7} ] (s) + \frac {9} {140} \mathcal {M} _\mu [\frac {\mu^5} { (\mu^2 - 1 )^7} ] (s) + \frac {11} {420} \mathcal {M} _\mu [\frac {\mu^3} { (\mu^2 - 1 )^7} ] (s), \end{equation} ....

So, since the required Mellin transforms do not all seem explicitly implementable, it appears to me that the Nemo goal of obtaining a (complete) representation of $f(\mu,b)$ (using the Matt. F. parameters) does not seem realizable.

$\endgroup$

Your Answer

By clicking "Post Your Answer", you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.