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Consider the generalized hypergeometric function: $$ {_3F_2}\left(1,\frac12,\frac12;\ \frac76,\frac{11}{6};\ \frac{4z}{27}\right) $$

I know it can be done with the beta integral as shown below: What is $_3F_2\left(1,\frac32,2;\ \frac43,\frac53;\ \frac4{27}\right)$ as an integral?

the harder problem is how to evaluate the 3f2 expression using the Euler integral . this hard because the integrals are elliptic , involve fractions of 1/3 or 1/6, and there are double integrals. I am guessing one makes substitutions, but there are many possible approaches to this.

CarP24
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  • I am faced to a very similar question in these links https://math.stackexchange.com/questions/4872230/how-to-prove-this-difficult-double-integral-identity and https://mathoverflow.net/questions/466853/how-to-prove-these-identities-for-log2-based-on-3f-2-integrals It is very difficult, I do not have an answer already. – Jorge Zuniga Mar 14 '24 at 00:16

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