Proof of a closed-form of $\prod_{n=1}^{\infty} \frac{1}{e} \left(1+\frac{1}{3n}\right)^{3n+1/2}$

I'm looking for a proof of the following equality:

$$\prod_{n=1}^{\infty} \frac{1}{e} \left(1+\frac{1}{3n}\right)^{3n+1/2}=\sqrt{\frac{\Gamma\left(\frac{1}{3}\right)}{2\pi}}\frac{3^{13/24} \exp \left[1+\frac{2\pi^2-3\psi_1 \left(\frac{1}{3}\right)}{12\pi \sqrt{3}} \right]}{A^4}$$ Where $\psi_1$ is the trigamma function, and $A$ is the Glaisher-Kinkelin constant.

This product is the last entry of this Wolfram Mathworld page. The product is stated to have been found by Gosper, however, his proof is not listed in the references. I have checked the OEIS entry of its decimal expansion also, however, the closed-form is not referenced there either. Assuming, Gosper refers to Bill Gosper, nothing about this product is stated on his Wikipedia page either.

Solution 1:

The $\log$ of your product is $$ =\sum_{n\ge 1} -1+(3n+1)\log(3n+1)-3n\log 3n-\frac12\log(3n+1)- \frac12\log(3n) $$ Up to a non-complicated constant coming from the regularization terms this is a mix of $\zeta(-1),\zeta'(-1),\zeta(0),\zeta'(0),\log\Gamma(1/3),L'(-1,\chi_3)$.

• $\chi$ is a an odd Dirichlet character so $L(-1,\chi_3)=0$ and the functional equation relates $L'(-1,\chi_3)$ with $L(2,\chi_3)$, it doesn't have a closed-form because $\chi_3$ is odd, which is where your $\psi_1(1/3)$ term comes from

• $\zeta'(0),\zeta'(-1)$ cause your $A$ term