Constants connection
We have the following well known connection between favourite constants such as...
Integrals: $$-{\pi\over 2}\ln{2}=\int_{0}^{\pi/2}\ln{\cos(x)}\mathrm dx$$
$$-{\sqrt{\pi}\over 4}(\gamma+\ln{4})=\int_{0}^{\infty}e^{-x^2}\ln{x}\mathrm dx$$
Infinite products and others $$e^{\gamma}=\left({2\over 1}\right)^{1/2}\left({2^2\over 1\cdot 3}\right)^{1/3}\left({2^3\cdot 4}\over {1\cdot 3^3}\right)^{1/4}\cdots$$
I have not see integrals, series, continued fraction and infinite products that is connecting the two favourite constants $\gamma$ and $\phi$ together.
My question is: Are there such integrals, series, continued fraction and infinite products showing these non-trivial connection?
I am not interested in trivial connection.
An example of trivial
$${\pi\over 4}=\int_{0}^{\infty}{1\over 1+x^2}\cdot{\mathrm dx\over 1+x^n}\tag1$$
Setting $n=\gamma\phi$
I can't remember who posted $(1)$ or the link