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Constants connection

We have the following well known connection between favourite constants such as...

Integrals: $$-{\pi\over 2}\ln{2}=\int_{0}^{\pi/2}\ln{\cos(x)}\mathrm dx$$

$$-{\sqrt{\pi}\over 4}(\gamma+\ln{4})=\int_{0}^{\infty}e^{-x^2}\ln{x}\mathrm dx$$

Continued fractions

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

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    Why don't you add a series for $\gamma$ to a series for $\phi$ ? And we don't know if $\gamma$ is irrational, or algebraic, or transcendental. – 2017-02-01
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    I just want to know if there any series, integrals, infinite products, etc, that can be expressable in terms of $\gamma$ and $\phi$ – 2017-02-01
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    There is no connection, $\phi$ is golden ratio, it is an algebraic number unlike $\gamma$. It has no deep role in maths. – 2017-02-02

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