This morning I've deduced two identities that involve Gregory coefficients $G_n$ invoking the so-called Schröder's integral formula (this is the Wikipedia's article for Gregory coefficients).
The first is $$\frac{1}{\varphi}\sum_{n=1}^\infty|G_n|\frac{F_n}{\varphi^n}=\int_0^\infty\frac{1+x}{(1+x)^2\varphi^2-(1+x)\varphi-1}\cdot\frac{dx}{\pi^2+\log^2 x},\tag{1}$$ where $\varphi$ is the golden ratio and $F_n$ are the Fibonacci numbers (with the standard definition), and this second identity that I believe that holds for reals $|y|<1$ was
$$\sum_{k=1}^\infty\sum_{n=1}^\infty\mu(n)\frac{G_k}{k}\cdot\frac{y^{kn}}{1-y^{kn}}=\int_0^\infty\log\left(1+\frac{y}{1+x}\right)\cdot\frac{dx}{\pi^2+\log^2 x},\tag{2}$$ where $\mu(n)$ denotes the Möbius function and the resulting identity is also a combination with the Lambert series for the Möbius funciton (a combination , with Schröder's integral formula).
My numerical check of such identities was very poor, and my doubt is that to create $(1)$ and $(2)$ I've swapped the summation that is implicit and the integral in each RHS of such identities.
Question. I think that the corresponding RHS of each/both equations can be justified using the dominated convergence theorem (or absolute convergence). Am I right? Many thanks.
If you detect numerically that some identity is false, please add such evidence.