4 problems

For $x\in[0,1]$, let $?(x)$ denote the Minkowski question mark function, defined for an irrational continued fraction $x=[a_1,a_2,\ldots]$ by … and for rational $x$ by the correspo…

Alternating moment identity.

Conjectural derivative identity.

Refined conjecture. For every $\beta\in[0,2]\setminus\{1\}$, there exists a constant $C(\beta)\geq 0$ such that