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In the study of the $2$-adic sum of digits function $S_2\left(n\right)$, the arithmetical function $u\left(0\right)=0$ and $u\left(n\right)={(-1)}^{n-1}$ for $n\ge 1$ plays a very important role. In this paper, we firstly generalize the relation between $S_2\left(n\right)$ and $u\left(n\right)$ to a bijective relation between arithmetical functions. And as an application, we investigate some aspects of the sum of digits functions $S_{𝒢}\left(n\right)$ induced by binary infinite Gray codes $𝒢$. We can show that the difference of the sum of digits function, $S_{𝒢}\left(n\right)-S_{𝒢}(n-1)$, is realized by an automaton. And the summation formula of the sum...