I was reading about BBP type formulas and there was a lot about $\pi$ and some $\log$'s. I started searching for some other constants and could find $2$ formulas for the catalan constant and learned that there is no formula for $e$. I want to know if there is a BBP type formula for $\gamma=0.577...$ but couldn't find anything about it. Such formula exists?
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$\begingroup$ “there is no formula for e” Source? $\endgroup$– user76284Commented Oct 18, 2023 at 16:41
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$\begingroup$ @user76284 mathoverflow.net/questions/219816/… $\endgroup$– PintecoCommented Oct 18, 2023 at 21:24
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$\begingroup$ I believe that says suspected, not proven. $\endgroup$– user76284Commented Oct 18, 2023 at 22:09
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If "having a BBP-type formula for $\alpha\in\mathbb R$" is defined as "there is $r\in\mathbb Q(x)$ and $b\in\mathbb R_{\mathrm{alg}}$ (the field of real $\mathbb Q$-algebraic numbers) such that $\alpha=\sum_{n\ge0}b^{-n}r(n)$", it seems that the answer is either "no" or inaccesible. For this definition, it's easy to see using fractional decomposition and some manipulations with power series, that if $\alpha\in\mathbb R$ has a BBP-type formula, then it's a Kontsevich--Zagier period (i.e. is expressible in a form $\int_Sq(x_1,\ldots,x_k)\,dx_1\wedge\cdots\wedge dx_k$ for some $\mathbb R_{\mathrm{alg}}$-semialgebraic set $S\in\mathbb R^k$, and $q\in\mathbb R_{\mathrm{alg}}(x)$). But it is currently not known whether $\gamma$ is a KZ-period. By the way, there are strong suspects that it isn't (in particular, it seems that there is no BBP-type formula for $\gamma$)
