A BBP-type formula for an irrational number $\alpha$ in the integer base $b\geq 2$ is a formula in the form $\alpha=\Sigma_{k=0}^{\infty}\frac{1}{b^k}\frac{p(k)}{q(k)}$ ($p, q$ are polynomials in integer coefficients) that allows computing the $n$th digit of $\alpha$ in the basis $b$ directly and without any need to computing the preceding $n-1$ digits of $\alpha$. It is also connected to the problem of normality of irrational numbers.

1- For which irrational numbers do we have a known Bailey–Borwein–Plouffe formula? For which of them do we have such a formula in the base $10$? (Please add a reference to a list if there is any).

2- Is there any known BBP formula for $e$ in an integer base $b$?

3- For which irrational numbers, the normality of the number in a given base is proved using a BBP formula?


Concerning (1), there is a large compendium of BBP formulas in this paper by Bailey. Regarding (2), it is strongly suspected (but not proved) that there is no BBP formula for $e$. Here is a quote from the paper by Bailey.

Powers of $e$ are specified here because, as far as anyone can tell (although this has not been rigorously proven), $e$ is not a polylogarithmic constant in the sense of this paper, and thus it and its powers are not expected to satisfy BBP-type linear relations (this assumption is confirmed by extensive experience using the author’s PSLQ programs).


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