arXiv subjects
Publications and source records attributed to Jaume Oliver Lafont.
Let $α=(1+\sqrt 5)/2$, the golden ratio, and $β=-1/α=(1 - \sqrt 5)/2$. Let $F_n$ and $L_n$ be the Fibonacci and Lucas numbers, defined by $F_n=(α^n -β^n)/\sqrt 5$ and $L_n=α^n + β^n$, for all non-negative integers. We derive base~$α$ expansions of $\log F_n$, $\log L_n$, $\arctan\dfrac1{F_n}$ and $\arctan\dfrac1{L_n}$ for all positive integers $n$.
BBP-type formulas are usually discovered experimentally, one at a time and in specific bases, through computer searches. In this paper, however, we derive directly, without doing any searches, explicit digit extraction BBP-type formulas in general binary bases $b=2^{12p}$, for $p$ positive odd integers. As particular examples, new binary formulas are presented for $π\sqrt 3$, $π\sqrt 3\log 2$, $\sqrt 3\;{\rm Cl}_2(π/3)$ and a couple of other polylogarithm constants. A variant of the formula for $π\sqrt 3\log 2$ derived in this paper has been known for over ten years but was hitherto unproved. Binary BBP-type formulas for the logarithms of an infinite set of primes and binary BBP-type representations for the arctangents of an infinite set of rational numbers are also presented. Finally, new binary BBP-type zero relations are established.