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$$P(s,b,n,A)=\sum_{k=0}^\infty \frac1{b^k}\sum_{j=1}^n \frac{a_j}{(nk+j)^s}$$

I would like to understand how this generic type of BBP formula relates to the famous BBP formula:

$$\pi=\sum_{k=0}^\infty \frac1{16^k}\left[\frac4{8k+1}-\frac2{8k+4}-\frac1{8k+5}-\frac1{8k+6}\right]$$

I only understand that $b$ is base $= 16$. The variables I'm not sure about are $n = 8$ and $s = 1$. $P(s,b,n,A) = P(1,16,8,A)$

I don't know what the $P$,$A$,$a$ and $j$ variables stand for or if I got $s$ and $n$ correct.

Based on this particular math paper (note they substitute the letter $l$ for $n$ in that paper), they say that $A$ is a vector of integers. I'm not really sure what that means but my guess is that $A$ in this case is four numbers: $4,-2,-1,-1$.

I would appreciate if someone could clarify or explain the meaning of these variables for me and what it is that I haven't understood correctly.

J. M. ain't a mathematician
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Neil
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1 Answers1

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This generic formula is a compact Bailey’s $P$-notation

$$P(\color{green}s,b,\color{red}n,A)=\sum_{k=0}^\infty \frac1{\color{blue}b^k}\sum_{j=1}^{\color{red}n} \frac{a_j}{(\color{red}nk+j)^\color{green}s}$$

Where $\color{green}s$, $\color{blue}b$ (base) and $\color{red}n$ are integers and $A=(a_1,a_2,...a_{\color{red}n})$ is a vector of integers.

using this notation we can write " famous BBP formula" more compactly as follows :

$$\pi=\sum_{k=0}^\infty \frac1{\color{blue}{16}^k}\bigg(\frac4{(\color{red}8k+1)^\color{green}1}-\frac2{(\color{red}8k+4)^\color{green}1}-\frac1{(\color{red}8k+5)^\color{green}1}-\frac1{(\color{red}8k+6)^\color{green}1}\bigg)= P(\color{green}1,\color{blue}{16},\color{red}8,(4,0,0,−2,−1,−1,0,0))$$

For instance,

$$\log3=\sum_{k=0}^{\infty}\frac{1}{\color{blue}4^k}\bigg(\frac{1}{(\color{red}2k+1)^\color{green}1}\bigg)= \frac{1}{2}P(\color{green}1,\color{blue}{4},\color{red}2,(1,0))$$

$$G=\frac{1}{2^{10}}P(\color{green}2,\color{blue}{212},\color{red}{24}, (210,210,−29,−3·210,−256,−211,−256,−9·27,−5·26,64,64,0,−16,64,8,−72,4,−8,4,−12,5, 4,−1,0))$$ where $G$ is the Catalan's Constant

How to use the BBP formula see here.

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