Miscellaneous mathematics / personal and anonymous formulas

Reference of the On-Line Encyclopedia of Integer Sequences for Pi digits: A000796

Plenty of formulas !

This is the odds and ends of mathematical formulas !
Actually, I gathered here the stuff that was difficult to put somewhere else. Some personal stuff at first, often related to Plouffe formulas. Then, some formulas of unknown origin, which use very interesting methods though. Finally, I present some useful methods for computation of pi digits.

• Hypergeometric series, harmonic series and Pi
  A HUGE page trying to summarize averything that has happened in the field of BBP formulas, polylogarithms, polygamma, binomial serie, harmonic series for six years. Obviously, this quest is endless, but it is kind of an introduction, say... I'm grateful to Gery Huvent and Jesus Guillera for their relevant help and comments, given that roughly half of these pages were written by them. Notice that the following pages stem from a 118 pages LaTeX document automatically translated by the great software TeX4HT of Eitan Gurari (thanks to him and P. Viton for their help and support in some desperation times...). Despite the power of TeX4HT, the result is not perfect and I'm not a specialist of the settings of this soft. Sorry, guys ! Especially Netscape owners will have dirty pages because of their processing of the style sheets, contrary to internet Explorer (so, now Microsoft is more rigourous, incredible...). Yet, I hope these pages will entitle you to send me comments, critics and suggestions !
  More than 40 pages in a row ! Here is the summary, which is detailed in the home page then the table of contents.
  
  1. Brief historical overview
  2. Introduction to hypergeometric series
  3. Psi function : a basis for BBP-like of Machin-like formulas
  4. Machin-like formulas
  5. BBP formulas: basics
  6. BBP formulas in base 2
  7. BBP formulas in base 3
  8. So what about the other bases ?
  9. Polygamma and Clausen
  10. Introducing factorials and binomial coefficients
  11. Central binomial series
  12. Others binomial coefficients
  13. Harmonic series
  14. Higher orders for factorials : Ramanujan, Borwein, Chudnovsky....
  15. Others hypergeometric formulas related to Pi

• Pi and randomness
  Several things to discuss and three pages

1. How Pi appears in the random phenomenon
   Probabilistic theorems, Brownian Motion...
2. Pi and the definitions of randomness theory
   Normal numbers, random sequences of Martin-Löf, computational complexity theory, BBP formulas...
3. Statistical properties of Pi digits
   Fractal dimension, tests on digits, Khintchine constant...
   
   
• Some personal ideas of formulas for Pi
  BBP formulas...
  
  
• Chebychev polynomials
  a very personal idea to generate some BBP-like formulas with the Chebychev polynomials !
  
  
• Spigot algorithm
  A recent and original method to compute the digits of Pi by hand, one after the other
  
  
• My old page Attic
  A potpourri of apparitions of Pi in some mathematical problems. Hope I'll renovate this page one day...
  
  
• C(n,k) Triangle
  A nice study about an integer triangle (like Pascal triangle) and which gives an approximation of Pi, of course ! This page definitely deserved to be taken out of the attic !
  
  
• Arithmetic functions and Pi
  Gauss, Euler and Moëbius functions, and several relations with Zeta function, for instance
  
  
• e and Pi in a mirror
  A beautiful brainwave of Benoit Cloitre about the similarity between two algorithms converging towards these two constants

• Bernoulli's numbers
  The definition used in this website

Visit the page about mathematicians to find pages related to these tools, or follow the links provided in the above pages