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Boris Gourévitch
The world of Pi - V2.57
modif. 13/04/2013



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The hypergeometric series, harmonic and Pi

Boris Gourevitch

2 february 2006
Content Page
1 A small quick historical recap
2 Introduction to hypergeometric series
 2.1 Definition
 2.2 A few properties and simple examples
3 The Psi function: Basis of formlae of the kind Machin or BBP
 3.1 Definition
 3.2 Differential equation
4 Formulae of kind Machin
5 Formulae BBP : The technic
 5.1 s = 1,  v = 0  : The polylogarithms
 5.2 Link between integrals, BBP formulae and polylogarithms
 5.3 Integrals and BBP formulae
 5.4 Function Y  and polylogarithms
 5.5 Interest in BBP formulae
6 BBP formulae in base 2 : s  (-  N  ,    p
v = q  , x = 21n  in Y
 6.1 The considered integrals
 6.2 The method
 6.3 Formulae  for p  , ln(2)  , ln(3)  and ln(5)
 6.4 Case of polylogarithms of order 2  : Formulae of order 2
 6.5 Case of polylogarithms of order 3
 6.6 Case of polylogarithms of order 4
 6.7 Case of polylogarithms of order 5
7 s  fixed integers,    p
v = q  , x = 31n  : BBP in base 3
 7.1 Formulae for   V~ -
p  3
 7.2 Formulae of order 2 : Integrals with ln(y)
 7.3 Formulae of order 3 : Integrals with ln2(y)
8 And the other bases then ? ?
9 Polygamma and Clausen
 9.1 Polygamma functions
 9.2 The function digamma
 9.3 Polygamma of order m > 1
 9.4 Combinations of Kölbig
10 Introduction to factorials and combinations
 10.1 A first example
 10.2 Umbral calculus
11 Central binomial series
 11.1 Inversion of combinations
 11.2 Useful deveoppements
 11.3 First direct formulae
 11.4 Formulae of greater order
12 Other binomial coefficients
 12.1 Primitive formulae
 12.2 Polynomiales factorial formulae
 12.3 Then, what does this mean ?
 12.4 Proof
 12.5 Fast combinations: factorial BBP formulae
 12.6 Then, what does this mean ?
 12.7 Typical proof
 12.8 Product of combinations
13 Harmonic series
 13.1 Proximity of harmonic series and of polylogarithms
 13.2 Study of  k      + sum  oo  Hkn n
fp(x) = n=1 np x  and of  k     + oo  sum   Hkn   n+1
gp(x) = n=1 (n+1)px
 13.3 Applications of the calculations of certain series
 13.4 Generalisations
14 Series of greater factorial : Ramanujan, Borwein, Chudnovsky....
 14.1 Squared central binomial coefficients : Elliptical formulae
 14.2 Cubed central binomial coefficients: Ramanujan's identity
15 Other hypergeometric formulae concerning p
References


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