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



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The Psi function: The basis to formulae of type Machin or BBP

3.1 Definition

We want to be able to combine in a p+1Fp  serie the combination and the terms (an+ b) = a (n + b)
               a . For this we will introduce the Psi function, or Lerch's transcendance, which is written

           sum  oo   xn
Y(x,s,v) =   (n-+v)s
          n=0
(16)

for v  (-  R,s  (-  N  . The convergence radius is of 1  . We can write this function in a fairly simple hypergeometric function :

               (        s fois          )
                      v,v,...,v,1
Y(x,s,v) =s+1 Fs  v+ 1,v+ 1,...,v + 1 ,x
                   ------- -------
                        s fois
(17)


This function has the advantage of regrouping a good part of classical functions of analysis, it's not that suprising since the relations between them are many!

So, we get

pict

where z(s,v) =  sum o o --1--
         n=0 (n+v)s  is the Zêta function of Hurwitz, y  (v) = (-1)m+1(m!)z(m + 1,v)
 m  is the PolyGamma function and         '
Y(v) = GG((vv))  is the Digamma function.

z(s) =  sum o o  1
        n=1 ns  is the famous Zêta function, L (x) =  sum o o  xn
 s       n=1 ns  is the polylogarithm of order s, b(s) =  sum o o -(-1)n
        n=0(2n+1)s  is the Bêta function of Dirichlet, and finaly         sum o o  cos(nx)
Cs(x) =   n=1--ns--  and         sum o o  sin(nx)
Ss(x) =  n=1--ns-  are Clausen's functions.

3.2 Differential Equations

The function Y  is a hypergeometric serie, so we can fire at it any kind of differential equation. Among all of those, let's take :

pict

but also by iterating

  --------s fois--------
 (  d    )   (  d    )
x x dx-+v  ... x dx + v f(x) = 0
(26)


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