[seqfan] Re: formula for Pi

[Simon Plouffe]

Hello,

  sorry for the delayed answer,

  first of all those series are sums of psi or digamma
functions with rational arguments, this is well known
and with the use of the GAUSS formula you can get
the general form of almost any Psi(a/b) when a/b are
rationals. In other words, whenever you have a sum
wihch can be craked into partial fractions then
you can have the direct and explicit sum for it.

The reference about that trick is the classic Abramowitz
and Stegun, see the gamma function chapter at the end,
there is a section on how to sum rational series of
degree 2,3,4... with the help of the psi, digamma or
trigamma function.

For the gauss formula, you can search the mathworld
site for digamma gauss formula or Psi formula of Gauss,
there is an explicit formula.

Now about BBP types of series, this is another story
since BBP type of series are geometric and polynomial
at the same time : we call it polylog series, they are
DIFFERENT from the one you have found.

For the Gauss formula : here is a Maple program :

DI := proc(s)
local un, de, j, p, q;
     p := numer(s);
     q := denom(s);
     un := -gamma - 1/2*Pi*cot(p*Pi/q) - ln(q);
     de := sum(cos(2*Pi*j*p/q)*ln(2*sin(j*Pi/q)), j = 1 .. q - 1);
     RETURN(un + de)
end:


this will calculate explicitely any Psi(a/b), just input
DI(3/5) and it will give you the explicit value of Psi(3/5).



For example Psi(3/5) =

/         1/2       1/2 1/2      /         1/2\
|  gamma 2    (5 + 5   )         |        5   |
|- ------------------------ + Pi |- 1/4 + ----|
\             2                  \         4  /

                   1/2       1/2 1/2
      - 1/2 ln(5) 2    (5 + 5   )

        /       1/2\
        |      5   |  1/2       1/2 1/2
      - |1/4 + ----| 2    (5 + 5   )    ln(2) -
        \       4  /

     /       1/2\
     |      5   |  1/2       1/2 1/2
     |1/4 + ----| 2    (5 + 5   )
     \       4  /

         1/2       1/2 1/2
        2    (5 - 5   )
     ln(------------------)
                4

        /         1/2\
        |        5   |  1/2       1/2 1/2
      + |- 1/4 + ----| 2    (5 + 5   )    ln(2) +
        \         4  /

     /         1/2\
     |        5   |  1/2       1/2 1/2
     |- 1/4 + ----| 2    (5 + 5   )
     \         4  /

         1/2       1/2 1/2 \
        2    (5 + 5   )    |  1/2   /       1/2 1/2
     ln(------------------)| 2     /  (5 + 5   )
                4          /      /

I used the trick of converting the cos(2*Pi/5) into
radicals for it...

This was obtained with the function,
now as you may know if you try Psi(3/19) it won't
give you the explicit formula in radicals and logs
of course because sin(Pi/19) has no radical expression,
this is also a result of Gauss by the way, but you
my know that already ?

I wish this is of some help,


simon plouffe


-- 
Simon Plouffe
06 63 17 47 73
0970 460 350
Château Bois Briand
10 rue Bois Briand
44300 Nantes