Following the first findings that were
inspired from
Ramanujan Notebooks and other articles,
I followed the same idea that the Zeta function (as well as the
Psi and Gamma functions) have nice properties. With the help of
MapleV, Pari-Gp and my Inverter ,I have
found those (new?) identities.
Here, Psi(k,x) is the function is the k'th
logarithmic derivative of the Gamma function at x. Some closed
expressions are known for some values of the Psi(k,x) function but
in general Psi(k,1/3) with k>0 is of unknown nature compared to
known constants.
The first identity was not very difficult to
find using my Inverter.
Following the same vein of pattern the second
identity follows and then the third. I could not establish the
same for Zeta(9) even with 256 digits of precision.
The next step is to find a closed expresion
for the alternate sums. We know from Apéry's 1979 article
that it can be expressed in terms of Zeta(3). If we omit the
constant factor for short, we have :
But unfortunately, there are no apparent
patterns for Zeta(5), Zeta(7), etc. despite my numerous efforts
for those sums and previous tests of D. H. Bailey and others are
equally inconclusive so far. The mystery remains until someone
finds the clue to all this.
An indication for further discoveries in that
direction might be given with these remarks:
There are many representations of the same
thing, for example,
The sum is a linear combination of Psi(p/q)
and if we simplify we find that
That is, a simple linear combination of
log(2)*sqrt(5) and log(3+sqrt(5))*sqrt(5). Which is in fact this
number,
This identity is well known and not difficult
to find. So, the case n^5 should be simple and in terms of sqrt(5)
and Psi(3,1/5) ?. This is strange since the other formulas were
with Pi*sqrt(3) only. It could be also that the sqrt(5) factor is
an artefact. Normaly the sum for n^5 should be expressible with
Psi(4,k/5) , k=1..4 and/or Psi(4,1/3)*Pi*sqrt(3) but numerical
evidence shows that it is NOT the case.
References
-Apéry, R. ``Irrationalité de
Zeta(2) et Zeta(3), Astérisque 61, 11-13, 1979.
-Berndt, B. C. Ramanujan's Notebooks: Part
I-IV. New York: Springer-Verlag, 1985 to now.
-van der Poorten, A. ``A Proof that Euler
Missed... Apéry's Proof of the Irrationality of .'' Math.
Intel. 1, 196-203, 1979.
-Zeilberger, D. ``The Method of Creative
Telescoping.'' J. Symb. Comput. 11, 195-204, 1991.