! I believe Fred refers to gosper.org/facfun.pdf
Er ... uh ... yeah, right! [ Suspicious palimpsest on the final page --- is there something we haven't been told? ] WFL On 4/5/16, Bill Gosper <billgosper@gmail.com> wrote:
On 2016-04-01 11:12, Fred Lunnon wrote:
Hogamous, higamous, Plouffe is polygammous. Higamous, hogamous, Gosper monogammous.
WFL
! I believe Fred refers to gosper.org/facfun.pdf But I occasionally succumb to infidelity: gosper.org/filds.pdf --rwg
On 4/1/16, Simon Plouffe <simon.plouffe@gmail.com> wrote:
Hello,
the Psi function is one of my favorite function, it has so many aspects and formulas, very interesting,
for example, you can express primes with it,
691 = Psi(11,1/2)/(Pi^12*2^8).
or Euler numbers, (absolute value), I hate that - sign all over. ;-).
E(2n) = -1/2*(Psi(k,1/8)-Psi(k,3/8)+Psi(k,5/8)-Psi(k,7/8))*Pi^(-k-1)*4^(-k).
we can also use Psi(n,1/4) and Psi(n,3/4) for the same.
another example is 67 =
1/64*(Psi(5,1/5)-Psi(5,2/5)-Psi(5,3/5)+Psi(5,4/5))*5^(1/2)/Pi^6
a very good question is : are there other primes like that ?
I have such a formula for 17, 31, 41, 61, 67, ...this type of numbers include prime euler numbers and (with some arrangement with the denominator) prime bernoulli numbers as well.
Another direction with that idea , a fact known to Ramanujan was that (for example)
24 * sum(n^673/(exp(2*Pi*n)-1),n=1..infinity) = 156344681616153723364... 465357092320036059151 a prime of 1077 digits, that number is in fact close to a Bernoulli number in disguise, B(674) we can express that infinite series with the polygamma function if we like instead.
also a fact which I find quite interesting, we have here an expression for some primes which is made with a sum of irrational numbers.
other representations are with binomial sums which are simplified if we use the polygamma function with a negative argument, , integer relations with known constants.
you may look at this document for more formulas here :
http://plouffe.fr/The%20many%20faces%20of%20the%20polygamma%20function.pdf
one interesting finding is that I have a polygamma expression for Li1(1/2), Li2(1/2), Li3(1/2), which are <the same> in term of polygamma function, i could not find yet an expression for Li4(1/2), but I am working on it since the first 3 values are very similar. see the formulas on page 3.
all formulas are with that same polygamma function, and yes : one last point, the one I use is the one implemented in Maple and not mathematica mainly because the mathematica program uses the wrong (in my opinion) generalization of polygamma with negative arguments, the correct one is the one of Espinosa and Moll. (see the references).
bonne lecture,
Simon plouffe
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