Oops, for F(1/3), F(1/9), etc., you'll also need 1/(F(Y)^9*F(9*Y)^3)+9*E^-(PI*Y)/(F(Y)^6*F(9*Y)^6)+27*E^-(2*PI*Y)/(F(Y)^3*F(9*Y)^9) = 1/F(3*Y)^12 and 1/(F(Y)^72*F(3*Y)^12) = -12*E^-(PI*Y)/(F(Y)^48*F(2*Y)^24*F(3*Y)^12)-196608*E^-(2*PI*Y)/(F(Y)^24*F(2*Y)^48*F(3*Y)^12)-16777216*E^-(3*PI*Y)/(F(2*Y)^72*F(3*Y)^12)+196830*E^-(2*PI*Y)/(F(Y)^36*F(2*Y)^24*F(3*Y)^24)+19131876*E^-(3*PI*Y)/(F(Y)^24*F(2*Y)^24*F(3*Y)^36)+387420489*E^-(4*PI*Y)/(F(Y)^12*F(2*Y)^24*F(3*Y)^48)+1/(F(Y)^60*F(2*Y)^24) , --rwg On Wed, Feb 2, 2011 at 3:13 PM, Bill Gosper <billgosper@gmail.com> wrote:
Simon, all you need is Jacobi's imaginary transformation F(y) = F(4/y)*e^(pi*(1/y-y/4)/6)*sqrt(y)/sqrt(2)
or his aequatio satis abstrusa 1/F(2*y)^24 = 1/(F(y)^16*F(4*y)^8)+16*e^-(pi*y)/(F(y)^8*F(4*y)^16)
to get F(1/2) from the results you quote. These, with 1/(F(y)^18*F(2*y)^8) = 256*e^-(pi*y)/(F(2*y)^24*F(5*y)^2)+1/(F(y)^24*F(5*y)^2)-250*e^-(pi*y)/(F(y)^12*F(2*y)^8*F(5*y)^6)-3125*e^-(2*pi*y)/(F(y)^6*F(2*y)^8*F(5*y)^12)
and maybe
1/F(y)^6 = 1/(F(y/5)^5*F(5*y))+5*e^-(pi*y/5)/(F(y/5)^4*F(5*y)^2)+15*e^-(2*pi*y/5)/(F(y/5)^3*F(5*y)^3)+25*e^-(3*pi*y/5)/(F(y/5)^2*F(5*y)^4)+25*e^-(4*pi*y/5)/(F(y/5)*F(5*y)^5)
should give you exact values for all the rest. --rwg
Hello,
I am collecting these approximations, I have some which are of interest ,
let's say that F is the partition function of Euler,
infinity --------' ' | | 1 F(x) = | | ------ | | n
| | 1 - x n = 1 now I put x-> exp(-Pi*x),
then I have this EXACT formula (well tested to 1000 digits).
5 20 28
F(1/5) F(4/5) F(2) ---------------- = ------------ 5 31 7 F(2/5) F(1) F(4)
Now here is the funny thing,
4 2 F(1/12) F(1/36) ----------------- = exp(Pi) 2 4 F(1/9) F(1/24)
This formula is valid up to 48 decimal digits.
From the theory of partitions and other theories
about these kind of functions, the values of F(1), F(2) and F(4) are well known (sort of).
Here are the EXACT known values :
3/8 2 GAMMA(3/4) --------------- = F(1)
Pi 1/4 exp(----) Pi 24
1/2 2 GAMMA(3/4) --------------- = F(2)
Pi 1/4 exp(----) Pi 12
7/8 2 GAMMA(3/4) --------------- = F(4)
Pi 1/4 exp(----) Pi 6 But on the other hand for 1/2 :
16 F(1/2) = 1/65536 exp(5 Pi)
is an approximation to 6 decimals.
Interesting isn't ?
I made a search for simpler formulas, I could not, these are the simplest terms I know.
Bonne journée, Simon Plouffe