rcs> Simon Plouffe was one of the discoverers of the nice pi formula
inf 4 2 1 1 -k pi = sum (---- - ---- - ---- - ----) 16 k=0 8k+1 8k+4 8k+5 8k+6
which allows computing individual bits of pi without computing the preceding part.
It seems to me that there should be another such formula, in which the mod 8 residues are rearranged a little. Perhaps with 8k+3 and 8k+7 replacing 1 & 5, or maybe just 1&3 instead of 1&5, or with 8k+2 in place of 8k+6.
Has anyone seen variations like this?
rwg> You don't need lattice numerology for these. Here are the closed forms--
rub them together symbolically:
inf ==== \ 1 sqrt(2)
------------- = (sqrt(2) log(sqrt(2) + 1) + 2 sqrt(2) atan(-------) / k 2 ==== (8 k + 1) 16 k = 0 [...]
I believe it was discussed here that Simon (et al?) later discovered that six of these series were bisections of inf ==== \ 1 log(5) + 2 atan(2) > ---------------- = ------------------, / k 4 ==== (4 k + 1) (- 4) k = 0 inf ==== \ 1 > ---------------- = acot(2), and / k ==== (4 k + 2) (- 4) k = 0 inf ==== \ 1 log(5) - 2 atan(2) > ---------------- = - ------------------, / k 2 ==== (4 k + 3) (- 4) k = 0 which are all you need. Also, apropos the 1st and 7th eqns in his inspired3.pdf, sum(n/(%e^(%pi*n)-1),n,1,inf) = gamma(1/4)^4/(64*%pi^3)-1/(4*%pi)+1/24 4 inf 1 ==== 4 (-)! \ n 1 4 1 > ---------- = - ---- + ------- + -- / n pi 4 pi 3 24 ==== %e - 1 pi n = 1 sum(n/(%e^(2*%pi*n)-1),n,1,inf) = 1/24-1/(8*%pi) inf ==== \ n 1 1 > ------------ = -- - ---- / 2 n pi 24 8 pi ==== %e - 1 n = 1 sum(n/(%e^(4*%pi*n)-1),n,1,inf) = -gamma(1/4)^4/(256*%pi^3)-1/(16*%pi)+1/24 4 inf 1 ==== (-)! \ n 1 4 1 > ------------ = - ----- - ----- + -- / 4 n pi 16 pi 3 24 ==== %e - 1 pi n = 1 and the middle of these is 1/8 of his 1st minus 1/24 of his 7th. His 1st can also be written 1/%pi = -16*sum(n/(%e^(2*%pi*n)+1),n,1,inf)+12*sum(n/(%e^(%pi*n)+1),n,1,inf)-4*sum(n/(%e^(%pi*n)-1),n,1,inf) inf inf inf ==== ==== ==== 1 \ n \ n \ n -- = - 16 > ----------- + 12 > --------- - 4 > --------- pi / 2 n pi / n pi / n pi ==== e + 1 ==== e + 1 ==== e - 1 n = 1 n = 1 n = 1 Also, in my old q-trig paper are a bunch of relations, for general q, between etas and logderivative(etas), e.g. sum((2*n-1)*q^(2*n-1)/(1-q^(2*n-1)),n,1,inf) = 2*eta(q^4)^8/(3*eta(q^2)^4)+(eta(q^2)^20/(eta(q)^8*eta(q^4)^8)-1)/24 20 2 eta (q ) inf ---------------- - 1 ==== 2 n - 1 8 4 8 8 4 \ (2 n - 1) q 2 eta (q ) eta (q) eta (q )
------------------ = ---------- + -------------------- / 2 n - 1 4 2 24 ==== 1 - q 3 eta (q ) n = 1
sum(n*q^n/(1-q^n),n,1,inf)-3*sum(n*q^(3*n)/(1-q^(3*n)),n,1,inf) = eta(q)^2*eta(q^3)^2*(3*eta(q^6)^8/eta(q^3)^4+eta(q^2)^8/eta(q)^4)^2/(12*eta(q^2)^4*eta(q^6)^4)-1/12 8 6 8 2 2 2 3 3 eta (q ) eta (q ) 2 inf inf eta (q) eta (q ) (---------- + --------) ==== n ==== 3 n 4 3 4 \ n q \ n q eta (q ) eta (q) 1
------ - 3 > -------- = ----------------------------------------- - -- / n / 3 n 4 2 4 6 12 ==== 1 - q ==== 1 - q 12 eta (q ) eta (q ) n = 1 n = 1
You probably don't recall, but the q=e^-pi case of the trisectands of this last series are improbably complicated. --rwg PS, I fixed the algebra drill color collisions in http://www.tweedledum.com/rwg/squares.htm . RISTOCETIN TRISECTION PREDYNASTIC CANDYSTRIPE SHATTERABLE BLAST-HEATER BARDOLATRIES LABRADORITES