Here's a peculiarity--an identically zero 12F11[823543/843750 = 7^7/2/3^3/5^6] with quintic surds of a parameter: Sum[((7348*k^5 + (-20040*a - 19360)* k^4 + (-20240*a^2 - 3300*a + 6730)*k^3 + (-6000*a^3 + 5060*a^2 + 4920*a + 245)* k^2 + (-680*a^4 + 1580*a^3 + 650*a^2 - 275*a -108)*k - 32*a^5 + 80*a^4 - 40*a^3 - 20*a^2 +12*a)*(7*k + 2*a - 4)!)/ (31250^k*(k - 3/5)!*(k - 2/5)!*k!*(k + a)!*(3*k + a + 1/2)!), {k, 0, Infinity}] == 0 Why always 7^7 and not some lesser power? --rwg On Sat, Jan 28, 2012 at 3:20 PM, Bill Gosper <billgosper@gmail.com> wrote:
Sum[((j + 1)*(22781*j^4 + 90037*j^3 + 130738*j^2 + 82472*j + 19032)*(-1)^j*(2*j)!*(5*j + 3)!)/ (3^j*(7*j + 9)!), {j, 0, Infinity}] == Pi/Sqrt[3] - 3/2
Woohoo, 11F10[-2^2 5^5/3/7^7] . --rwg (With Z.H. help.)
On Fri, Jan 27, 2012 at 7:42 PM, Bill Gosper <billgosper@gmail.com> wrote:
Here's a much simpler (and rapidly convergent) 14F13[-432/7^7]:
Sum[((-1)^k*2^(-1 - 4*k)*f[k]*(4*k)!*(-2/3 + 3*k)!)/(19/3 + 7*k)!, {k, 0,Infinity}] == (Sqrt[3]*Pi + 3*Log[2])/494385
with f[k] -> k^6 + ((1598477814*k^5 + 2136105918*k^4 + 1462457619*k^3 + 538646562*k^2 + 100644207*k + 7398460)/480542220)
I actually have it with two parameters = 3F2[1,a,a+1/2;b,b+1/2]. --rwg
On Wed, Jan 18, 2012 at 1:46 AM, Bill Gosper <billgosper@gmail.com>wrote:
There are endlessly many identities of the form
pFp-1[rational] = pile of Gammas(rationals),
but, as Dick Askey pointed out to me decades ago, the rationals all seem to be made out of primes < 7. I've always known how to construct one with 7s, but the hardware (and my stomach) were never up to it. Just to get it over with, Sum[(((j^13 + ((946722645184512*j^12 + 3397234214391808*j^11 + 7324312847932416*j^10 + 10575837135607296*j^9 + 10789977229862400*j^8 + 7994097957085696*j^7 + 4349780625735744*j^6 + 1736115415231328*j^5 + 501325200097728*j^4 + 101649833417016*j^3 + 13682157776100*j^2 + 1093720984200*j + 39131717625)/ 119811560185856))*(-1)^j*((j - 1/2)!)^2*(6*j + 1/2)!*(7*j + 1/2)!)/((j + 1)!* (7*j + 6)!*(7*j + 7)!)), {j, 0, Infinity}] == Pi/14625434593
This is thirteen or so 16F15[-6^6/7^7]s. The XP Macsyma (compiled for a 286 in NT) that disgorged this was so effete that afterward, it claimed 2F1[1/2,1/2;2] = 0, and when asked for the pFp-1s for the above, said Error: FACTOR ran out of primes. --rwg