more hype : 2F1[1, a , 1/2, 1/4]

[Wouter Meeussen]

2F1[1, a , 1/2, 1/4] doesn't evaluate with FullSimplify (Mma 4.0.1)
but written as a sum from 1 to oo :
1+ Sum[Pochhammer[a,k]/4^k/Pochhammer[1/2,k], {k,1,\[Infinity]}],{a,1,24}]
does break down :


 4/3 + (2*Pi)/(9*Sqrt[3]),
 2 + (4*Pi)/(9*Sqrt[3]),
 3 + (20*Pi)/(27*Sqrt[3]), 
 40/9 + (280*Pi)/(243*Sqrt[3]),
 13/2 + (140*Pi)/(81*Sqrt[3]), 
 47/5 + (616*Pi)/(243*Sqrt[3]),
 202/15 + (8008*Pi)/(2187*Sqrt[3]), 
 134/7 + (11440*Pi)/(2187*Sqrt[3]),
 757/28 + (48620*Pi)/(6561*Sqrt[3]), 
 1595/42 + (1847560*Pi)/(177147*Sqrt[3]), 
 531/10 + (2586584*Pi)/(177147*Sqrt[3]), 
 12203/165 + (10816624*Pi)/(531441*Sqrt[3]), 
 30491/297 + (135207800*Pi)/(4782969*Sqrt[3]), 
 20320/143 + (20801200*Pi)/(531441*Sqrt[3]), 
 196379/1001 + (86176400*Pi)/(1594323*Sqrt[3]), 
 4057832/15015 + (1068587360*Pi)/(14348907*Sqrt[3]), 
 1352459/3640 + (1469307620*Pi)/(14348907*Sqrt[3]), 
 450785/884 + (6050090200*Pi)/(43046721*Sqrt[3]), 
 1853129/2652 + (223853337400*Pi)/(1162261467*Sqrt[3]), 
 617687/646 + (306325619600*Pi)/(1162261467*Sqrt[3]), 
 8441507/6460 + (1255935040360*Pi)/(3486784401*Sqrt[3]), 
 362978341/203490 + (15430059067280*Pi)/(31381059609*Sqrt[3]), 
 120991273/49742 + (7013663212400*Pi)/(10460353203*Sqrt[3]), 
 1895513363/572033 + (28664536607200*Pi)/(31381059609*Sqrt[3])

again, no general form for these coefficients...

a propos, one more term for
 Hypergeometric2F1[10, 11, 19/2, 1/4] == (2722720*Pi)/(177147*Sqrt[3])

wouter.


Dr. Wouter L. J. MEEUSSEN
w.meeussen.vdmcc at vandemoortele.be
eu000949 at pophost.eunet.be