oo n 5 ==== (n + a) (11 n + 6 a + 1) 4 (n + a - -)! (n + 2 a - 2)! (2 n + a - 1)! \ 6
---------------------------------------------------------------------- / 2 ==== (n + a - -)! (3 n + 2 a + 1)! n = 0 3
1 3 3 (-)! (a - -)! 3 2 = ---------------, 2/3 4
oo ==== 2 2 \ 3 k + (3 a + 1) k + a > --------------------------------- = 1, / 2 k + a + 1 ==== (k + a) (k + a + 1) ( ) k = 0 k
It is interesting that Maple can do the second one, but for the first one gives the following answer,
sum((n + a)*(11*n + 6*a + 1)*4^n*(n + a - 5/6)!*(n + 2*a - 2)!*(2*n + a - 1)!/(n + a - 2/3)!/(3*n + 2*a + 1)!,n=0..infinity);
/ 16 7/360 (1/6)! |60 hypergeom([1/2, 1, 1, 7/6], [4/3, 4/3, 5/3], --) \ 27 16 \ + 11 hypergeom([3/2, 2, 2, 13/6], [7/3, 7/3, 8/3], --)| 27 / / 1/2 GAMMA(a - 1/2) / (Pi (1/3)!) / Comparing two answers, we get the following identity, 60*hypergeom([1/2, 1, 1, 7/6],[4/3, 4/3, 5/3],16/27)+ 11*hypergeom([3/2, 2, 2, 13/6],[7/3, 7/3, 8/3],16/27)=120*Pi*sqrt(3)/7 Alec Mihailovs http://math.tntech.edu/alec/