While seeking to accelerate the zeta(2) series, I specialized a very general path-invariant system of 3x3 matrices to a very special case of Dixon's theorem. Out[75]= {{k, {{(1/2 + k)^2/((k + 1/2 (1 - 2 n)) (k + 1/2 (1 + 2 n))), 0, -((n^2 (-1 + 2 n))/( 4 k (k + 1/2 (1 - 2 n))))}, {0, (1/2 + k)^2/((k + 1/2 (1 - 2 n)) (k + 1/2 (1 + 2 n))), -(n^2/( 2 (1/2 + k) (k + 1/2 (1 - 2 n))))}, {0, 0, 1}}}, {n, {{( n^2 (1/2 (1 - 2 k) + n))/((1 + n)^2 (1/2 (1 + 2 k) + n)), 0, n}, {0, (n^2 (1/2 (1 - 2 k) + n))/((1 + n)^2 (1/2 (1 + 2 k) + n)), 1}, {0, 0, 1}}}} (Strictly speaking, Dixon's 3F2 doesn't cover zeta(2), but unlike pFq, matrix products are not required to sum from n=0.) When your 3x3 matrices all become triangular, as above, you can separate the system into two systems of 2x2s by deleting one or the other of the top two rows, and the corresponding column from each matrix in the system. Corey's path-invariant matrix package contains the indispensabie function sidestep, which takes a matrix and "conjugates" it with all the matrices in the current system in such a way as to preserve path invariance. Merely by replacing Inverse with PseudoInverse, sidestep accepts nonsquare matrices and performs the dimension reduction! To delete the top rows and left columns, In[76]:= sidestep[{{0, 0}, {1, 0}, {0, 1}}] Out[76]= {{k, {{(1 + 2 k)^2/( 1 + 4 k + 4 k^2 - 4 n^2), -(( 2 n^2)/((1 + 2 k) (1 + 2 k - 2 n)))}, {0, 1}}}, {n, {{(n^2 (1 - 2 k + 2 n))/((1 + n)^2 (1 + 2 k + 2 n)), 1}, {0, 1}}}} Path invariance check: In[77]:= picheck[%] Out[77]= True A triangular contour in this 2x2 system leads immediately to Out[350]= 10 HypergeometricPFQ[{1/2, 1/2, 1/2}, {3/2, 3/2}, -(1/4)] In[351]:= N[%/\[Pi]^2] Out[351]= 1. (Recall the "Chinese proverb" pi^2 =10.) Congratulations to Mma for knowing how to evaluate %350, and not doing so automatically: In[352]:= FunctionExpand[%%] Out[352]= 10 (\[Pi]^2/6 + 2 (-(\[Pi]^2/10) - ArcCsch[2]^2) - 2 (-(\[Pi]^2/10) + ArcCsch[2]^2 + 1/2 (\[Pi]^2/15 - ArcCsch[2]^2)) - Log[1/2 + Sqrt[5]/2]^2 + 2 Log[1/2 + Sqrt[5]/2] Log[3/2 + Sqrt[5]/2]) In[353]:= FullSimplify[%] Out[353]= \[Pi]^2 --rwg