Using path-invariance, we can, in polynomial vs exponential time, exhaustively search a taxicab-metric "sphere" for the direction of fastest convergence. Starting with the matrices i: {{(i + k) (i + n)/((1 + i) (i + j + k + n)), -j k n}, {0, 1}}, j: {{(j + k) (j + n)/((1 + j) (i + j + k + n)), -i k n}, {0, 1}}, k: {((i + k) (j + k)/((1 + k) (i + j + k + n)), i j n}, {0, 1}}, n: {{(i + n) (j + n)/((1 + n) (i + j + k + n)), i j k}, {0, 1}} which in the {n,k} plane give Gauß's identity 2F1(a,b,c;1)=Gamma Gamma/Gamma/Gamma, and adding the h dimension via Sunday's "cheap trick", h: {{(h (h + i - j))/((h + k) (h + i + n)), -1 + j + k + n}, {0, 1}}}, i: {{((h + i - j) (i + n))/((1 + i - j - k) (h + i + n)), ((-1 + h + k) (-1 + j + k + n))/(-1 - i + j + k)}, {0, 1}}}, j: {{((i - j - k) (j + n))/((-1 + h + i - j) (j + k + n)), ((-1 + h + k) (-1 + h + i + n))/(-1 + h + i - j)}, {0, 1}}}, k: {{(k (-i + j + k))/((h + k) (j + k + n)), -1 + h + i + n}, {0, 1}}}, n: {{((i + n) (j + n))/((h + i + n) (j + k + n)), -1 + h + k}, {0, 1}}} then searching the neighborhood of starting point n=1,i=0,j=0,h=1,k=1 at a radius of 18 says that "best" is to take 6 n steps, 6 h steps, and 6 k steps, yielding the 6 bits/term identity (combining just one step of n, h, and k) π^2/6==Sum[(13 + 21*k)*π^(3/2)*k!^3/(64^(k + 1)*(1/2 + k)!^3), {k, 0, ∞}] which Mathematica converts to a useless pair of 4F3s. Notice how notationally and computationally inferior are both Sum and pFq notations vs this teensy matrix product In[893]:= {{(2/j + 4)^-3, 21 j/8 - 1}, {0, 1}} In[894]:= Dot @@ Table[%, {j, 3}] Out[894]= {{1/2744000, 23687/14400}, {0, 1}} In[895]:= 23687/14400 - π^2/6. Out[895]= -3.51129*10^-6 (~3 ppm from 3 terms). The Sum identity appears to express π^2 in terms of π^(3/2) ! But of course you don't need π to evaluate the summand. I quoted "best" above because, e.g., we could have combined the n, h, and k steps with a coordinate transformation, yielding fewer steps with messier matrices. Subjectivity persists. --rwg Soon: the "best" matrices (from this system) for csc πx, sin πx, π, and 1/π .