I said,
But I wanted [ n x 1 ] inf [ ----- - 0 ] /===\ [ n + 2 2 ] [ 0 0 z(x) ] | | [ ] [ ] | | [ n x ] = [ 0 0 y(x) ], | | [ 0 ----- 1 ] [ ] n = 1 [ n + 2 ] [ 0 0 1 ] [ ] [ 0 0 1 ]
and we used the TLINREL function to determine log(1 - x) (log(1 - x) (x - 1) - x) z(x) = -----------------------------------, 3 x 2 (log(1 - x) (x - 1) - x) y(x) = - --------------------------. 2 x I thought you'd never ask! This product is interesting because it is prod(N(2,n),n,1,inf), where [ n x 1 ] [ ----- - 0 ] [ n + k k ] [ ] N(k, n) := [ n x ], [ 0 ----- 1 ] [ n + k ] [ ] [ 0 0 1 ] which with K(k, n) := [ k x (n + 1) x + k (x - 1) - n 1 ] [ --------------- - ------------------------- ------------------ ] [ (n + k) (x - 1) 2 2 ] [ (k + 1) (n + k) (x - 1) k (k + 1) (x - 1) ] [ ] [ k x 1 ] [ 0 --------------- - ----- ], [ (n + k) (x - 1) x - 1 ] [ ] [ 0 0 1 ] forms a path-invariant pair: N(k,n) K(k,n+1) = K(k,n) N(k+1,n). Equating alternate paths from k=a, n=b around two sides of the infinite rectangle with opposite corner approaching k=oo, n=oo gives an identity relating "triangular" sums, one in powers of x, and the other in powers of x/(x-1). This corresponds to the "linear" transformation of ordinary hypergeometrics, A&S 15.3.4 More interestingly, we can take a diagonal path from (a,b), N(a,b) K(a,b+1) N(a+1,b+1) K(a+1,b+2) = J(0) J(1) J(2) ..., where J(j) = N(j+a,j+b) K(j+a,j+b+1), which with a=1, b=1 is [ 2 ] [ j x x (2 x - 1) j (2 x - 2) - 1 ] [ ----------------- - -------------------------- - -------------------- ] [ 1 1 2 1 2 ] [ 4 (j + -) (x - 1) 4 (j + -) (j + 1) (x - 1) 2 j (j + -) (x - 1) ] [ 2 2 2 ] [ ] [ 2 ] [ j x j (x - 2) - 1 ] [ 0 ----------------- ----------------- ] [ 1 1 ] [ 4 (j + -) (x - 1) 2 (j + -) (x - 1) ] [ 2 2 ] [ ] [ 0 0 1 ] This is analogous the the "quadratic transformation" (A&S 15.3.15), and computes [ 1 log(1 - x) (log(1 - x) + 2) 1 ] [ 0 0 - + --------------------------- - ----- ] [ x 2 x - 1 ] [ 2 x ] [ ] [ log(1 - x) ] [ 0 0 - ---------- ] . [ x ] [ ] [ 0 0 0 ] The symbolic expansion at x=0 takes half as many terms as prod(N(1,n),n,1,inf) or prod(K(k,1),k,1,inf). Numerically it wins big at x=-1 and breaks even at x=4/5, and extends the lower limit of convergence from -1 to -2-2 sqrt(2). My previous 3by3 systems came from grafting together contiguous 2by2s, e.g. the "3F2 Rosetta Stone". What's new here is targeting functions that require 3x3 to begin with. Presumably, th1s technique extends to functions more interesting than log(1-x)^2. --rwg