Benoit Cloitre sent me a draft containing a recurrence identity implying a convergent matrix product: [ 2 2 ] [ --------- --------- ] [ sqrt(%pi) sqrt(%pi) ] [ ] = [ %pi 2 %pi 2 ] [ sqrt(---) - sqrt(---) sqrt(---) - sqrt(---) ] [ 2 %pi 2 %pi ] inf [ sqrt(n + 1) ] /===\ [ ------------- 1 ] | | [ n sqrt(n + 2) ] | | [ ] , | | [ sqrt(n) ] n = 1 [ ----------- 0 ] [ sqrt(n + 2) ] giving the c.f. (if you'll pardon my ascii), _ /2 /3 2 /2 / - / - v V 1 V 1 ------ = ---- + ------------------ . pi - 2 1 /3 /4 / - / - V 2 V 2 ---- + ----------- 2 /4 /5 / - / - V 3 V 3 ---- + ---- 3 . . . Nontriangular 2x2 products usually diverge, and you get the c.f. as the limit of the top row over the bottom. If we can find decent asymptotics to c.f. numerators (and denominators), we might scale them out of the matrix to produce more interesting identities of which the c.f. is a projection (which, in the case above, conceals a sqrt(pi)). --Bill Gosper P.S., note the near "recurrence" of our old friend, pi/2+2/pi, the expected magnitude of a 3D rotation, and once-believed optimum "sofa problem" area.