Fred Lunnon wrote:
I had noticed that h(q) = h(p), where p = (2-2q)/(2-q) [snip] Latest data:
h(-sqrt2) = h(+sqrt2) = 0; h^2(-4/3) = h^2(7/5) = 59/450, h^2(-1) = h^2(4/3) = 11/18, h^2(-2/3) = h^2(5/4) = 287/288, h^2(-1/2) = h^2(6/5) = 231/200, h^2(-1/3) = h^2(8/7) = 1139/882, h^2(0) = h^2(1) = 3/2, h^2(1/4) = h^2(6/7) = 2511/1568, h^2(1/3) = h^2(4/5) = 731/450, h^2(2/5) = h^2(3/4) = 1311/800, h^2(1/2) = h^2(2/3) = 119/72.
Hmm. 59/450 = 1/9 + 1/50 and 11/18 = 1/2 + 1/9. However, 287/288 = 7*(1/9 + 1/32); the denominators clearly come from squaring q and p, but where does the 7 come from? Aha! 1/9 = 1 - (-4/3)^2/2, 1/50 = 1 - (7/5)^2/2; and then 1 - (-2/3)^2/2 = 7/9 and 1 - (5/4)^2/2 = 7/32. After checking the rest, it seems your data is produced by the formula h^2(q) = 2 - q^2/2 - p^2/2, where p is given by the formula above. [It wasn't as easy as I make it look above; I've omitted a lot of pointless thrashing about.] -- Fred W. Helenius fredh@ix.netcom.com