I said

I know some kids who go to fancy private schools--maybe they'll let me sneak a peek at their texts. I just peeked at a *public* (Palo Alto, vs Los Altos) 7th grader's Algebra 1 text, and it's ten times better. The pictures are relevant, there is no crediting an affirmative action committee, and the only braino I found was the caption of a photo of Saturn claiming that the reason it has an average surface temperature is that it gives off more heat than the sun provides.(!) The book factors polynomials and derives the quadratic formula. I assume it's "honors algebra" instead of "dishonored algebra", but there's nothing on the respective covers suggesting anything except they're both for California standards. I guess this is to protect the self-esteem of the poor slobs carrying around the dishonorable versions. (I assume there's an honors course at Los Altos that my young friend wasn't offered.)

Enough of that. A Bruce Berndt question about 1/pi series prompted me to try reconstructing my derivation (in the ancient ftp://publications.ai.mit.edu/ai-publications/pdf/AIM-304.pdf) of Ramanujan's dyadic rational inf 2 k 3 ==== (42 k + 5) ( ) 1 \ k -- = > ----------------- pi / 12 k + 4 ==== 2 k = 0 F[1/64] series directly from Gauss's 2F1[1]. But I spazzed and got inf 2 i ==== (2 i + 1) (6 i + 7) (7 i + 9) ( ) \ i pi - 3 = 8 > ----------------------------------- / 2 2 4 i + 4 2 ==== (2 i + 2) (4 i + 5) ( ) i = 0 2 i + 2 instead. As usual, pFq notation is ridiculous, (c300) make1hyper(%) (d300) %pi - 3 = 3 3 3 13 16 7 7 9 9 7 9 1 7 hyper_f ([1, 1, -, -, -, --, --], [-, -, -, -, -, -], --) 7, 6 2 2 2 6 7 4 4 4 4 6 7 64 -------------------------------------------------------------- 50 (c301) dfloat(%) (d301) 0.14159265358979d0 = 0.14159265358979d0 but at least the parameters are rational. I'm almost sure I found such in the 70s, but failed several times to rediscover it. Note how economical is the equivalent matrix product: (c302) factor(d170) [ 3 ] [ k (2 k + 1) (6 k + 1) (7 k + 2) ] [ ----------------------- ------------------- ] (d302) [ 2 2 450 ] [ 2 (4 k + 3) (4 k + 5) ] [ ] [ 0 1 ] (c303) prud(%,k,1,9.0d0) [ 2.60286523404292d-19 0.14159265358979d0 ] (d303) [ ] [ 0.0d0 1.0d0 ] I'm afraid this comes a few months too late for another young friend who needed a pi-3 series for some kind of programming contest. Perhaps he should join this list, if we're done with prom(iscuity.-) Note that if we squeeze out one more term, we get an exact series for pi - 3.14: [ 3 ] [ k (2 k + 1) (6 k + 1) (7 k + 2) ] [ ----------------------- ------------------- ] (d304) [ 2 2 132300 ] [ 2 (4 k + 3) (4 k + 5) ] [ ] [ 0 1 ] (c305) prud(%,k,2,9.0d0) [ 7.6524237880862d-17 0.00159265358979d0 ] (d305) [ ] [ 0.0d0 1.0d0 ] Here is the real indictment of pFq notation. From 2F1[a,-a;1] = sin(pi a)/(pi a) I derived the more general F[a|1/64], which puts a cubic(a,k) in the UR of d302, but intractable, needless surds in the pFq form. Correcting my derivation spaz and retrying, inf 3 2 i + 1 3 ==== (i + 1) (14 i - 3) ( ) 1 \ i -- = 3 > ------------------------------. pi / 2 12 i + 2 ==== (2 i - 3) (2 i - 1) 2 i = 0 It's 1/pi, dyadic, and F[1/64]. But it's weird. Try again. inf 2 k - 1 3 ==== (42 k + 5) ( ) 1 \ k 35 -- = > --------------------- - --. pi / 12 k + 1 16 ==== 2 k = 0 Finally. This is equivalent to Ramanujan's. (Nice exercise.) Now I can apply the exact same process to Heine's series instead of Gauss's and get a q-extension (maybe Mizan knows a prettier one), 3 2 oo 2 6k 2k+1 3 6k+1 6k+1 6k+3 ==== (q; q ) q ((q + 1) (1 - q ) - q (1 - q )) \ k

----------------------------------------------------------- / 3 3 ==== 4 4 2 k = 0 (q ; q ) (- q; q ) k k + 1 2 2 (q; q ) 1/4 2 oo q (1 - q ) = ---------- = -------------. 2 pi(q) 2 2 (q ; q ) oo Test: (c479) taylor(makeprod(lhs(%)),q,0,24)

2 3 4 5 6 7 8 9 (d479)/T/ 1 - 2 q + 3 q - 6 q + 11 q - 18 q + 28 q - 44 q + 69 q - 104 q 10 11 12 13 14 15 16 + 152 q - 222 q + 323 q - 460 q + 645 q - 902 q + 1254 q 17 18 19 20 21 22 23 - 1722 q + 2343 q - 3174 q + 4278 q - 5722 q + 7601 q - 10056 q 24 2 3 4 5 6 7 + 13250 q + . . . = 1 - 2 q + 3 q - 6 q + 11 q - 18 q + 28 q - 44 q 8 9 10 11 12 13 14 15 + 69 q - 104 q + 152 q - 222 q + 323 q - 460 q + 645 q - 902 q 16 17 18 19 20 21 22 + 1254 q - 1722 q + 2343 q - 3174 q + 4278 q - 5722 q + 7601 q 23 24 - 10056 q + 13250 q + . . . (c480) 1/lhs(%) 2 3 4 6 7 9 10 11 12 (d480)/T/ 1 + 2 q + q + 2 q + 2 q + 3 q + 2 q + 2 q + 2 q + 2 q + q 13 15 16 18 20 21 22 24 + 2 q + 2 q + 4 q + 2 q + q + 4 q + 2 q + 2 q + . . . which will obey nonlinear, three term relations among pi(q), pi(q^a), and pi(q^b), for rational a and b. (Actually, we want Pi(q):=pi(q)/(1-q^2), which changes q^(6k^2) to q^(6(k^2-1/24)) in our series.) --rwg --------------------------------- Building a website is a piece of cake. Yahoo! Small Business gives you all the tools to get online.