ZW Sun has posted three conjectured series for 1/pi to the NMBRTHRY list. I posted his first series to MathFun a couple of weeks ago. Since then, he's come up with two more. -- Rich ----- Forwarded message from zwsun@nju.edu.cn ----- Date: Sat, 29 Jan 2011 12:30:17 -0600 From: Zhi-Wei SUN <zwsun@nju.edu.cn> Reply-To: Number Theory List <NMBRTHRY@listserv.nodak.edu>, Zhi-Wei SUN <zwsun@nju.edu.cn> Subject: One more conjectural series for 1/pi To: NMBRTHRY@listserv.nodak.edu Dear number theorists, For integers b and c let T_n(b,c) denote the coefficient of x^n in the expansion of (x^2+bx+c)^n. I have announced two conjectural series for 1/pi: Sum_{k=0,1,2,...}(30k+7)binom(2k,k)^2*T_k(1,16)/(-256)^k = 24/pi and Sum_{k=0,1,2,..}(15k+2)binom(2k,k)binom(3k,k)T_k(18,6)/972^k = 45*sqrt(3)/(4*pi). Here I release a new conjectural series for 1/pi. CONJECTURE (Zhi-Wei Sun, Jan. 29, 2011). We have Sum_{k=0,1,2,...}(40k+3)binom(4k,2k)binom(2k,k)T_k(98,1)/12544^k = 70*sqrt(21)/(9*pi). Note that this new series also converges rapidly. Let s(n)=sum_{k=0}^n(40k+3)binom(4k,2k)binom(2k,k)T_k(98,1)/12544^k. By Mathematica, if n is at least 340 then |s(n)*9pi/(70*sqrt(21))-1|<10^{-100}. I have included this conjecture and the related congruences in the latest version of my preprint (arXiv:1101.0600) which will be available from the next Tuesday. In general, I suggest the investigation of series for 1/pi of the following forms (with a,b,c,d,m integers, d positive and m(b^2-4c) nonzero): Type I. Sum_{k=0,1,2,...}(a+d*k)binom(2k,k)^2*T_k(b,c)/m^k; Type II. Sum_{k=0,1,2,...}(a+d*k)binom(2k,k)binom(3k,k)*T_k(b,c)/m^k; Type III. Sum_{k=0,1,2,...}(a+d*k)binom(4k,2k)binom(2k,k)*T_k(b,c)/m^k; Type IV. Sum_{k=0,1,2,...}(a+d*k)binom(6k,3k)binom(3k,k)*T_k(b,c)/m^k. My three conjectural series belong to type I,II,III respectively. I have not been able to find a series for 1/pi of type IV. Since T_k(b,c) (with b^2-4c nonzero) is involved, series of the above types are not hyper-geometric and this is one of the main reasons why such conjectural series for 1/pi are more difficult than Ramanujan-type series. There should be VERY FEW series for 1/pi of the above 4 types. If there are some such series that I have missed, you are welcome to pick them up. In my opinion, the three conjectural series for 1/pi might remain open for many years. Zhi-Wei Sun http://math.nju.edu.cn/~zwsun ----- End forwarded message -----