[seqfan] FYI - A conjectural series for 1/pi of a new type from: Zhi-Wei SUN

[Alexander P-sky]

---------- Forwarded Message ----------
From: Zhi-Wei SUN <zwsun at nju.edu.cn>
To: NMBRTHRY at LISTSERV.NODAK.EDU
Subject: A conjectural series for 1/pi of a new type
Date: Thu, 13 Jan 2011 09:34:58 -0600

Dear number theorists,

  On Jan. 2, 2011 I found a new type series for 1/pi.

 CONJECTURE (Zhi-Wei Sun). We have

  Sum_{k=0,1,2,...}(30k+7)binom(2k,k)^2*a_k/(-256)^k = 24/pi,

where a_k=T_k(1,16) is the coefficient of x^k in (x^2+x+16)^k.

 I have included this conjecture in an article of mine (to be expanded
later) available from
http://arxiv.org/abs/1101.0600

 Since a_k=T_k(1,16) ~ 0.75*9^k/sqrt(k*pi) as k tends to the infinity,
we have

  binom(2k,k)^2*a_k/(-256)^k ~ 0.75(-9/16)^k/(k*pi)^{1.5}

and hence the series in the conjecture converges rapidly.

 I have contacted several famous experts at pi-series or modular forms,
they have never seen such a series for 1/pi before and none of them
could prove the conjecture. It seems that all known methods used to
prove Ramanujan-type series for 1/pi (including the current theory of
modular functions and the WZ method) do not work for this curious
series. Such a new series for 1/pi should be very rare!

 I consider the conjecture particularly difficult and very challenging.
It might appeal for a powerful tool or a new theory.

 Any comments are welcome!

Zhi-Wei Sun
http://math.nju.edu.cn/~zwsun