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Re: A conjectural series for 1/pi of a new type
Posted:
Jan 18, 2011 9:37 PM


From: Alexander Psky <apovolotATgmailDOTcom> Date: Tue, 18 Jan 2011 20:51:08 0500 Subject: Re: A conjectural series for 1/pi of a new type To: NMBRTHRY@listserv.nodak.edu Cc: ZhiWei SUN <zwsun@nju.edu.cn>, Vladimir Shevelev <shevelev@bgu.ac.il>
WolframAlpha unfortunately times out on the attempt to verify suggested identity (transformed into the following explicit expression with the help from Vladimir Shevelev)
sum((30*k+7)*binom(2k,k)^2*(Hypergeometric2F1[1/2  k/2, k/2, 1, 64])/(256)^k,k=0...infinity)
Alternate form:
sum_(k=0)^infinity((1)^(k) 2^(4 k) (30 k+7) _2F_1((1k)/2, k/2;1;64) Gamma(k+1/2)^2)/(pi Gamma(k+1)^2)
Regards, Alexander R. Povolotsky  Forwarded message  From: "pevnev@juno.com" <pevnev@juno.com> Date: Wed, 19 Jan 2011 01:34:38 GMT Subject: Re: Fw: A conjectural series for 1/pi of a new type To: shevelev@bgu.ac.il Cc: apovolot@gmail.com
Thanks Vladimir,
WolframAlpha interpreted your sum (in Maple format) as
sum((16^j*C(k, j)*C(kj, j)),j=0...floor(k/2)) = = Hypergeometric2F1[1/2  k/2, k/2, 1, 64]
http://www.wolframalpha.com/input/?i=sum%28%2816^j*C%28k%2c+j%29*C%28kj%2c+j%29%29%2cj%3d0...floor%28k%2f2%29%29&s=6&incTime=true
Thanks, Alex ================================ From: Vladimir Shevelev <shevelev@bgu.ac.il> To: "pevnev@juno.com" <pevnev@juno.com> Subject: Re: Fw: A conjectural series for 1/pi of a new type Date: Tue, 18 Jan 2011 22:45:35 GMT
Proof: coef_(x^k) (x^2+x+16)^k=coef_(x^k)(x*(x+1)+16)^k=Sum{i=0,...,k}coef_(x^(k i))C(k,i)*(x+1)^i*16^(ki)=Sum{i=0,...,k}C(k,i)*C(i,ki)*16^(ki)=[k i=j]=Sum{j=0,...,floor(k/2)} 16^j*C(k, j)*C(kj, j).
 Forwarded message  From: "pevnev@juno.com" <pevnev@juno.com> Date: Tue, 18 Jan 2011 16:12:02 GMT Subject: Fw: A conjectural series for 1/pi of a new type To: shevelev@bgu.ac.il Cc: apovolot@gmail.com
Dear Vladimir  FYI,
> a_k=T_k(1,16) is the coefficient of x^k in (x^2+x+16)^k
Is there a way to express in closed (or otherwise symbolically formulated) form ?
Thanks, Best Regards, Alexander R. Povolotsky
 Forwarded Message  From: ZhiWei SUN <zwsun@nju.edu.cn> To: NMBRTHRY@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 (ZhiWei 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 piseries 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 Ramanujantype 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!
ZhiWei Sun http://math.nju.edu.cn/~zwsun



