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

Plain Text

From: Alexander P-sky <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: Zhi-Wei 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((1-k)/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(k-j, 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%28k-j%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^(k-i)=Sum{i=0,...,k}C(k,i)*C(i,k-i)*16^(k-i)=[k-
i=j]=Sum{j=0,...,floor(k/2)}
16^j*C(k, j)*C(k-j, 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: Zhi-Wei 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 (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