[Maxima] Stepen Lucas's "Integral approximations to Pi with nonnegative integrands"

Alexander Povolotsky apovolot at gmail.com
Fri Feb 20 10:52:46 CST 2009


Thanks Dear Robert,

Perhaps to somewhat clarify what it is I am thinking about (to be in the
program, which is probably more than what Stephen Lukas currently has in his
program ?),
let me quote my latest guess statement (in my today's email to Stephen
Lukas, Grant Kady, Jonathan Borwein and David Bailey)

> With reference to
> http://www.research.att.com/~njas/sequences/A002486
> http://www.research.att.com/~njas/sequences/A002485
> Let
> i(n), k(n), l(n) and m(n) be some integer functions of n
>  (to be found, where i(n) appears to be various powers of 2 )
> in the following (using Maple's notations) guessed identity
>
> Pi =
> A002485(n)/A002486(n) - 1/(i*l)*Int(x^m*(1-x)^m*(k+ (k+l)*x^2)/(1+x^2),x =
0 .. 1)
>
> for n =2,3, ...,
> (per n's definition as index in the aforementioned A002485(n) and
A002486(n) )

Of course above guess may be not accurate and it may be that the  actual
experimental finding might reveal something else (or nothing at all ?)

Cheers,
Alex

On Fri, Feb 20, 2009 at 11:36 AM, Robert Dodier <robert.dodier at gmail.com>wrote:

> Seems like it would be straightforward to translate to Maxima
> the Maple program shown in the referenced paper.
> CVS Maxima seems to be able to compute the integrals
> (5.17.1 asks about the sign of parameters).
> The "type" stuff seems to be a set membership or maybe just
> a pattern-matching test; maybe freeof could replace it.
> Haven't looked at the paper in any detail. Have fun!
>
> Robert Dodier
>
>
> ---------- Forwarded message ----------
> From: "Alexander R.Povolotsky" <apovo... at gmail.com>
> Date: Feb 19, 5:49 pm
> Subject: Stepen Lucas's "Integral approximations to Pi with
> nonnegative integrands"
> To: sci.math.symbolic
>
>
> Hi,
>
> S. K. Lukas in
>
> http://www.math.jmu.edu/~lucassk/Papers/more%20on%20pi.pdf<
> http://www.math.jmu.edu/%7Elucassk/Papers/more%20on%20pi.pdf>
>
> derived several identities, which relate Pi (via definite integrals)
> with
> the several few Pi fractional convergents, which denominators and
> numerators
> are described in
>
> http://www.research.att.com/~njas/sequences/A002486<
> http://www.research.att.com/%7Enjas/sequences/A0024865>
> http://www.research.att.com/~njas/sequences/A002485<
> http://www.research.att.com/%7Enjas/sequences/A0024865>
>
> I raised the issue re the possibility of deriving generalized ("n"
> parameter
> based ) definite integral identity relating Pi with ALL (each at its
> own
> value of n) Pi fractional convergents (referenced in above sequences)
> - see my exchange with S. K. Lukas below.
>
>  Unfortunately I do not have sufficient computational resources (I do
> not have access to Maple or Mathematica, instead I have Pari/GP
> installed
> on my very old home computer) to take advantage of Stephen's generous
> offer to play with his Maple program, which he wrote and which is
> listed in
>
>  http://www.math.jmu.edu/~lucassk/Papers/more%20on%20pi.pdf<
> http://www.math.jmu.edu/%7Elucassk/Papers/more%20on%20pi.pdf>
>
> The program is on page 9 of the linked to pdf and
> that it is literally only 18 lines of code.
> May one of you could help me running
> Stephen's program towards experimental attempt of deriving desired
> generalized
> identity?
>
> Thanks,
> Best Regards,
> Alexander R. Povolotsky
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