Re: Stepen Lucas's "Integral approximations to Pi with nonnegative integrands"



On Feb 19, 7:49 pm, "Alexander R.Povolotsky" <apovo...@xxxxxxxxx>
wrote:
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

May be the search should be based around finding
i(n), k(n), l(n) and m(n) as some integer functions of n, where i(n)
appears to represent the various powers of 2 ) in the following (using
Maple's notations) guessed (by me ) 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)

Does above guesstimate help in writing a program ?

Of course my above identity guess may be not accurate and it may be
that in the end the actual experimental finding might reveal something
else (or nothing at all ;-) )

Cheers,
Alex (Alexander R. Povolotsky )
.


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