[seqfan] Feigenbaum Function Coefficients
Paul D Hanna
pauldhanna at juno.com
Sun Aug 15 20:34:06 CEST 2010
SeqFans,
I would like to generate the Feigenbaum function coefficients (for N=2).
The constants that form the coefficients are not found in the OEIS,
and it would be nice to have more digits.
At the bottom of this email, I copy the coefficients that I know, but they are dated 1982
(Lanford, "A COMPUTER-ASSISTED PROOF OF THE FEIGENBAUM CONJECTURES"
http://www.ams.org/journals/bull/1982-06-03/S0273-0979-1982-15008-X/S0273-0979-1982-15008-X.pdf).
Where can I find more accurate coefficients?
More importantly, how can I generate these for myself?
These references seem to indicate a continued fraction method of generating the function and thus the coefficients, but I can't make enough sense out of the method to implement into a program:
(*) "A priori bounds for anti-Herglotz functions with positive real parts"
http://www.lboro.ac.uk/departments/ma/research/preprints/papers02/02-38.pdf
(**) "Continued fractions and solutions of the Feigenbaum-Cvitanovic equation"
http://info.lut.ac.uk/departments/ma/research/preprints/papers02/02-04.pdf
Can anyone assist?
Thanks,
Paul
----------------------------------------------------------------------------
Feigenbaum Function
g(x) = 1/a*g(g(a*x), where a = g(1) = -1/2.5029078750... with g(0)=1
is an even function:
g(x) = Sum C(n)*x^(2*n)
where the coefficients begin:
{C=[1,
-1.527632997036301454035890310240,
+0.104815194787303733216742613801,
+0.026705670525193354032652094944,
-0.003527409660908709170234190769,
+0.000081600966547531745172190486,
+0.000025285084233963536176262552,
-2.5563171662784938463532541*10^-6,
-9.6512715508912032163725768*10^-8,
+2.8193463974504091370756629*10^-8,
-2.77305116079901172437*10^-10,
-3.02842702213056632983*10^-10,
+2.67058928074807555396*10^-11,
+9.96229164102848231059*10^-13,
-3.62420298290415608455*10^-13,
+2.17965774482707047701*10^-14,
+1.52923289948096260560*10^-15,
-3.184728789952775*10^-16,
+1.134672106211871*10^-17,
+1.881676056825439*10^-18,
-2.275612564632121*10^-19,
-9.822447629421762*10^-22,
+2.064129756004508*10^-21,
-1.249320059243689*10^-22,
-1.0770612046*10^-23,
+1.8727468082*10^-24,
-2.5777082101*10^-26,
-1.5541904560*10^-26,
+1.2804434650*10^-27,
+5.5850587986*10^-29,
-1.5278346925*10^-29,
+5.0417426639*10^-31,
+1.0165368070*10^-31,
-1.00690*10^-32,
-5.24253*10^-34,
+1.72437*10^-34,
-1.31439*10^-35,
-1.85830*10^-38,
+8.05506*10^-38,
-6.26717*10^-39,
+1.76882*10^-40
]}
[END]
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