[seqfan] Jacobi (cn,sn,dn) rational addition sequences
Brad Klee
bradklee at gmail.com
Fri Aug 4 18:58:55 CEST 2017
Hi Seqfans,
Another thread (http://list.seqfan.eu/pipermail/seqfan/2017-July/017805.html)
mentions the possibility to elaborate upon A066770 & A066771 using elliptic
functions and addition rules. One simple example using Jacobi Elliptic
functions (http://mathworld.wolfram.com/JacobiEllipticFunctions.html)
follows, with shorthand: (x,y,z) = (cn,sn,dn).
The identities hold: x^2 + y^2 = 1; (k*y)^2 + z^2 = 1. Pythagorean pair
(x,y)=(3/5,4/5) satisfies the first equation, while choosing k=3/4 and
z=4/5 satisfies the second. This sets the initial condition for iteration
via the addition rules:
P0 = (x0,y0,z0) = (3/5,4/5,4/5); where k = 3/4.
PN = (xN,yN,zN) = N*P0 = (N-1)*P0+P0 = P0+P0 . . . +P0
Where operation "+" of P3 = P1 + P2 acts according to the addition rules
for the Jacobi Elliptic Functions
x3 = (x1*x2 - y1*y2*z1*z2) / (1 - (k*y1*y2)^2)
y3 = (y1*x2*z2 + y2*x1*z1) / (1 - (k*y1*y2)^2)
z3 = (z1*z2 -k^2*x1*x2*y1*y2) / (1 - (k*y1*y2)^2)
Note that as k |---> 0 and every z |--->1, the addition rules are the same
as for (x,y) = (cos,sin), making clear the analogy to A066770, A066771.
Again these addition rules are integrable, especially via polar
coordinates, and with real period EllipticK. Iterating the recurrence for
PN on the chosen P0, we obtain three rational sequences:
3-4-5 iteration of the Jacobi Elliptic Functions (sn, cn, dn) with k=3/4 .
==x==
Rational: 3/5, -(31/481), -(534717/742085), -(23223396479/23668072321),
-(8804610237917757/18618861255059525)
Numerators: 3, -31, -534717, -23223396479, -8804610237917757
Denominators: 5, 481, 742085, 23668072321, 18618861255059525
==y==
Rational: 4/5, 480/481, 514556/742085, -(4566344640/23668072321),
-(16405512274644476/18618861255059525)
Numerators: 4, 480, 514556, -4566344640, -16405512274644476
Denominators: 5, 481, 742085, 23668072321, 18618861255059525
==z==
Rational: 4/5, 319/481, 633844/742085, 23418981121/23668072321,
13973914121544724/18618861255059525
Numerators: 4, 319, 633844, 23418981121, 13973914121544724
Denominators: 5, 481, 742085, 23668072321, 18618861255059525
Superseeker search assistant offers generous praise, saying: "Even though
there are a large number of sequences in the OEIS, at least one of yours is
not there!" I'm not sure if these sequences are "of general interest", but
as a stepping stone to ECC, they could have some practical value.
Thanks,
Brad
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