Weierstrass Elliptic Function Polynomials
The inspiration of my research was a solution of the
congruent number 5 problem on pp. 419-427 in the book:
Uspensky and Heaslet, Elementary Number Theory, 1939.
I placed the sequences A129206,..,A129209 in the OEIS.
My original code dates back to about 2002 partly based
on Kerawala's 1947 article on Poncelet's porism.
The sequences are homogenous polynomials in the
variables X,Y,Z with scale factors of W,x0,y0,z0
for the four sequences respectively. Thus, the w
sequence terms all have a factor of W, while the x
sequence terms all have a factor of x0, and so on.
The n-th sequence terms all have a factor of Q^n^2.
Note that w is an elliptic divisibility sequence.
The w sequence is an odd sequence while the x,y,z
sequences are all even sequences. Thus, w is an
analog of theta_1, while x,y,z are analogs of the
other three Jacobi theta functions. Also, w is an
analog of the Weierstrass sigma function, while
x,y,z are analogs of the other sigma functions.
More precisely, given numbers t and |q|<1, while
x0 = theta_2(0, q), y0 = theta_3(0, q),
z0 = theta_4(0, q), W = theta_1(t, q),
X = theta_2(t, q)/x0, Y = theta_3(t, q)/y0,
Z = theta_4(t, q)/z0, and Q = 1, then we get
wn(n) = theta_1(n*t, q), xn(n) = theta_2(n*t, q),
yn(n) = theta_3(n*t, q), zn(n) = theta_4(n*t, q).
There is a similar result for the four Weierstrass
sigma functions. I have versions of the Weierstrass
zeta function and its first few derivatives which I
notate beginning with the letter "w", but they are
rational functions. I have also polynomial versions
beginning with the letter "W" which are more closely
related to the the sigma function polynomial sequences.
Note that I have introduced several constants with
more or less arbitrary names. The g2,g3 are just the
Weierstrass invariants. The ex,ey,ez correspond to
the Weierstrass e1,e2,e3. The DD is the discriminant
Delta. The j,J correspond to Klein's modular function.
The p1,p2,p3,p4,p5 are my invariants of generalized
Note that it is possible to take the "derivative" of
any expression using a derivative function that I
named Du, and use it to verify identities for the
derivatives of the Weierstrass elliptic functions
and the Jacobi elliptic functions including analogs
of the Jacobi Zeta and Epsilon functions.
Note that not all possible identities between the
elliptic functions are satisfied by these polynomial
sequences. For exmaple, the important identity for
the 4th power of theta null functions does not hold.
That is, the identity 0 = x0^4 +y0^4 -z0^4 is clearly
not valid since the variables x0, y0, z0 are free.
Note that this is a work in progress and there may
be slight changes in detail but almost everything
in here is in a workable state which is unlikely to
change in future. Please inform me of any errors you
find. I may write fuller documentation if there is
any real interest in my work detailed here.
Note the following special case: if
W = u^2-v^2, X = (u^2+v^2)/2, Y = Z = u*v,
x0 = 2, y0 = z0 = Q = 1, then we have
wn(n) = (u*v)^(n^2-n) * (u^(2*n)-v^(2*n)),
xn(n) = (u*v)^(n^2-n) * (u^(2*n)+v^(2*n)),
and yn(n) = zn(n) = (u*v)^(n^2).
My results are comparable to the ones in a 1950
article in the American Journal of Mathematics by
Morgan Ward, yet are quite different in origin
and scale. More directly comparable are formulas
of the Chudnovsky's from an article in Advances
in Applied Mathematics from 1986 on page 418.
In 1878 Edouard Lucas published a memoir on the
arithmetization of elliptic functions. He may have
been working towards the generalization of this
special case without success. He does refer to the
memoir of Moutard but apparently did not really
understand its implications for his quest.
Currently, my results are written in the form of
computer program scripts in two versions. The
This page has URL http://grail.eecs.csuohio.edu/~somos/wxyz.html
Last updated 09 Dec 2017
by Michael Somos <firstname.lastname@example.org>