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
   Somos-4 sequences.

   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 <ms639@georgetown.edu>