WXYZ Math Project

   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 = 1/sqrt(a*b), X = (a+b)/2/sqrt(a*b),
      x0 = 2, Y = Z = y0 = z0 = Q = 1,
   then we have the equations
      wn(n) = (a^n-b^n)/(a-b) * W^n,
      xn(n) = (a^n + b^n) * W^n,
      yn(n) = zn(n) = 1.
   These are multiples of the usual Lucas sequences
      U_n = wn(n)/W^n, V_n = xn(n)/W^n.

   My results are comparable to those given in 1948
   and 1950 articles by Morgan Ward in the American
   Journal of Mathematics, but different in origin
   and scale. For exmpale, his 1950 article in Amer.
   J. Math. has polynomial sequences A_n, B_n, C_n,
   and D_n with Table V. initial values on page 291.
   They are comparable to xn(n), wn(n), yn(n), zn(n).
   More directly comparable are the formulas of the
   Chudnovsky's from an article in Advances in Applied
   Mathematics from 1986 on page 418.

   Dickson, _History of the Theory of Numbers_, volume
   II, page 468 mentions Lucas, etc. work on congruent
   numbers. In particular on the congruent number 5:
   "The equations x^2-5y^2=v^2, x^2+5y^2=v^2" and
   "X=u^2x^2+5v^2y^2, U=u^2x^2-5v^2y^2, V=u^4-2x^4,
   Y=2xyuv," A similar formula appears on page 469.

   In 1878 Edouard Lucas published a memoir _Theorie
   Des Fonctions Numeriques Simplement Periodiques_.
   On page 35 he notes that certain formulas for his
   U_n, V_n sequences resemble similar ones for the
   Jacobi theta functions.  He refers to a memoir of
   Moutard that was included in the 1862 volume 1 of
   Poncelet's book _Applications ..._ pp. 535-560,
   but apparently he did not really understand its
   implications for his quest.

   More details about this work of Lucas is contained
   in Hugh C. Williams, _E'douard Lucas and Primality
   Testing_, 1998, Wiley, pp. 451-457. On page 453 is
   h{m+n}h{m-n}=h{m+1}h{m-1}h(n)^2-h{n+1}h{n-1}h(m)^2
   which is similar to one of my own equations and
   this quote "Lucas mentioned that these formulas
   belonged to the theory of elliptic functions,...".

   A good introduction to this is the article _Some
   Extensions of the Lucas Functions_ co-authored by
   R. K. Guy, E. Roettger and H. C. Williams, at URL
    while much more details
   are in the 2009 Ph.D. thesis of Eric L. F.  Roettger
   available at URL
   .

   The memoir by Moutard on page 542 defined three
   sequences of homogeneous polynomials using two sets
   of duplication equations. Equation (6) for the even
   and (7) for odd cases. For example, b_{2p}+c_{2p} =
   2b^2_pc^2_p.  The three sequences are nonhomogeneous
   special cases of x_n, y_n, z_n. Thus, a_n = x_n
   with Q = x_0 = y_0 = z_0 = 1, x_1 = a, y_1 = b,
   z_1 = c. This 
   page 542 is viewable on the web.

   Notice that B. C. Carlson in 2004 published an article
   _Symmetry in c, d, n of Jacobian elliptic functions._
   which uses Glaisher's abbreviations for the Jacobian
   elliptic functions. The idea is that {p,q,r} = {c,d,n}
   in any order so (s,p,q,r) corresonds to my (w,x,y,z).
   He also introduced Delta(p,q) := ps^2-qs^2 which is
   the same as my XY,XZ,YZ.

   Note that Karl Weierstrass in 1840 wrote a paper on
   elliptic functions that used A,B,C,D to denote what I
   have with Z,W,X,Y. This eventually appeared in the
   Table of Integrals, Series and Products by Gradshteyn
   and Ryshik as section 8.148 which refers to a Russian
   Book from 1941. The original reference to Weierstrass
   is given in a answer to a Stack Exchange question at
   
   Where does the Weierstrass expansion of sn come from?.