This page has URL http://grail.eecs.csuohio.edu/~somos/wxyz.htmlThe 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 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. The Moutard memoir on page 542 are defined three sequences of homogeneous polynomials using two sets of duplication equations. Equation (6) for even and (7) for odd cases. for example, b_{2p}+c_{2p} = 2b^2_pc^2_p. These sequences are just slightly 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. On page 545 is the important equation A_{p-q}A_{p+q}=A^2_qA_{p-1}A_{p+1}-A^2_qA_{q-1}A_{q+1} More details about the 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 our own equations and is the same as Moutard's equation mentioned before. Also this quote "Lucas mentioned that these formulas belonged to the theory of elliptic functions,...".

Last updated 24 Dec 2018

by Michael Somos <ms639@georgetown.edu>