ELLIPTIC FUNCTIONS

     I have found the Maclaurin/Laurent power series expansions of the twelve basic Jacobian elliptic functions and the four basic Jacobian theta functions, in which the coefficients are q-series. Some of these have already been communicated by email to a few mathematicians. I will be communicating them to a few others and submitting them, with their proofs, for publications in journals. I have recorded these expansions,  without proofs, in a 3-page document (pdf), and placed it in my Google drive. Readers of this blog can access them by copying the following link and pasting it onto Google search. https://drive.google.com/file/d/1qMghKmQUU5tESaIcAn6u32hbLIL8Z8RY/view?usp=sharing Somjit Datta, Ph.D Calcutta, India

  In the past year, I have found numerous results involving Jacobian theta functions and elliptic functions, some of which I intend to post in this blog.   While studying these functions, I have expected to find certain results involving them in the literature, but have failed to do so. Consequently, I have endeavoured to derive them on my own, and in many cases, though not in all, I have succeeded in my effort.

    In an earlier post I have discussed the elliptic exponential function                                    E(iu, k) = cn(u, k) +i sn(u, k)      By replacing u by -iu in this function we obtain the function E(u, k). When q approaches 0,  E(iu, k)  reduces to the exponential function e^iu = cos u + i sin u and E(u, k) reduces to e^u.  As is well-known, the inverse of the exponential function v = e^u is the logarithm function ln v =u, such that ln(1+v) = Σ (-1)^{n-1}. v^n /n                                                                                                       (1...

      Contd.   3.      4.((θ_2)^5) ((θ_3)^4) ((θ_4)^2) + (θ_2) ((θ_3)^8) ((θ_4)^2) + 6.(θ_2^(2)) (θ_4) (θ_4^(2))                                                            - 6.(θ_2)((θ_4^(2))^2) + (θ_2) (θ_4) (θ_4^(4))                                                               = (θ_2^(4)) ((θ_4)^2)   4.      5. (θ_1^(1)) ( θ_4) ( θ_4^(4)) + 10.( θ_1^(3))( θ_4^(2) ) ( θ_4) - 30. ( θ_1^(1)) ( θ_4^(2) )^2            + ( θ_2)(( θ_3)^9) (( θ_4)^3) + 14. (( θ_2)^5) (( θ_3)^5) (( θ_4)^3) + (( θ_2)^9) ( θ_3) (( θ_4)^3)                      ...

  Let  θ_1(z, q),  θ_2(z, q),  θ_3(z, q)  and θ_4(z, q) denote, as usual, the four basic Jacobian theta functions.      Let θ_2, θ_3, and θ_4 denote θ_2(0, q), θ_3(0, q) and θ_4(0, q); and let θ_1^(n), θ_2^(n), θ_3^(n) and θ_4^(n) denote, respectively, the values of the n-th derivatives of the four functions at z=0 .  Note that (θ_2)^m, (θ_3)^m , and (θ_4)^m denote the ordinary m-th powers. ( Bracketed exponents denote orders of derivatives and un-bracketed exponents denote powers, as usual.)   The following identity of Jacobi is well-known.                                                       (θ_2) (θ_3) (θ_4) = θ_1^(1)      This belongs to an infinite class of identities involving the four theta functions and their n-th derivatives for various values of n.   ...

As q approaches 0, the following q-series reduce, in the limit, to the well-known Euler numbers  E(0) = 1, E(1) = 0, E(2) = -1, E(3) = 0, E(4) = 5, E(5) = 0, E(6) = -61, E(8) = 1385, etc. ( i.e the constant term of ε (n) is the Euler number E(n).) ε (0) = 1, ε (1) = 0, ε (2) = -1 - 2 q^9 + 16 q^10 - 76 q^11 + 288 q^12 - 950 q^13 + 2832 q^14 - 7812 q^15 + 20256 q^16 - .... .... ε (3) = 0, ε (4) = 5 - 64 q + 512 q^2 - 2816 q^3 + 12288 q^4 - 45952 q^5 + 153600 q^6 - 470528 q^7 + 1343488 q^8 - 3619114 q^9 + 9280352 q^10 - 22808204 q^11 + 54016448 q^12 - 123808734 q^13 + 275619936 q^14 - 597712996 q^15 + 1265867712 q^16 - .... .... ε (5)= 0, ε (6) = - 61 +1216 q - 13824 q^2 + 119040 q^3 - 856064 q^4 + 5329536 q^5 - 29313024 q^6 + 144861696 q^7 - 652836864 q^8 + 2716941850 q^9 - 10551777680 q^10 + 38573871772 q^11 - 133685438240 q^12 + 441849019198 q^13 - 1399644240272 q^14 + 4267104538996 q^15 - 12565129215776 q^16 + .... .... ε (7) = 0, ε (8) = 1385 -...

ELLIPTIC BERNOULLI q-SERIES AND BERNOULLI NUMBERS (contd.) Continued from the last post. β(4)  = -1/30.(1 + 344q - 5376q^2 + 67200q^3 - 587776q^4 + 4060224 q^5 - 23632896 q^6 +  120578304q^7 - 553598976q^8 + ....), β(5)  = 0, β(6)  = 1/42.(1 - 108q + 54912q^2 - 873152q^3 + 10997760q^4 - ....) β(7) = 0, β(8) = -1/30.(1 + 1288q - 324416q^2 + 8416512q^3 - 142155776q^4 + ....)     The coefficients are getting progressively larger and larger, rendering further calculation prohibitively difficult.  The calculations for  β(6) and  β(8)   were so formidable that I could not progress beyond q^4.  But the general pattern is clear.       As promised, the Bernoulli numbers B(0) = 1, B(1) = -1/2, B(2) = 1/6, B(3) = 0, B(4) = -1/30, B(5) = 0, B(6) = 1/42 have been shown to be the constant terms of these q-series. As q approaches 0,  β(n) reduces to B(n ) in the limit. Somjit Datta, PhD J...