ELLIPTIC FUNCTIONS

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...

ELLIPTIC BERNOULLI q-SERIES AND BERNOULLI NUMBERS The well-known Bernoulli numbers B(n), n = 0, 1, 2, 3, 4.... are defined as follows: u / {e(u) -1} = B(0) + B(1).u/1! + B(2).u^2/2! + B(3). u^3/3! + B(4).u^4/4! + B(5).u^5/5! + B(6).u^6/6! + .... .     It is known that 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. We recall from an earlier blogpost the elliptic exponential function  E(iu, τ)  := cn (u, τ) + i sn(u, τ).  It is clear that lim  E(iu, τ) = e(iu) = cos u + i sin u as q approaches 0, where q:=e(iπτ)  .  Replacing u by -iu, we get the functions E(u,  τ) and e(u) , where lim  E(u,  τ) = e(u) as q approaches 0. We refer to each of  E(iu,  τ) and E(u,  τ) as the elliptic exponential function, since the context indicates which one is meant.   We have   lim  u / {E(u, τ) - 1} = u / {e(u)-1} as q approaches 0. Therefore...

THE ELLIPTIC PI  I hereby report a certain number of significant interest that I have recently found in the course of my investigations in the theory of elliptic functions and theta functions.   It is well-known that  sin(u+2π) = sin u ,  cos (u+2π) = cos u ; sn(u+4K, τ) = sn(u, τ) ,  c n(u+4K, τ ) = cn(u, τ) ;  and   lim   sn(u, τ) = sin u ,   lim  c n(u, τ) = cos u   as q approaches 0, where q:=e(iπτ), Im τ > 0.  It can be shown that  lim 2K = π  as q approaches 0. Therefore 2K can be regarded as the 'elliptic pi'. It is known that  K= (1/2).π.θ(3)^2,  where θ(3) is the special value of the third Jacobi theta function θ(3)(z,τ) at z=0.  Hence we have the elliptic pi 2K(τ) =   π.θ(3)^2,  where we have written  K(τ) for K. It is well-known that  π = 3.14159265359 (approximately).  I have recently calculated the following object: T...