IDENTITIES INVOLVING JACOBIAN THETA FUNCTIONS AND THEIR DERIVATIVES I

  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.

      I have recently found the following identities of this infinite class.

1.             3.(θ_2) (θ_3) (θ_4^(2)) - (θ_2) ((θ_3)^5) (θ_4) - ((θ_2)^5) (θ_3) (θ_4)
                                                             
                                                              = θ_1^(3)

2.     4. ((θ_2)^4) ((θ_3)^5) ((θ_4)^2) + ((θ_2)^8) (θ_3) ((θ_4)^2) + 6.(θ_3^(2)) (θ_4) (θ_4^(2))
                 
                                         - 6.(θ_3) ((θ_4^(2))^2) + (θ_3) (θ_4) (θ_4^(4))

                                                              = (θ_3^(4)) ((θ_4)^2)




Somjit Datta, Ph.D
Calcutta, India
October 24, 2018