Permutation symmetry for theta functions

AbstractThis paper does for combinations of theta functions most of what Carlson (2004) [1] did for Jacobian elliptic functions. In each case the starting point is the symmetric elliptic integral RF of the first kind. Its three arguments (formerly squared Jacobian elliptic functions but now squared combinations of theta functions) differ by constants. Symbols designating the constants can often be used to replace 12 equations by three with permutation symmetry (formerly in the letters c, d, n for the Jacobian case but now in the subscripts 2, 3, 4 for theta functions). Such equations include derivatives and differential equations, bisection and duplication relations, addition formulas (apparently new for theta functions), and an example of pseudoaddition formulas