Permutation symmetry for theta functions (Q631828)

This article does for combinations of theta functions most of what the author [\textit{B.C. Carlson}, ``Symmetry in \(c\), \(d\), \(n\) of Jacobian elliptic functions,'' J. Math. Anal. Appl. 299, No.\,1, 242--253 (2004; Zbl 1061.33019)] did for Jacobian elliptic functions. In each case the starting point is the symmetric elliptic integral \(R_F\) 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 subscript \(2, 3, 4\) 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. The formulas in Section 2.2. are known; the other formulas are concise expressions for old and new theta function formulas. A most interesting paper.