Elliptic function based algorithms to prove Jacobi theta function relations. (November 2018) - Ye, Liangjie
Abstract: In this paper we prove identities involving the classical Jacobi theta functions of the form ∑ c ( i 1, i 2, i 3, i 4 ) θ 1 ( z | τ ) i 1 θ 2 ( z | τ ) i 2 θ 3 ( z | τ ) i 3 θ 4 ( z | τ ) i 4 = 0 with c ( i 1, i 2, i 3, i 4 ) ∈ K [ Θ ], where K is a computable field and Θ : = { θ 1 ( 2 k + 1 ) ( 0 | τ ) : k ∈ N } ∪ { θ j ( 2 k ) ( 0 | τ ) : k ∈ N  and  j = 2, 3, 4 } . We give two algorithms that solve this problem. The second algorithm is simpler and works in a restricted input class.