On the equivalence of two fundamental theta identities

Abstract

Two fundamental theta identities, a three-term identity due to Weierstrass and a five-term identity due to Jacobi, both with products of four theta functions as terms, are shown to be equivalent. One half of the equivalence was already proved by R.J. Chapman in 1996. The history and usage of the two identities, and some generalizations are also discussed.

v3: 15 pages, minor errors corrected, references added, appendix on four-term theta identities added, accepted by Analysis and Applications

Cited by in corpus (9)

• Theta vocabulary I
• On the tenth-order mock theta functions
• On Ramanujan's lost notebook and new tenth-order like identities for second-, sixth-, and eighth-order mock theta functions
• Zeros of the i.i.d. Gaussian Laurent series on an annulus: weighted Szegő kernels and permanental-determinantal point processes
• Connection problems for quantum affine KZ equations and integrable lattice models
• Splitting Appell functions in terms of single quotients of theta functions
• Theta vocabulary II. Multidimensional case
• Excursion Processes Associated with Elliptic Combinatorics
• Short proofs of Ramanujan-like identities for the eighth order mock theta function $V_0(q)$