On the equivalence of two fundamental theta identities. (English) Zbl 1379.11049
Summary: 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 [“A \(q\)-trigonometric identity: solutions of problem 10226”, Am. Math. Mon. 103, No. 2, 175–177 (1996; doi:10.2307/2975119)]. The history and usage of the two identities, and some generalizations are also discussed.
MSC:
| 11F27 | Theta series; Weil representation; theta correspondences |
| 01A55 | History of mathematics in the 19th century |
Keywords:
theta functions; identities involving sums of products of theta functions; history of theta functions; applications of theta functionsSoftware:
DLMFReferences:
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