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arXiv · 2412.06081

Extended Landen's formulas and double product theta functions

Kiyoshi Sogo
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Abstract

For an arbitrary positive integer $p$, Landen's formula is extended to express theta function with modulus $p\tau$ by $p$ product of theta functions with $\tau$, which is applied to several examples. Next it is shown that double product of theta functions of genus $g=1$ is written by a sum of $g=2$ theta functions, which is a subset having a special period matrix of $\tau_{11}=\tau_{22}$. Several applied examples are shown, which include the cubic identity of Borwein and Borwein.

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Kiyoshi Sogo. 2024-12-08. Extended Landen's formulas and double product theta functions. https://arxiv.org/abs/2412.06081

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