Extended Landen's formulas and double product theta functions
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arXiv
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| Format: | Preprint |
| Publié: |
2024
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| _version_ | 1866916514278408192 |
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| author | Sogo, Kiyoshi |
| author_facet | Sogo, Kiyoshi |
| contents | For an arbitrary positive integer $p$, Landen's formula is extended to express theta function with modulus $pτ$ by $p$ product of theta functions with $τ$, 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 $τ_{11}=τ_{22}$. Several applied examples are shown, which include the cubic identity of Borwein and Borwein. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2412_06081 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Extended Landen's formulas and double product theta functions Sogo, Kiyoshi Mathematical Physics Number Theory 33E05 For an arbitrary positive integer $p$, Landen's formula is extended to express theta function with modulus $pτ$ by $p$ product of theta functions with $τ$, 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 $τ_{11}=τ_{22}$. Several applied examples are shown, which include the cubic identity of Borwein and Borwein. |
| title | Extended Landen's formulas and double product theta functions |
| topic | Mathematical Physics Number Theory 33E05 |
| url | https://arxiv.org/abs/2412.06081 |