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| Main Authors: | , , , |
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| Format: | Preprint |
| Published: |
2023
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| Subjects: | |
| Online Access: | https://arxiv.org/abs/2311.06489 |
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| _version_ | 1866909339389788160 |
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| author | Hasegawa, Takehiro Saigo, Hayato Saito, Seiken Sugiyama, Shingo |
| author_facet | Hasegawa, Takehiro Saigo, Hayato Saito, Seiken Sugiyama, Shingo |
| contents | We extend a certain type of identities on sums of $I$-Bessel functions on lattices, previously given by G. Chinta, J. Jorgenson, A. Karlsson and M. Neuhauser. Moreover we prove that, with continuum limit, the transformation formulas of theta functions such as the Dedekind eta function can be given by $I$-Bessel lattice sum identities with characters. We consider analogues of theta functions of lattices coming from linear codes and show that sums of $I$-Bessel functions defined by linear codes can be expressed by complete weight enumerators. We also prove that $I$-Bessel lattice sums appear as solutions of heat equations on general lattices. As a further application, we obtain an explicit solution of the heat equation on $\mathbb{Z}^n$ whose initial condition is given by a linear code. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2311_06489 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Lattice sums of $I$-Bessel functions, theta functions, linear codes and heat equations Hasegawa, Takehiro Saigo, Hayato Saito, Seiken Sugiyama, Shingo Mathematical Physics Information Theory Number Theory Primary 33C10, Secondary 11F27, 94B05, 35K05 We extend a certain type of identities on sums of $I$-Bessel functions on lattices, previously given by G. Chinta, J. Jorgenson, A. Karlsson and M. Neuhauser. Moreover we prove that, with continuum limit, the transformation formulas of theta functions such as the Dedekind eta function can be given by $I$-Bessel lattice sum identities with characters. We consider analogues of theta functions of lattices coming from linear codes and show that sums of $I$-Bessel functions defined by linear codes can be expressed by complete weight enumerators. We also prove that $I$-Bessel lattice sums appear as solutions of heat equations on general lattices. As a further application, we obtain an explicit solution of the heat equation on $\mathbb{Z}^n$ whose initial condition is given by a linear code. |
| title | Lattice sums of $I$-Bessel functions, theta functions, linear codes and heat equations |
| topic | Mathematical Physics Information Theory Number Theory Primary 33C10, Secondary 11F27, 94B05, 35K05 |
| url | https://arxiv.org/abs/2311.06489 |