A Fourier analysis approach to disprove the weak Shanks conjecture
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arXiv
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| Format: | Preprint |
| Published: |
2025
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| _version_ | 1866914000585883648 |
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| author | Geronimo, Jeffrey S. Woerdeman, Hugo J. |
| author_facet | Geronimo, Jeffrey S. Woerdeman, Hugo J. |
| contents | We derive formulas for the Fourier coefficients of $|f|^2$, where $f(z_1,z_2)=(1-\frac{z_1+z_2}{r})^{-α}$, in terms of hypergeometric functions. Using these formulas we provide additional counterexamples to the weak Shanks conjecture, which was recently disproven by Bénéteau, Khavinson and Seco. The obtained formulas allow for (numerical) optimization over the parameters $α$ and $r$. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2508_15938 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | A Fourier analysis approach to disprove the weak Shanks conjecture Geronimo, Jeffrey S. Woerdeman, Hugo J. Complex Variables Classical Analysis and ODEs Functional Analysis Primary 42B05, Secondary 32A60, 32A70 We derive formulas for the Fourier coefficients of $|f|^2$, where $f(z_1,z_2)=(1-\frac{z_1+z_2}{r})^{-α}$, in terms of hypergeometric functions. Using these formulas we provide additional counterexamples to the weak Shanks conjecture, which was recently disproven by Bénéteau, Khavinson and Seco. The obtained formulas allow for (numerical) optimization over the parameters $α$ and $r$. |
| title | A Fourier analysis approach to disprove the weak Shanks conjecture |
| topic | Complex Variables Classical Analysis and ODEs Functional Analysis Primary 42B05, Secondary 32A60, 32A70 |
| url | https://arxiv.org/abs/2508.15938 |