Lerch's $Φ$ and the Polylogarithm at the Positive Integers
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arXiv
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| Natura: | Preprint |
| Pubblicazione: |
2020
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| _version_ | 1866914620946513920 |
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| author | Sousa, Jose Risomar |
| author_facet | Sousa, Jose Risomar |
| contents | We review the closed forms of the partial Fourier sums associated with $\HP_k(n)$ from a previous paper and create an asymptotic expression for $\HP(n)$ as a way to obtain formulae for the full Fourier series (if $|b|<1$, one obtains a surprising pattern, $\HP(n) \sim H(n)-\sum_{k\ge 2}(-1)^kζ(k)b^{k-1}$). Finally, the derived Fourier series formulae are used to obtain a formula for the Lerch transcendent function, $Φ(e^z,k,b)$, and by extension the polylogarithm, $\mathrm{Li}_{k}(e^{z})$, at the positive integers $k$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2006_08406 |
| institution | arXiv |
| publishDate | 2020 |
| record_format | arxiv |
| spellingShingle | Lerch's $Φ$ and the Polylogarithm at the Positive Integers Sousa, Jose Risomar Number Theory 11-XX We review the closed forms of the partial Fourier sums associated with $\HP_k(n)$ from a previous paper and create an asymptotic expression for $\HP(n)$ as a way to obtain formulae for the full Fourier series (if $|b|<1$, one obtains a surprising pattern, $\HP(n) \sim H(n)-\sum_{k\ge 2}(-1)^kζ(k)b^{k-1}$). Finally, the derived Fourier series formulae are used to obtain a formula for the Lerch transcendent function, $Φ(e^z,k,b)$, and by extension the polylogarithm, $\mathrm{Li}_{k}(e^{z})$, at the positive integers $k$. |
| title | Lerch's $Φ$ and the Polylogarithm at the Positive Integers |
| topic | Number Theory 11-XX |
| url | https://arxiv.org/abs/2006.08406 |