论文标题
Lerch的$φ$和正整数的Polyogarithm
Lerch's $Φ$ and the Polylogarithm at the Positive Integers
论文作者
论文摘要
We review the closed-forms of the partial Fourier sums associated with $HP_k(n)$ and create an asymptotic expression for $HP(n)$ as a way to obtain formulae for the full Fourier series (if $b$ is such that $|b|<1$, we get a surprising pattern, $HP(n) \sim H(n)-\sum_{k\ge 2}（-1）^kζ（k）b^{k-1} $）。最后，我们使用找到的傅立叶系列公式来获得LERCH超越函数的值，$φ（E^M，K，B）$，并扩展到Polygarithm，$ \ Mathrm {li} _ {k} _ {k}（e^{m}）$，在正整数$ k $上。
We review the closed-forms of the partial Fourier sums associated with $HP_k(n)$ and create an asymptotic expression for $HP(n)$ as a way to obtain formulae for the full Fourier series (if $b$ is such that $|b|<1$, we get a surprising pattern, $HP(n) \sim H(n)-\sum_{k\ge 2}(-1)^kζ(k)b^{k-1}$). Finally, we use the found Fourier series formulae to obtain the values of the Lerch transcendent function, $Φ(e^m,k,b)$, and by extension the polylogarithm, $\mathrm{Li}_{k}(e^{m})$, at the positive integers $k$.
