论文标题
Eisenstein系列的Eichler积分作为$ Q $ - 加权$ t $ -hook函数的分区功能
Eichler integrals of Eisenstein series as $q$-brackets of weighted $t$-hook functions on partitions
论文作者
论文摘要
我们考虑分区上的$ t $ -hook函数$ f_ {a，t}：\ mathcal {p} \ rightarrow \ rightarrow \ mathbb {c} $由$ f_ f_ {a，t}（λ）（λ）：= t^t^{a-1} \ sum_ {h \ in \ Mathcal {h} _t（λ）} \ frac {1} {h^a}，$$，其中$ \ mathcal {h} _t _t（λ）$是$ t $的倍数的分区挂钩数字的多组。 Bloch-Okounkov $ q $ -brackets $ \ langle f_ {a，t} \ rangle_q $包括经典Eisenstein系列的Eichler积分。对于$ a \ geq 2 $，我们表明这些$ q $ - 支架是天然的重量$ 2-a $ sesquiharmonic和谐波maass表格，而对于奇数$ a \ leq -1，我们表明它们是Holomorphic量子量子模块的形式。我们使用这些结果来获得Chowla-Selberg类型的新公式，以及涉及Riemann Zeta功能和Bernoulli数字值的渐近扩展。我们利用Berndt，Han和Ji和Zagier的作品。
We consider the $t$-hook functions on partitions $f_{a,t}: \mathcal{P}\rightarrow \mathbb{C}$ defined by $$ f_{a,t}(λ):=t^{a-1} \sum_{h\in \mathcal{H}_t(λ)}\frac{1}{h^a}, $$ where $\mathcal{H}_t(λ)$ is the multiset of partition hook numbers that are multiples of $t$. The Bloch-Okounkov $q$-brackets $\langle f_{a,t}\rangle_q$ include Eichler integrals of the classical Eisenstein series. For even $a\geq 2$, we show that these $q$-brackets are natural pieces of weight $2-a$ sesquiharmonic and harmonic Maass forms, while for odd $a\leq -1,$ we show that they are holomorphic quantum modular forms. We use these results to obtain new formulas of Chowla-Selberg type, and asymptotic expansions involving values of the Riemann zeta-function and Bernoulli numbers. We make use of work of Berndt, Han and Ji, and Zagier.
