Remarks on MacMahon's $q$-series
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arXiv
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| Autori principali: | , |
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| Natura: | Preprint |
| Pubblicazione: |
2024
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| _version_ | 1866911879453999104 |
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| author | Ono, Ken Singh, Ajit |
| author_facet | Ono, Ken Singh, Ajit |
| contents | In his important 1920 paper on partitions, MacMahon defined the partition generating functions \begin{align*} A_k(q)=\sum_{n=1}^{\infty}\mathfrak{m}(k;n)q^n&:=\sum_{0< s_1<s_2<\cdots<s_k} \frac{q^{s_1+s_2+\cdots+s_k}}{(1-q^{s_1})^2(1-q^{s_2})^2\cdots(1-q^{s_k})^2},\\ C_k(q)=\sum_{n=1}^{\infty} \mathfrak{m}_{odd}(k;n)q^n&:=\sum_{0< s_1<s_2<\cdots<s_k} \frac{q^{2s_1+2s_2+\cdots+2s_k-k}}{(1-q^{2s_1-1})^2(1-q^{2s_2-1})^2\cdots(1-q^{2s_k-1})^2}. \end{align*} These series give infinitely many formulas for two prominent generating functions. For each non-negative $k$, we prove that $A_k(q), A_{k+1}(q), A_{k+2}(q),\dots$ (resp. $C_k(q), C_{k+1}(q), C_{k+2}(q),\dots$) give the generating function for the 3-colored partition function $p_3(n)$ (resp. the overpartition function $\overline{p}(n)$). |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2402_08783 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Remarks on MacMahon's $q$-series Ono, Ken Singh, Ajit Combinatorics 05A17 In his important 1920 paper on partitions, MacMahon defined the partition generating functions \begin{align*} A_k(q)=\sum_{n=1}^{\infty}\mathfrak{m}(k;n)q^n&:=\sum_{0< s_1<s_2<\cdots<s_k} \frac{q^{s_1+s_2+\cdots+s_k}}{(1-q^{s_1})^2(1-q^{s_2})^2\cdots(1-q^{s_k})^2},\\ C_k(q)=\sum_{n=1}^{\infty} \mathfrak{m}_{odd}(k;n)q^n&:=\sum_{0< s_1<s_2<\cdots<s_k} \frac{q^{2s_1+2s_2+\cdots+2s_k-k}}{(1-q^{2s_1-1})^2(1-q^{2s_2-1})^2\cdots(1-q^{2s_k-1})^2}. \end{align*} These series give infinitely many formulas for two prominent generating functions. For each non-negative $k$, we prove that $A_k(q), A_{k+1}(q), A_{k+2}(q),\dots$ (resp. $C_k(q), C_{k+1}(q), C_{k+2}(q),\dots$) give the generating function for the 3-colored partition function $p_3(n)$ (resp. the overpartition function $\overline{p}(n)$). |
| title | Remarks on MacMahon's $q$-series |
| topic | Combinatorics 05A17 |
| url | https://arxiv.org/abs/2402.08783 |