Certain infinite products in terms of MacMahon type series
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arXiv
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| Natura: | Preprint |
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2024
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| _version_ | 1866911946510434304 |
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| author | Jin, Seokho Pandey, Badri Vishal Singh, Ajit |
| author_facet | Jin, Seokho Pandey, Badri Vishal Singh, Ajit |
| contents | Recently, Ono and the third author discovered that the reciprocals of the theta series $(q;q)_\infty^3$ and $(q^2;q^2)_\infty(q;q^2)_\infty^2$ have infinitely many closed formulas in terms of MacMahon's quasimodular forms $A_k(q)$ and $C_k(q)$. In this article, we use the well-known infinite product identities due to Jacobi, Watson, and Hirschhorn to derive further such closed formulas for reciprocals of other interesting infinite products. Moreover, with these formulas, we approximate these reciprocals to arbitrary order simply using MacMahon's functions and {\it MacMahon type} functions. For example, let $Θ_{6}(q):=\frac{1}{2}\sum_{n\in\mathbb{Z}} χ_6(n) n q^{\frac{n^2-1}{24}}$ be the theta function corresponding to the odd quadratic character modulo $6$. Then for any positive integer $n$, we have $$\frac{1}{Θ_{6}(q)}= q^{-\frac{3n^2+n}{2}}\sum_{\substack{k=r_1\\ k\equiv n\hspace{-0.2cm}\pmod{2}}}^{r_2}(-1)^{\frac{n-k}{2}}A_{k}(q)C_{\frac{3n-k}{2}}(q)+O(q^{n+1}),$$ where $r_1:=\lfloor\frac{3n-1-\sqrt{12n+13}}{3}\rfloor+1$ and $r_2:=\lceil\frac{3n-1+\sqrt{12n+13}}{3}\rceil-1$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2407_04798 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Certain infinite products in terms of MacMahon type series Jin, Seokho Pandey, Badri Vishal Singh, Ajit Number Theory 05A17, 11P81 Recently, Ono and the third author discovered that the reciprocals of the theta series $(q;q)_\infty^3$ and $(q^2;q^2)_\infty(q;q^2)_\infty^2$ have infinitely many closed formulas in terms of MacMahon's quasimodular forms $A_k(q)$ and $C_k(q)$. In this article, we use the well-known infinite product identities due to Jacobi, Watson, and Hirschhorn to derive further such closed formulas for reciprocals of other interesting infinite products. Moreover, with these formulas, we approximate these reciprocals to arbitrary order simply using MacMahon's functions and {\it MacMahon type} functions. For example, let $Θ_{6}(q):=\frac{1}{2}\sum_{n\in\mathbb{Z}} χ_6(n) n q^{\frac{n^2-1}{24}}$ be the theta function corresponding to the odd quadratic character modulo $6$. Then for any positive integer $n$, we have $$\frac{1}{Θ_{6}(q)}= q^{-\frac{3n^2+n}{2}}\sum_{\substack{k=r_1\\ k\equiv n\hspace{-0.2cm}\pmod{2}}}^{r_2}(-1)^{\frac{n-k}{2}}A_{k}(q)C_{\frac{3n-k}{2}}(q)+O(q^{n+1}),$$ where $r_1:=\lfloor\frac{3n-1-\sqrt{12n+13}}{3}\rfloor+1$ and $r_2:=\lceil\frac{3n-1+\sqrt{12n+13}}{3}\rceil-1$. |
| title | Certain infinite products in terms of MacMahon type series |
| topic | Number Theory 05A17, 11P81 |
| url | https://arxiv.org/abs/2407.04798 |