Congruences for sums of MacMahon's $q$-Catalan polynomials
Fuente:
arXiv
Saved in:
| Main Authors: | , |
|---|---|
| Format: | Preprint |
| Published: |
2024
|
| Subjects: | |
| Online Access: | |
| Tags: |
Add Tag
No Tags, Be the first to tag this record!
|
| _version_ | 1866914837989163008 |
|---|---|
| author | Amdeberhan, Tewodros Tauraso, Roberto |
| author_facet | Amdeberhan, Tewodros Tauraso, Roberto |
| contents | One variant of the $q$-Catalan polynomials is defined in terms of Gaussian polynomials by $\mathcal{C}_k(q)=\genfrac{[}{]}{0pt}{}{2k}{k}_q-q\genfrac{[}{]}{0pt}{}{2k}{k+1}_q$. Liu studied congruences of the form $\sum_{k=0}^{n-1} q^k\mathcal{C}_k$ modulo the cyclotomic polynomial $Φ_n(q)^2$, provided that $n\equiv\pm 1\pmod3$. Apparently the case $n\equiv 0\pmod3$ has been missing from the literature. It is our primary purpose to fill this gap by the current work. In addition, we discuss certain fascinating link to Dirichlet character sum identities. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2406_12332 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Congruences for sums of MacMahon's $q$-Catalan polynomials Amdeberhan, Tewodros Tauraso, Roberto Number Theory Combinatorics 11B65, 11A07, 05A10, 05A19 One variant of the $q$-Catalan polynomials is defined in terms of Gaussian polynomials by $\mathcal{C}_k(q)=\genfrac{[}{]}{0pt}{}{2k}{k}_q-q\genfrac{[}{]}{0pt}{}{2k}{k+1}_q$. Liu studied congruences of the form $\sum_{k=0}^{n-1} q^k\mathcal{C}_k$ modulo the cyclotomic polynomial $Φ_n(q)^2$, provided that $n\equiv\pm 1\pmod3$. Apparently the case $n\equiv 0\pmod3$ has been missing from the literature. It is our primary purpose to fill this gap by the current work. In addition, we discuss certain fascinating link to Dirichlet character sum identities. |
| title | Congruences for sums of MacMahon's $q$-Catalan polynomials |
| topic | Number Theory Combinatorics 11B65, 11A07, 05A10, 05A19 |
| url | https://arxiv.org/abs/2406.12332 |