A polynomial approach to Carlitz's $q$-Bernoulli numbers
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arXiv
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| Auteurs principaux: | , |
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| Format: | Preprint |
| Publié: |
2025
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| _version_ | 1866912504638078976 |
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| author | Mouzaia, Mohamed Farhi, Bakir |
| author_facet | Mouzaia, Mohamed Farhi, Bakir |
| contents | This paper investigates $q$-analogues of the classical Bernoulli polynomials and numbers. We introduce a new polynomial sequence ${\left(B_{n , q}(X)\right)}_{n \in \mathbb{N}_0}$, defined via the Jackson integral, and explore its connections with Carlitz's $q$-Bernoulli polynomials and numbers. Specifically, we prove that the numbers $B_{n , q}(0)$ are exactly the Carlitz $q$-Bernoulli numbers and that the polynomials $B_{n , q}(X)$ are genuine $q$-analogues of the classical Bernoulli polynomials. This approach leverages the Jackson integral to reformulate Carlitz's $q$-Bernoulli numbers in terms of classical polynomial structures, offering new insights into their properties. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2507_20384 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | A polynomial approach to Carlitz's $q$-Bernoulli numbers Mouzaia, Mohamed Farhi, Bakir Number Theory Primary 05A30, Secondary 11B68 This paper investigates $q$-analogues of the classical Bernoulli polynomials and numbers. We introduce a new polynomial sequence ${\left(B_{n , q}(X)\right)}_{n \in \mathbb{N}_0}$, defined via the Jackson integral, and explore its connections with Carlitz's $q$-Bernoulli polynomials and numbers. Specifically, we prove that the numbers $B_{n , q}(0)$ are exactly the Carlitz $q$-Bernoulli numbers and that the polynomials $B_{n , q}(X)$ are genuine $q$-analogues of the classical Bernoulli polynomials. This approach leverages the Jackson integral to reformulate Carlitz's $q$-Bernoulli numbers in terms of classical polynomial structures, offering new insights into their properties. |
| title | A polynomial approach to Carlitz's $q$-Bernoulli numbers |
| topic | Number Theory Primary 05A30, Secondary 11B68 |
| url | https://arxiv.org/abs/2507.20384 |