$q$-analogues of sums of consecutive powers of natural numbers and extended Carlitz $q$-Bernoulli numbers and polynomials
Fuente:
arXiv
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| Natura: | Preprint |
| Pubblicazione: |
2025
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| _version_ | 1866915091969998848 |
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| author | Farhi, Bakir |
| author_facet | Farhi, Bakir |
| contents | In this paper, we investigate a specific class of $q$-polynomial sequences that serve as a $q$-analogue of the classical Appell sequences. This framework offers an elegant approach to revisiting classical results by Carlitz and, more interestingly, to establishing an important extension of the Carlitz $q$-Bernoulli polynomials and numbers. In addition, we establish explicit series representations for our extended Carlitz $q$-Bernoulli numbers and express them in terms of $q$-Stirling numbers of the second kind. This leads to a novel formula that explicitly connects the Carlitz $q$-Bernoulli numbers with the $q$-Stirling numbers of the second kind. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2501_02499 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | $q$-analogues of sums of consecutive powers of natural numbers and extended Carlitz $q$-Bernoulli numbers and polynomials Farhi, Bakir Number Theory Combinatorics Primary 05A30, Secondary 11B68, 11B73, 05A40 In this paper, we investigate a specific class of $q$-polynomial sequences that serve as a $q$-analogue of the classical Appell sequences. This framework offers an elegant approach to revisiting classical results by Carlitz and, more interestingly, to establishing an important extension of the Carlitz $q$-Bernoulli polynomials and numbers. In addition, we establish explicit series representations for our extended Carlitz $q$-Bernoulli numbers and express them in terms of $q$-Stirling numbers of the second kind. This leads to a novel formula that explicitly connects the Carlitz $q$-Bernoulli numbers with the $q$-Stirling numbers of the second kind. |
| title | $q$-analogues of sums of consecutive powers of natural numbers and extended Carlitz $q$-Bernoulli numbers and polynomials |
| topic | Number Theory Combinatorics Primary 05A30, Secondary 11B68, 11B73, 05A40 |
| url | https://arxiv.org/abs/2501.02499 |