Hypergeometric Bernoulli Polynomials Defined on Simplicial $d$-Polytopic Numbers
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arXiv
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| Format: | Preprint |
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2026
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| _version_ | 1866914433009188864 |
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| author | Orozco, Ronald |
| author_facet | Orozco, Ronald |
| contents | We introduce an ${\rm S}_d$-analogue of the hypergeometric Bernoulli polynomials and study their properties. To achieve this goal, we introduce a calculus defined on the simplicial $d$-polytopic numbers. Two definitions of the ${\rm S}_d$-derivatives are given. These two definitions allow us to derive an identity relating Kummer confluent hypergeometric function and Touchard polynomials. This calculus is closely related to the $d$-Hoggatt binomial coefficients. ${\rm S}_d$-analogs of the exponential function and the hypergeometric functions are given. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2603_28940 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Hypergeometric Bernoulli Polynomials Defined on Simplicial $d$-Polytopic Numbers Orozco, Ronald Combinatorics Number Theory 11B68, 33C15 We introduce an ${\rm S}_d$-analogue of the hypergeometric Bernoulli polynomials and study their properties. To achieve this goal, we introduce a calculus defined on the simplicial $d$-polytopic numbers. Two definitions of the ${\rm S}_d$-derivatives are given. These two definitions allow us to derive an identity relating Kummer confluent hypergeometric function and Touchard polynomials. This calculus is closely related to the $d$-Hoggatt binomial coefficients. ${\rm S}_d$-analogs of the exponential function and the hypergeometric functions are given. |
| title | Hypergeometric Bernoulli Polynomials Defined on Simplicial $d$-Polytopic Numbers |
| topic | Combinatorics Number Theory 11B68, 33C15 |
| url | https://arxiv.org/abs/2603.28940 |