Binomial sums and properties of the Bernoulli transform
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arXiv
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| Autori principali: | , , |
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| Natura: | Preprint |
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2026
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| author | Elkhiri, Laid Mihoubi, Miloud Moulay, Meriem |
| author_facet | Elkhiri, Laid Mihoubi, Miloud Moulay, Meriem |
| contents | In this paper, we study the binomial sum $S_{n}(q):=% \overset{n}{\underset{k=0}{\sum }}a_{k}\binom{n}{k}\left( 1-q\right) ^{k}q^{n-k}$ for a given sequence $\left( a_{n}\right) $ of real or complex numbers. We express $S_{n}(q)$ in function of the powers of $q,$ and, we explicit it when the sequence $\left( a_{n}\right) $ is the sequence of Fibonacci numbers, Laguerre polynomials, Meixner polynomials, binomial coefficients and the sequence $\left[ n\right] _{p}.$ We establish later some properties, relations, probabilistic interpretations and generating functions between $S_{n}(q)$ and $S_{n}(x+q-xq).$ Further identities related to Appell polynomials are also given in the last of the paper. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2602_16096 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Binomial sums and properties of the Bernoulli transform Elkhiri, Laid Mihoubi, Miloud Moulay, Meriem Number Theory Combinatorics 11B50, 11C08 (Primary) 11B39, 11B83 (Secondary) G.1.1; G.2.1 In this paper, we study the binomial sum $S_{n}(q):=% \overset{n}{\underset{k=0}{\sum }}a_{k}\binom{n}{k}\left( 1-q\right) ^{k}q^{n-k}$ for a given sequence $\left( a_{n}\right) $ of real or complex numbers. We express $S_{n}(q)$ in function of the powers of $q,$ and, we explicit it when the sequence $\left( a_{n}\right) $ is the sequence of Fibonacci numbers, Laguerre polynomials, Meixner polynomials, binomial coefficients and the sequence $\left[ n\right] _{p}.$ We establish later some properties, relations, probabilistic interpretations and generating functions between $S_{n}(q)$ and $S_{n}(x+q-xq).$ Further identities related to Appell polynomials are also given in the last of the paper. |
| title | Binomial sums and properties of the Bernoulli transform |
| topic | Number Theory Combinatorics 11B50, 11C08 (Primary) 11B39, 11B83 (Secondary) G.1.1; G.2.1 |
| url | https://arxiv.org/abs/2602.16096 |