Faulhaber's formula, Bernoulli numbers, power sums of natural numbers and totatives and the functional equation $f(x)+x^k=f(x+1)$
Fuente:
arXiv
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| Formato: | Preprint |
| Publicado: |
2024
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| _version_ | 1866914514675433472 |
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| author | Wu, Chai Wah |
| author_facet | Wu, Chai Wah |
| contents | In modern usage the Bernoulli numbers and Bernoulli polynomials follow Euler's approach and are defined using generating functions. We consider the functional equation $f(x)+x^k=f(x+1)$ and show that a solution can be derived from Faulhaber's formula for the sum of powers that provide a characterization of Bernoulli numbers and related results. We then use these results to study sums of powers of totatives of $n$ that are less than $\frac{n}{2}$. In particular, we show that, like the case of the sum of powers of all totatives, the sum of powers of this half of the totatives can also be expressed as a linear combinations of Dirichlet inverses of Jordan totients of odd degrees. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2403_00760 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Faulhaber's formula, Bernoulli numbers, power sums of natural numbers and totatives and the functional equation $f(x)+x^k=f(x+1)$ Wu, Chai Wah Number Theory Combinatorics History and Overview 11B68, 11A05 In modern usage the Bernoulli numbers and Bernoulli polynomials follow Euler's approach and are defined using generating functions. We consider the functional equation $f(x)+x^k=f(x+1)$ and show that a solution can be derived from Faulhaber's formula for the sum of powers that provide a characterization of Bernoulli numbers and related results. We then use these results to study sums of powers of totatives of $n$ that are less than $\frac{n}{2}$. In particular, we show that, like the case of the sum of powers of all totatives, the sum of powers of this half of the totatives can also be expressed as a linear combinations of Dirichlet inverses of Jordan totients of odd degrees. |
| title | Faulhaber's formula, Bernoulli numbers, power sums of natural numbers and totatives and the functional equation $f(x)+x^k=f(x+1)$ |
| topic | Number Theory Combinatorics History and Overview 11B68, 11A05 |
| url | https://arxiv.org/abs/2403.00760 |