Recurrence Relations for k-Fold Nested Power Sums
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866917093642862592 |
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| author | Povolotsky, Alexander R. |
| author_facet | Povolotsky, Alexander R. |
| contents | We consider the $k$-nested sum of integer powers, $F(n,m,k)$, defined as repeated partial sums of the classical Faulhaber polynomials. We provide an explicit recurrence relation relating $F(n,m,k)$ to sums of lower power $m-1$ and higher nesting level $k+1$. This identity is derived from a core algebraic relation on the binomial coefficients that form the kernel of the nested sum's representation. We discuss the relevance to the 2010 paper by S.~Butler and P.~Karasik, ``A Note on Nested Sums'' (JIS, Vol.~13, Article~10.4.4), which studies nested sums of powers of integers that generalize Faulhaber-type sums. We also discuss the equivalence to a related recurrence previously established in the context of hypersums of powers of integers by J.~L.~Cereceda. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2511_15729 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Recurrence Relations for k-Fold Nested Power Sums Povolotsky, Alexander R. Combinatorics Number Theory We consider the $k$-nested sum of integer powers, $F(n,m,k)$, defined as repeated partial sums of the classical Faulhaber polynomials. We provide an explicit recurrence relation relating $F(n,m,k)$ to sums of lower power $m-1$ and higher nesting level $k+1$. This identity is derived from a core algebraic relation on the binomial coefficients that form the kernel of the nested sum's representation. We discuss the relevance to the 2010 paper by S.~Butler and P.~Karasik, ``A Note on Nested Sums'' (JIS, Vol.~13, Article~10.4.4), which studies nested sums of powers of integers that generalize Faulhaber-type sums. We also discuss the equivalence to a related recurrence previously established in the context of hypersums of powers of integers by J.~L.~Cereceda. |
| title | Recurrence Relations for k-Fold Nested Power Sums |
| topic | Combinatorics Number Theory |
| url | https://arxiv.org/abs/2511.15729 |