Asymptotic expansions of integrals and Nielsen's polylogarithms
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arXiv
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| Natura: | Preprint |
| Pubblicazione: |
2026
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| _version_ | 1866917388366118912 |
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| author | Kuba, Markus Levy, Moti |
| author_facet | Kuba, Markus Levy, Moti |
| contents | This article derives full asymptotic expansions for integrals of the form \[ \int_{0}^{1}f(u)(1+q\cdot u^{n})^{w/n}du \] as $n\rightarrow\infty$, with parameters real $w\neq 0$ and $q\in(-1,1]$, or positive $w$ for $q=-1$. We relate the coefficients of the asymptotic expansions to Nielsen's generalized polylogarithms. For $q=-1$, we obtain an expansion in terms of multiple zeta values, which in this setting, reduce to ordinary zeta values. A key point is that for $q=1$, the integrals typically produce alternating multiple zeta values; we formulate a precise symmetry constraint on the relevant coefficient sequence under which all coefficients reduce to polynomials in ordinary zeta values. We also translate this symmetry into a statement about a binomial transform, and we verify the condition for several classical Appell-type families, like Euler, Bernoulli, Genocchi, and Hermite. Finally, we obtain precise results about the convergence of norms of random variables. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2604_05895 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Asymptotic expansions of integrals and Nielsen's polylogarithms Kuba, Markus Levy, Moti Number Theory Combinatorics 11M32, 33B30, 60C05 This article derives full asymptotic expansions for integrals of the form \[ \int_{0}^{1}f(u)(1+q\cdot u^{n})^{w/n}du \] as $n\rightarrow\infty$, with parameters real $w\neq 0$ and $q\in(-1,1]$, or positive $w$ for $q=-1$. We relate the coefficients of the asymptotic expansions to Nielsen's generalized polylogarithms. For $q=-1$, we obtain an expansion in terms of multiple zeta values, which in this setting, reduce to ordinary zeta values. A key point is that for $q=1$, the integrals typically produce alternating multiple zeta values; we formulate a precise symmetry constraint on the relevant coefficient sequence under which all coefficients reduce to polynomials in ordinary zeta values. We also translate this symmetry into a statement about a binomial transform, and we verify the condition for several classical Appell-type families, like Euler, Bernoulli, Genocchi, and Hermite. Finally, we obtain precise results about the convergence of norms of random variables. |
| title | Asymptotic expansions of integrals and Nielsen's polylogarithms |
| topic | Number Theory Combinatorics 11M32, 33B30, 60C05 |
| url | https://arxiv.org/abs/2604.05895 |