Maximizing weighted sums of binomial coefficients using generalized continued fractions
Fuente:
arXiv
Enregistré dans:
| Auteurs principaux: | , |
|---|---|
| Format: | Preprint |
| Publié: |
2023
|
| Sujets: | |
| Accès en ligne: | |
| Tags: |
Ajouter un tag
Pas de tags, Soyez le premier à ajouter un tag!
|
| _version_ | 1866913367642341376 |
|---|---|
| author | Glasby, S. P. Paseman, G. R. |
| author_facet | Glasby, S. P. Paseman, G. R. |
| contents | Let $m,r\in\mathbb{Z}$ and $ω\in\mathbb{R}$ satisfy $0\leqslant r\leqslant m$ and $ω\geqslant1$. Our main result is a generalized continued fraction for an expression involving the partial binomial sum $s_m(r) = \sum_{i=0}^r\binom{m}{i}$. We apply this to create new upper and lower bounds for $s_m(r)$ and thus for $g_{ω,m}(r)=ω^{-r}s_m(r)$. We also bound an integer $r_0 \in \{0,1,\dots,m\}$ such that $g_{ω,m}(0)<\cdots<g_{ω,m}(r_0-1)\leqslant g_{ω,m}(r_0)$ and $g_{ω,m}(r_0)>\cdots>g_{ω,m}(m)$. For real $ω\geqslant\sqrt3$ we prove that $r_0\in\{\lfloor\frac{m+2}{ω+1}\rfloor,\lfloor\frac{m+2}{ω+1}\rfloor+1\}$, and also $r_0 =\lfloor\frac{m+2}{ω+1}\rfloor$ for $ω\in\{3,4,\dots\}$ or $ω=2$ and $3\nmid m$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2310_12517 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Maximizing weighted sums of binomial coefficients using generalized continued fractions Glasby, S. P. Paseman, G. R. Number Theory 05A10, 11B65, 11Y65 Let $m,r\in\mathbb{Z}$ and $ω\in\mathbb{R}$ satisfy $0\leqslant r\leqslant m$ and $ω\geqslant1$. Our main result is a generalized continued fraction for an expression involving the partial binomial sum $s_m(r) = \sum_{i=0}^r\binom{m}{i}$. We apply this to create new upper and lower bounds for $s_m(r)$ and thus for $g_{ω,m}(r)=ω^{-r}s_m(r)$. We also bound an integer $r_0 \in \{0,1,\dots,m\}$ such that $g_{ω,m}(0)<\cdots<g_{ω,m}(r_0-1)\leqslant g_{ω,m}(r_0)$ and $g_{ω,m}(r_0)>\cdots>g_{ω,m}(m)$. For real $ω\geqslant\sqrt3$ we prove that $r_0\in\{\lfloor\frac{m+2}{ω+1}\rfloor,\lfloor\frac{m+2}{ω+1}\rfloor+1\}$, and also $r_0 =\lfloor\frac{m+2}{ω+1}\rfloor$ for $ω\in\{3,4,\dots\}$ or $ω=2$ and $3\nmid m$. |
| title | Maximizing weighted sums of binomial coefficients using generalized continued fractions |
| topic | Number Theory 05A10, 11B65, 11Y65 |
| url | https://arxiv.org/abs/2310.12517 |