On the maximum of the weighted binomial sum $(1+a)^{-r}\sum_{i=0}^{r}\binom{m}{i}a^{i}$
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arXiv
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| Format: | Preprint |
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2023
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| _version_ | 1866910284093849600 |
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| author | Byun, Seok Hyun Poznanović, Svetlana |
| author_facet | Byun, Seok Hyun Poznanović, Svetlana |
| contents | Recently, Glasby and Paseman considered the following sequence of binomial sums $\{2^{-r}\sum_{i=0}^{r}\binom{m}{i}\}_{r=0}^{m}$ and showed that this sequence is unimodal and attains its maximum value at $r=\lfloor\frac{m}{3}\rfloor+1$ for $m\in\mathbb{Z}_{\geq0}\setminus\{0,3,6,9,12\}$. They also analyzed the asymptotic behavior of the maximum value of the sequence as $m$ approaches infinity. In the present work, we generalize their results by considering the sequence $\{(1+a)^{-r}\sum_{i=0}^{r}\binom{m}{i}a^{i}\}_{r=0}^{m}$ for integers $a \geq 1$. We also consider a family of discrete probability distributions that naturally arises from this sequence. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2305_16631 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | On the maximum of the weighted binomial sum $(1+a)^{-r}\sum_{i=0}^{r}\binom{m}{i}a^{i}$ Byun, Seok Hyun Poznanović, Svetlana Combinatorics 05A10 Recently, Glasby and Paseman considered the following sequence of binomial sums $\{2^{-r}\sum_{i=0}^{r}\binom{m}{i}\}_{r=0}^{m}$ and showed that this sequence is unimodal and attains its maximum value at $r=\lfloor\frac{m}{3}\rfloor+1$ for $m\in\mathbb{Z}_{\geq0}\setminus\{0,3,6,9,12\}$. They also analyzed the asymptotic behavior of the maximum value of the sequence as $m$ approaches infinity. In the present work, we generalize their results by considering the sequence $\{(1+a)^{-r}\sum_{i=0}^{r}\binom{m}{i}a^{i}\}_{r=0}^{m}$ for integers $a \geq 1$. We also consider a family of discrete probability distributions that naturally arises from this sequence. |
| title | On the maximum of the weighted binomial sum $(1+a)^{-r}\sum_{i=0}^{r}\binom{m}{i}a^{i}$ |
| topic | Combinatorics 05A10 |
| url | https://arxiv.org/abs/2305.16631 |