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arXiv · 2305.16631

On the maximum of the weighted binomial sum $(1+a)^{-r}\sum_{i=0}^{r}\binom{m}{i}a^{i}$

Seok Hyun Byun·Svetlana Poznanović
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Abstract

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.

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Seok Hyun Byun, Svetlana Poznanović. 2023-12-31. On the maximum of the weighted binomial sum $(1+a)^{-r}\sum_{i=0}^{r}\binom{m}{i}a^{i}$. https://arxiv.org/abs/2305.16631

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