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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

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.

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

Abstract

Recently, Glasby and Paseman considered the following sequence of binomial sums and showed that this sequence is unimodal and attains its maximum value at for . They also analyzed the asymptotic behavior of the maximum value of the sequence as approaches infinity. In the present work, we generalize their results by considering the sequence for integers . We also consider a family of discrete probability distributions that naturally arises from this sequence.
Paper Structure (10 sections, 16 theorems, 110 equations, 1 figure, 2 tables)

This paper contains 10 sections, 16 theorems, 110 equations, 1 figure, 2 tables.

Table of Contents

  1. Introduction
  2. Log-concavity of $\{f_{m,a}(r)\}_{r=0}^{m}$
  3. Peak location for $\{f_{m,a}(r)\}_{r=0}^{m}$
  4. Proof of Proposition \ref{['prop:firstinequality']}
  5. Proof of Proposition \ref{['prop:secondinequality']}
  6. Proof of Lemma \ref{['lem:longlemma']}
  7. Proof of Theorem $1.1(b)$
  8. Maximal value of $\{f_{m,a}(r)\}_{r=0}^{m}$
  9. The weighted average of $\{f_{m,a}(r)\}_{r=0}^{m}$
  10. Discussion

Key Result

Theorem 1.1

Let $a \in \mathbb{R}_{>0}$ and $m \in {\mathbb{Z}}_{\geq 2}$. Then Moreover, if $a \in {\mathbb{Z}}_{\geq 1}$, then

Figures (1)

  • Figure 1: An illustration of the logic of the proof of Proposition \ref{['prop:secondinequality']}.

Theorems & Definitions (33)

  • Theorem 1.1
  • Lemma 2.1
  • proof
  • proof : Proof of Theorem \ref{['1.1']} \ref{['1.1.(a)']}
  • Proposition 3.1
  • Proposition 3.2
  • Lemma 3.3
  • proof
  • Lemma 3.4
  • proof
  • ...and 23 more