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A proof of the only mode of a unimodal sequence

Jun Wan, Zuo-Ru Zhang

Abstract

In study the generalized Jacobsthal and Jaco-Lucas polynomials, Sun introduced the interesting numerical triangle Jaco-Lucas sequence $\{JL_{n,k}\}_{n\geq k\geq0}$. In this paper, we proved this sequence is log-concave with the only mode by computer algebra.

A proof of the only mode of a unimodal sequence

Abstract

In study the generalized Jacobsthal and Jaco-Lucas polynomials, Sun introduced the interesting numerical triangle Jaco-Lucas sequence . In this paper, we proved this sequence is log-concave with the only mode by computer algebra.
Paper Structure (5 sections, 12 theorems, 48 equations, 1 table)

This paper contains 5 sections, 12 theorems, 48 equations, 1 table.

Table of Contents

  1. Introduction
  2. The proof of log-concavity
  3. Recurrence relations of $JL_{n,k}$
  4. Bounds of $\frac{JL_{n+1,k}}{JL_{n,k}}$
  5. The mode of the sequence

Key Result

Theorem 1.1

For any positive integers $n$ and $k$ with $n\geq 2k> 0$, $JL_{n,k}$ is log-concavity and strictly increasing before the only mode $k^*=\lfloor \frac{n-4}{6} \rfloor+1$.

Theorems & Definitions (12)

  • Theorem 1.1
  • Lemma 2.1
  • Lemma 2.2
  • Lemma 2.3
  • Theorem 2.4
  • Theorem 3.1
  • Lemma 3.2
  • Proposition 3.3
  • Lemma 3.4
  • Lemma 3.5
  • ...and 2 more