On the modes of the Poisson distribution of order k

TL;DR

This paper derives improved bounds for the modes of the Poisson distribution of order k and proves a recent conjecture for specific values of k, advancing understanding of its modal properties.

Contribution

It provides sharper bounds for the modes of the Poisson distribution of order k and proves a recent conjecture for k=2,3,4,5, addressing an open problem since 1983.

Findings

Established sharper upper and lower bounds for the modes.

Proved a recent conjecture for k=2,3,4,5.

Solved an open problem from 1983 for specific k values.

Abstract

Sharp upper and lower bounds are established for the modes of the Poisson distribution of order k. The lower bound established in this paper is better than the previously established lower bound. In addition, for k = 2, 3, 4, 5, a recent conjecture is presently proved solving partially an open problem since 1983.