Unimodality of ordinary multinomials and maximal probabilities of convolution powers of discrete uniform distribution

TL;DR

This paper proves the unimodality of ordinary multinomials and their asymptotic properties, deriving formulas for maximal probabilities in convolution powers of discrete uniform distributions, and provides generating functions for related sequences.

Contribution

It introduces new proofs of unimodality and asymptotic strong unimodality for ordinary multinomials, and derives explicit expressions for maximal convolution probabilities.

Findings

Proved unimodality and asymptotic strong unimodality of ordinary multinomials.

Derived formulas for the smallest mode and maximal probabilities.

Provided generating functions for generalized sequences.

Abstract

We establish the unimodality and the asymptotic strong unimodality of the ordinary multinomials and give their smallest mode leading to the expression of the maximal probability of convolution powers of the discrete uniform distribution. We conclude giving the generating functions of the sequence of generalized ordinary multinomials and for an extension of the sequence of maximal probabilities for convolution power of discrete uniform distribution.