Convexity and log-concavity of the partition function weighted by the parity of the crank

TL;DR

This paper investigates the convexity and log-concavity properties of partition functions weighted by crank parity, providing new inequalities and bounds that refine classical results in partition theory.

Contribution

It establishes new convexity and log-concavity results for crank-weighted partition functions, extending classical convexity results for the partition function p(n).

Findings

Proves a convexity inequality for M_k(n) for n ≥ 39.

Shows M_0(n) and M_1(n) are log-concave for n ≥ 94.

Demonstrates higher order Turán inequalities hold for n ≥ 207.

Abstract

Let $M_0(n)$ (resp. $M_1(n)$) denote the number of partitions of $n$ with even (reps. odd) crank. Choi, Kang and Lovejoy established an asymptotic formula for $M_0(n)-M_1(n)$. By utilizing this formula with the explicit bound, we show that $M_k(n-1)+M_k(n+1)>2M_k(n)$ for $k=0$ or $1$ and $n\geq 39$. This result can be seen as the refinement of the classical result regarding the convexity of the partition function $p(n)$, which counts the number of partitions of $n$. We also show that $M_0(n)$ (resp. $M_1(n)$) is log-concave for $n\geq 94$ and satisfies the higher order Tur\'an inequalities for $n\geq 207$ with the aid of the upper bound and the lower bound for $M_0(n)$ and $M_1(n)$.