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

Janet J. W. Dong; Kathy Q. Ji

paper

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

arXiv:2307.02013

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 $M_{k} (n - 1) + M_{k} (n + 1) > 2 M_{k} (n)$ 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 Turán inequalities for $n \geq 207$ with the aid of the upper bound and the lower bound for $M_{0} (n)$ and $M_{1} (n)$ .

29 pages