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Journal Article

The second shifted difference of partitions and its applications

MPS-Authors
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Rolen,  Larry
Max Planck Institute for Mathematics, Max Planck Society;

External Resource

https://doi.org/10.1017/S0004972722000764
(Publisher version)

https://doi.org/10.48550/arXiv.2203.11608
(Preprint)

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Citation

Gomez, K., Males, J., & Rolen, L. (2023). The second shifted difference of partitions and its applications. Bulletin of the Australian Mathematical Society, 107(1), 66-78. doi:10.1017/S0004972722000764.


Cite as: https://hdl.handle.net/21.11116/0000-000B-07A0-8
Abstract
A number of recent papers have estimated ratios of the partition function
$p(n-j)/p(n)$, which appears in many applications. Here, we prove an
easy-to-use effective bound on these ratios. Using this, we then study second
shifted difference of partitions, $f(j,n):= p(n) -2p(n-j) +p(n-2j)$, and give
another easy-to-use estimate of $f(j,n)$. As applications of these, we prove a
shifted convexity property of $p(n)$, as well as giving new estimates of the
$k$-rank partition function $N_k(m,n)$ and non-$k$-ary partitions along with
their differences.