Asymptotics of the shifted finite differences of the overpatition function and a problem of Wang--Xie--Zhang

2025/12/29 by Mukherjee, Gargi
#05A16 #05A20 #11P82 #FOS: Mathematics #Number Theory (math.NT)

paper · doi:10.48550/arxiv.2512.23679

Abstract

Let p(n) denote the overpartition function, and for j∈ ℕ, Δ^r_j denote the r-fold applications of the shifted difference operator Δ_j defined by Δ_j(a)(n):=a(n)-a(n-j). The main goal of this paper is to derive an asymptotic expansion of Δ^r_j(p)(n) with an effective error bound which subsequently gives an answer to a problem of Wang, Xie, and Zhang. In order to get the asymptotics of Δ^r_j(p)(n), we derive an asymptotic expansion of the shifted overpartition function p(n+k) for any integer k≠ 0.

Citations

• Detailed asymptotic expansions for partitions into powers
• Higher order log-concavity of the overpartition function and its consequences
• Inequalities for higher order differences of the logarithm of the overpartition function and a problem of Wang-Xie-Zhang
• The second shifted difference of partitions and its applications
• THE SECOND SHIFTED DIFFERENCE OF PARTITIONS AND ITS APPLICATIONS
• Turán inequalities for the plane partition function
• Inequalities for the overpartition function
• Log-concavity of the overpartition function
• Finite Differences of the Logarithm of the Partition Function
• Log-Concavity of the Partition Function

Related

• Inequalities for higher order differences of the logarithm of the overpartition function and a problem of Wang-Xie-Zhang
• Asymptotics for the reciprocal and shifted quotient of the partition function
• On a nonlinear relation for computing the overpartition function
• Asymptotics and inequalities for the distinct partition function
• Polynomization of the Liu-Zhang inequality for overpartition function