Finite difference of the overpartition function

Abstract

Let p(n) denote the integer partition function. Good conjectured that Δrp(n) alternates in sign up to a certain value n=n(r), and then it stays positive. Gupta showed that for any given r and sufficiently large n, Δrp(n)>0. Odlyzko proved this conjecture and gave an asymptotic formula for n(r). Then, Almkvist, Knessel and Keller gave many contributions for the exact value of n(r). For the finite difference of log⁡p(n), DeSalvo and Pak proved that 0≤−△2log⁡p(n−1)≤log⁡(1+1n) and conjectured a sharper upper bound for −△2log⁡p(n). Chen, Wang and Xie proved this conjecture and showed the positivity of (−1)r−1△rlog⁡p(n), and further gave an upper bound for (−1)r−1△rlog⁡p(n). As for the overpartition function p‾(n), Engel recently proved that p‾(n) is log-concave for n≥2, that is, −△2log⁡p‾(n)≥0 for n≥2. Motivated by these results, in this paper we will prove the positivity of finite differences of the overpartition function and give an upper bound for △rp‾(n). Then we show that for any given r≥1, there exists a positive number n(r) such that (−1)r−1△rlog⁡p‾(n)>0 for n>n(r), where △ is the difference operator with respect to n. Moreover, we give an upper bound for (−1)r−1△rlog⁡p‾(n).