A Rademacher Type Formula for  Partitions and Overpartitions

Sills, Andrew V.

doi:10.1155/2010/630458

Cited by 15 publications

(7 citation statements)

References 58 publications

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“…Then they use (1.1) to get an upper and lower bound for C(T (n)) which leads to upper and lower bound for C(p(n)). In this paper we will derive a similar Statement for overpartitions by a careful analysis of the Rademacher-type series given by Sills [1].…”

Section: Introduction and Statement Of Resultsmentioning

confidence: 97%

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Log-concavity of the overpartition function

Engel¹

2016

Ramanujan J

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Section: Introduction and Statement Of Resultsmentioning

confidence: 97%

“…Commonly, the number of overpartitions of a positive integer n is denoted by p(n). Sills [1] rediscovered Zuckermann's [6] formula for the overpartition and pointed out that it is indeed a Rademacher-type series…”

Section: Introduction and Statement Of Resultsmentioning

confidence: 99%

Log-concavity of the overpartition function

Engel¹

2016

Ramanujan J

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“…A. Sellers [24,25], Kim [27], J. Lovejoy [29][30][31][32][33][34], K. Mahlburg [36], M. Merca [40,42] and A. V. Sills [46].…”

Section: (13)mentioning

confidence: 99%

Distinct partitions and overpartitions

Merca¹

2021

CJM

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In 1963, Peter Hagis, Jr. provided a Hardy-Ramanujan-Rademacher-type convergent series that can be used to compute an isolated value of the partition function $Q(n)$ which counts partitions of $n$ into distinct parts. Computing $Q(n)$ by this method requires arithmetic with very high-precision approximate real numbers and it is complicated. In this paper, we investigate new connections between partitions into distinct parts and overpartitions and obtain a surprising recurrence relation for the number of partitions of $n$ into distinct parts. By particularization of this relation, we derive two different linear recurrence relations for the partition function $Q(n)$. One of them involves the thrice square numbers and the other involves the generalized octagonal numbers. The recurrence relation involving the thrice square numbers provide a simple and fast computation of the value of $Q(n)$. This method uses only (large) integer arithmetic and it is simpler to program. Infinite families of linear inequalities involving partitions into distinct parts and overpartitions are introduced in this context.

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“…Recall an overpartition [5] of a nonnegative integer n is a partition of n where the first occurrence of each distinct part may be overlined. Let p(n) denote the number of overpartitions of n. Zukermann [21] gave a formula for the overpartition function, which is considered by Sills [19] as a Rademacher-type convergent series…”

Section: Introductionmentioning

confidence: 99%

Higher order log-concavity of the overpartition function and its consequences

Mukherjee¹,

Zhang²,

Zhang³

2022

Preprint

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Let p(n) denote the overpartition function. In this paper, we study the asymptotic higher order log-concavity property of the overpatition function in a similar framework done by Hou and Zhang for the partition function. This will enable us to move on further in order to prove log-concavity of overpartitions, explicitly by studying the asymptotic expansion of the quotient p(n − 1)p(n + 1)/p(n) 2 upto a certain order so that one can finally ends up with the phenomena of 2-log-concavity and higher order Turán property of p(n) by following a sort of unified approach.

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A Rademacher Type Formula for Partitions and Overpartitions

Abstract: A Rademacher-type convergent series formula which generalizes the Hardy-Ramanujan-Rademacher formula for the number of partitions of n and the Zuckerman formula for the Fourier coefficients of ϑ 4 (0 | τ ) −1 is presented.

Cited by 15 publications

References 58 publications

Log-concavity of the overpartition function

Log-concavity of the overpartition function

Distinct partitions and overpartitions

Higher order log-concavity of the overpartition function and its consequences

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