The spt-function of Andrews

Chen, William Y. C.

doi:10.1017/9781108332699.004

Cited by 23 publications

(18 citation statements)

References 117 publications

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“…The objective of this paper is to prove the higher order Turán inequalities for the partition function p(n) when n ≥ 95, as conjectured in [5]. The Turán inequalities and the higher order Turán inequalities are related to the Laguerre-Pólya class of real entire functions (cf.…”

Section: Introductionmentioning

confidence: 97%

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Higher order Turán inequalities for the partition function

Chen

Jia

Wang

2018

Trans. Amer. Math. Soc.

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The Turán inequalities and the higher order Turán inequalities arise in the study of Maclaurin coefficients of an entire function in the Laguerre-Pólya class. A real sequence {a n } is said to satisfy the Turán inequalities if for n ≥ 1, a 2 n − a n−1 a n+1 ≥ 0. It is said to satisfy the higher order Turán inequalities if for n ≥ 1, 4(a 2 n − a n−1 a n+1 )(a 2 n+1 − a n a n+2 ) − (a n a n+1 − a n−1 a n+2 ) 2 ≥ 0. A sequence satisfying the Turán inequalities is also called log-concave. For the partition function p(n), DeSalvo and Pak showed that for n > 25, the sequence {p(n)} n>25 is log-concave, that is, p(n) 2 − p(n − 1)p(n + 1) > 0 for n > 25. It was conjectured by Chen that p(n) satisfies the higher order Turán inequalities for n ≥ 95. In this paper, we prove this conjecture by using the Hardy-Ramanujan-Rademacher formula to derive an upper bound and a lower bound for p(n + 1)p(n − 1)/p(n) 2 . Consequently, for n ≥ 95, the Jensen polynomials g 3,n−1 (x) = p(n − 1) + 3p(n)x + 3p(n + 1)x 2 + p(n + 2)x 3 have only real zeros. We conjecture that for any positive integer m ≥ 4 there exists an integer N (m) such that for n ≥ N (m), the polynomials m k=0 m k p(n + k)x k have only real zeros. This conjecture was independently posed by Ono.

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Section: Introductionmentioning

confidence: 97%

“…It was conjectured in [5] that for large n, the partition function p(n) satisfies many inequalities pertaining to invariants of a binary form. Here we mention two of them.…”

Section: Introductionmentioning

confidence: 99%

Higher order Turán inequalities for the partition function

Chen

Jia

Wang

2018

Trans. Amer. Math. Soc.

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“…As an illustration of these techniques, we also prove a recent conjecture of Chen which involves an inequality of polynomials in ratios of close partition numbers. Theorem 1.2 (Conjecture 6.13 in [1]). Let u n = p(n + 1)p(n − 1)/p(n) 2 .…”

Section: More Precisely Ifmentioning

confidence: 99%

Hyperbolicity of the partition Jensen polynomials

Larson

Wagner

2019

Res. number theory

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Given an arithmetic function a : N → R, one can associate a naturally defined, doubly infinite family of Jensen polynomials. Recent work of Griffin, Ono, Rolen, and Zagier shows that for certain families of functions a : N → R, the associated Jensen polynomials are eventually hyperbolic (i.e., eventually all of their roots are real). This work proves Chen, Jia, and Wang's conjecture that the partition Jensen polynomials are eventually hyperbolic as a special case. Here, we make this result explicit. Let N (d) be the minimal number such that for all n ≥ N (d), the partition Jensen polynomial of degree d and shift n is hyperbolic. We prove that N (3) = 94, N (4) = 206, and N (5) = 381, and in general, that N (d) ≤ (3d) 24d (50d) 3d 2 .

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“…Recently, Dawsey and Masri [8] gave an effective asymptotic formula of the Andrews spt-function due to the algebraic formula [1] for the spt-function. According to this asymptotic formula, they proved some inequalities on the spt-function conjectured by Chen [4].…”

Section: Introductionmentioning

confidence: 94%