Asymptotics for the Fourier coefficients of eta-quotients

Chern, Shane

doi:10.1016/j.jnt.2018.11.008

Cited by 14 publications

(10 citation statements)

References 21 publications

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“…The proof of this theorem will rely on the work of Chern [8], which is based on the circle method. We remark here that for t being a prime and δ ∈ (0, 24/(t − 1)], an asymptotic formula for c (δ) t (n) with different form was given by Schlosser and Zhou [15] through the circle method.…”

Section: Introduction and Main Resultsmentioning

confidence: 99%

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Sign Changes of Coefficients of Powers of the Infinite Borwein Product

Wang¹

2021

Preprint

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We denote by c (m) t(n) the coefficient of q n in the series expansion of (q; q) m ∞ (q t ; q t ) −m ∞ , which is the m-th power of the infinite Borwein product. Let t and m be positive integers with m(t − 1) ≤ 24. We provide asymptotic formula for c (m) t (n), and give characterizations of n for which c (m) t (n) is positive, negative or zero. We show that c (m) t (n) is ultimately periodic in sign and conjecture that this is still true for other positive integer values of t and m. Furthermore, we confirm this conjecture in the cases (t, m) = (2, m), (p, 1), (p, 3) for arbitrary positive integer m and prime p.

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Section: Introduction and Main Resultsmentioning

confidence: 99%

“…To deduce the asymptotic formula of c (m) t (n), we need a result of Chern [8]. Before stating it, we introduce some notations in Chern's work [8].…”

Section: Asymptotic Formula and Nonvanishing Coefficientsmentioning

confidence: 99%

Sign Changes of Coefficients of Powers of the Infinite Borwein Product

Wang¹

2021

Preprint

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“…Remark. In [6], Chern considered the case where Σ ≤ 0 in (4.1) and obtained a similar asymptotic formula for g(n).…”

Section: Proof Of Theorem 13mentioning

confidence: 99%

“…, F 2 (q) = E(q 2 ) 2 E(q 8 )E(q 12 )E(q 16 ) E(q)E(q 4 ) 4 E(q 24 ) , F 3 (q) = E(q 2 ) 2 E(q 16 ) E(q)E(q 4 )E(q 8 ) 2 , F 4 (q) = E(q 2 )E(q 6 )E(q 24 ) E(q)E(q 8 )E(q 12 ) 2 , F 5 (q) = E(q 2 ) 3 E(q 12 ) E(q)E(q 4 ) 3 E(q 6 ) , F 6 (q) = E(q 2 ) 4 E(q ) 2 E(q)E(q 4 ) 6 , F 7 (q) = q 2 E(q 2 ) 2 E(q 8 ) 2 E(q 12 )E(q 48 ) 2 E(q)E(q 4 ) 4 E(q 16 )E(q 24 ) 2 .…”

Section: Introductionmentioning

confidence: 99%

Some generating functions and inequalities for the andrews–stanley partition functions

Chen¹,

Chern²,

Fan³

et al. 2021

Proceedings of the Edinburgh Mathematical Society

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Let $\mathcal {O}(\pi )$ denote the number of odd parts in an integer partition $\pi$ . In 2005, Stanley introduced a new statistic $\operatorname {srank}(\pi )=\mathcal {O}(\pi )-\mathcal {O}(\pi ')$ , where $\pi '$ is the conjugate of $\pi$ . Let $p(r,\,m;n)$ denote the number of partitions of $n$ with srank congruent to $r$ modulo $m$ . Generating function identities, congruences and inequalities for $p(0,\,4;n)$ and $p(2,\,4;n)$ were then established by a number of mathematicians, including Stanley, Andrews, Swisher, Berkovich and Garvan. Motivated by these works, we deduce some generating functions and inequalities for $p(r,\,m;n)$ with $m=16$ and $24$ . These results are refinements of some inequalities due to Swisher.

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“…Finally, we estimate the main contribution Σ 1 . To do so, we need the following evaluation of an integral, which is a special case of Lemma 2.4 in [6]. For the sake of completeness, we sketch a brief proof.…”

Section: Major Arcsmentioning

confidence: 99%