Hyperbolicity of Appell polynomials of functions in the $$\delta $$-Laguerre–Pòya class

Iskander, Jonas; Jain, Vanshika

doi:10.1007/s40993-020-00202-4

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Explicit Bounds For Bessel Functions1

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2021

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“…Remark. We note that similar estimates needed for Lemma 2.2 (1) also appear in Section 4.1 of [24]. For the reader's convenience, we provide a proof here.…”

Section: Explicit Bounds For Bessel Functionsmentioning

confidence: 79%

See 1 more Smart Citation

Fractional partitions and conjectures of Chern–Fu–Tang and Heim–Neuhauser

Bringmann¹,

Kane

Rolen³

et al. 2021

Trans. Amer. Math. Soc. Ser. B

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Many papers have studied inequalities for partition functions. Recently, a number of papers have considered mixtures between additive and multiplicative behavior in such inequalities. In particular, Chern–Fu–Tang and Heim–Neuhauser gave conjectures on inequalities for coefficients of powers of the generating partition function. These conjectures were posed in the context of colored partitions and the Nekrasov–Okounkov formula. Here, we study the precise size of differences of products of two such coefficients. This allows us to prove the Chern–Fu–Tang conjecture and to show the Heim–Neuhauser conjecture in a certain range. The explicit error terms provided will also be useful in the future study of partition inequalities. These are laid out in a user-friendly way for the researcher in combinatorics interested in such analytic questions.

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“…Remark. We note that similar estimates needed for Lemma 2.2 (1) also appear in Section 4.1 of [24]. For the reader's convenience, we provide a proof here.…”

Section: Explicit Bounds For Bessel Functionsmentioning

confidence: 79%

“…This paper also tells us that these inequalities eventually hold for p α (n). However, one could make these bounds explicit similar to how we have done here, which may show when exactly the inequalities begin to hold (see for example [12,24]).…”

Section: Discussionmentioning

confidence: 99%