On the Conservation Conjectures of Kudla and Rallis

Sun, Binyong; Zhu, Chen-Bo

doi:10.1007/978-3-319-59728-7_19

Cited by 2 publications

(2 citation statements)

References 25 publications

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“…Motivated by Firoozbakht's conjecture on the strictly decreasing property of the sequence { n √ p n } n⩾1 , where p n is the n-th prime number (see [32, p.185]), Sun [36,37] studied the monotonicity of many combinatorial sequences of this type, which has received much attention. The goal of this section is to prove the log-concavity and monotonicity of the sequence { n √ B n } n⩾1 .…”

Section: Log-concavity and Monotonicity Of {mentioning

confidence: 99%

Inequalities associated with the Baxter numbers

Zhao¹

2023

Glas. Mat. Ser. III

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The Baxter numbers \(B_n\) enumerate a lot of algebraic and combinatorial objects such as the bases for subalgebras of the Malvenuto-Reutenauer Hopf algebra and the pairs of twin binary trees on \(n\) nodes. The Turán inequalities and higher order Turán inequalities are related to the Laguerre-Pólya (\(\mathcal{L}\)-\(\mathcal{P}\)) class of real entire functions, and the \(\mathcal{L}\)-\(\mathcal{P}\) class has a close relation with the Riemann hypothesis. The Turán type inequalities have received much attention. In this paper, we are mainly concerned with Turán type inequalities, or more precisely, the log-behavior, and the higher order Turán inequalities associated with the Baxter numbers. We prove the Turán inequalities (or equivalently, the log-concavity) of the sequences \(\{B_{n+1}/B_n\}_{n\geqslant 0}\) and \(\{\hspace{-2.5pt}\sqrt[n]{B_n}\}_{n\geqslant 1}\). Monotonicity of the sequence \(\{\hspace{-2.5pt}\sqrt[n]{B_n}\}_{n\geqslant 1}\) is also obtained. Finally, we prove that the sequences \(\{B_n/n!\}_{n\geqslant 2}\) and \(\{B_{n+1}B_n^{-1}/n!\}_{n\geqslant 2}\) satisfy the higher order Turán inequalities.

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Section: Log-concavity and Monotonicity Of {mentioning

confidence: 99%

Inequalities associated with the Baxter numbers

Zhao¹

2023

Glas. Mat. Ser. III

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“…Sun and the author in its full generality [50]) provide one other way to determine the vanishing/nonvanishing, which is external. (We refer the reader to [51] for a easy read on the conservation relations.) Note that vanish/non-vanishing of Θ(π) amounts to an inequality on the first occurrence index of π in a (generalized) Witt tower, and so will yield information on the nonvanishing/vanishing in another (generalized) Witt tower.…”

Section: Introductionmentioning

confidence: 99%

Local theta correspondence and nilpotent invariants

Zhu¹

2019

Representations of Reductive Groups

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We consider two types of nilpotent invariants associated to smooth representations, namely generalized Whittaker models, and associated characters (in the case of a real reductive group). We survey some recent results on the behavior of these nilpotent invariants under local theta correspondence, and highlight the special role of a certain double fiberation of moment maps.2000 Mathematics Subject Classification. 22E46 (Primary).

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On the Conservation Conjectures of Kudla and Rallis

Cited by 2 publications

References 25 publications

Inequalities associated with the Baxter numbers

Inequalities associated with the Baxter numbers

Local theta correspondence and nilpotent invariants

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