Bell Numbers, Log-Concavity, and Log-Convexity
By Nobuhiro Asai, I. Kubo, H. Kuo. 2001
55 citations recorded by the source corpus.
AbstractLet {bk(n)}n=0∞ be the Bell numbers of order k. It is proved that the sequence {bk(n)/n!}n=0∞ is log-concave and the sequence {bk(n)}n=0∞ is log-convex, or equivalently, the following inequalities hold for all n⩾0, $$1 \leqslant \frac{{b_k (n + 2)b_k (n)}}{{b_k (n + 1)^2 }} \leqslant \frac{{n + 2}}{{n + 1}}$$ . Let {α(n)}n=0∞ be a sequence of positive numbers with α(0)=1. We show that if {α(n)}n=0∞ is log-convex, then α(n)α(m)⩽α(n+m), ∀n,m⩾0. On the other hand, if {α(n)/n!}n=0∞ is log-concave, then $$\alpha (n + m) \leqslant \left( {\begin{array}{*{20}c} {n + m} \\ n \\ \end{array} } \right)\alpha (n)\alpha (m),{\text{ }}\forall n,m \geqslant 0$$ . In particular, we have the following inequalities for the Bell numbers $$b_k (n)b_k (m) \leqslant b_k (n + m) \leqslant \left( {\begin{array}{*{20}c} {n + m} \\ n \\ \end{array} } \right)b_k (n)b_k (m),{\text{ }}\forall n,m \geqslant 0$$ . Then we apply these results to characterization theorems for CKS-space in white noise distribution theory.
Subfields: Applied Mathematics, Mathematical Physics
View Paper: https://www.semanticscholar.org/paper/d207d915caa8a03acd045ea2d8821e55a4d9320b
How researchers see this paper
The researchers whose trusted networks rank it highest, strongest advocate first. Every researcher gets a different order.
In Applied Mathematics
Citations in the corpus
- Transferable AutoML by Model Sharing Over Grouped Datasets (2019, background): “3 directly is often unfeasible as the state number is a Bell number [2] and grows exponentially with the number of datasets, even considering the simplification of states by reducing the empty clusters.”