Loading...
Please wait, while we are loading the content...
Similar Documents
Bell Numbers, Log-Concavity, and Log-Convexity
| Content Provider | Semantic Scholar |
|---|---|
| Author | Asai, Nobuhiro Kubo, Izumi Kuo, Hui-Hsiung |
| Copyright Year | 2000 |
| Abstract | 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. |
| Starting Page | 79 |
| Ending Page | 87 |
| Page Count | 9 |
| File Format | PDF HTM / HTML |
| DOI | 10.1023/A:1010738827855 |
| Volume Number | 63 |
| Alternate Webpage(s) | http://math.lsu.edu/~preprint/1999/hhk1999b.ps |
| Alternate Webpage(s) | https://arxiv.org/pdf/math/0104137v1.pdf |
| Alternate Webpage(s) | http://arxiv.org/pdf/math.CO/0104137.pdf |
| Alternate Webpage(s) | http://arxiv.org/pdf/math/0104137v1.pdf |
| Alternate Webpage(s) | https://doi.org/10.1023/A%3A1010738827855 |
| Language | English |
| Access Restriction | Open |
| Content Type | Text |
| Resource Type | Article |
