Logarithmically concave sequence: Difference between revisions

Revision as of 01:28, 13 June 2021

Revision as of 16:12, 14 September 2021

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These conditions mirror the ones required for [[Logarithmically_concave_function|log-concave functions]].

Sequences that fulfill the three conditions are also called '''PòlyaPólya Frequency sequences of order 2''' ('''PF₂''' sequences). Refer to chapter 2 of <ref name="brenti">{{Cite book|last=Brenti|first=Francesco|url=|title=Unimodal, F. (1989). Unimodal Loglog-Concaveconcave and PòlyaPólya Frequencyfrequency Sequencessequences in Combinatorics. combinatorics|year=1989|publisher=[[American Mathematical Society]]|isbn=978-1-4704-0836-7|___location=Providence, R.I.|oclc=851087212}}</ref> for a discussion on the two notions. For instance, the sequence {{math|(1,1,0,0,1)}} satisfies the concavity inequalities but not the internal zeros condition.

For instance, the sequence {{math|(1,1,0,0,1)}} satisfies the concavity inequalities but not the internal zeros condition.

Examples of log-concave sequences are given by the [[binomial coefficient]]s along any row of [[Pascal's triangle]] and the [[Newton's inequalities|elementary symmetric means]] of a finite sequence of real numbers.

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{{Reflist}}

* {{cite journal|last=Stanley|first=R. P.|authorlink=Richard P. Stanley|title=Log-Concave and Unimodal Sequences in Algebra, Combinatorics, and Geometry|journal=[[Annals of the New York Academy of Sciences]]|date=December 1989|volume=576|pages=500–535|doi= 10.1111/j.1749-6632.1989.tb16434.x}}

==See also==

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[[Category:Sequences and series]]

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