Logarithmically concave sequence: Difference between revisions

Revision as of 01:16, 4 November 2013

Latest revision as of 13:47, 10 January 2025

(16 intermediate revisions by 12 users not shown)

Line 1:

{{Short description|Type of sequence of numbers}}

In [[mathematics]], a sequence {{math|''a''₀, ''a''₁, ..., ''a''_''n''}} of nonnegative real numbers is called a '''logarithmically concave sequence''', or a '''log-concave sequence''' for short, if {{math|''a''_''i''² > ''a''_''i''−1''a''_''i''+1}} holds for {{math|1 < ''i'' < ''n'' − 1}}.

[[File:Pascal's_triangle_5.svg|thumb|The rows of Pascal's triangle are examples for logarithmically concave sequences.]]

In [[mathematics]], a sequence {{math|''a''}} = {{math| (''a''₀, ''a''₁, ..., ''a''_''n'')}} of nonnegative real numbers is called a '''logarithmically concave sequence''', or a '''log-concave sequence''' for short, if {{math|''a''_''i''² >≥ ''a''_''i''−1''a''_''i''+1}} holds for {{math|10 < ''i'' < ''n'' − 1}}.

''Remark:'' some authors (explicitly or not) add two further conditions in the definition of log-concave sequences:

Examples of log-concave sequences are given by the [[binomial coefficient]]s along any row of [[Pascal's triangle]].

* {{math|''a''}} is non-negative

* {{math|''a''}} has no internal zeros; in other words, the [[Support (mathematics)|support]] of {{math|''a''}} is an interval of {{math|'''Z'''}}.

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

Sequences that fulfill the three conditions are also called '''Pólya Frequency sequences of order 2''' ('''PF₂''' sequences). Refer to chapter 2 of <ref name="brenti">{{Cite book|last=Brenti|first=Francesco|url=|title=Unimodal, log-concave and Pólya frequency sequences in 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.

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.

==References==

{{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]]|yeardate=December 1989|month=December|volume=576|issue=1 |pages=500–535|doi= 10.1111/j.1749-6632.1989.tb16434.x|bibcode=1989NYASA.576..500S }}

==See also==

Line 14 ⟶ 23:

[[Category:Sequences and series]]

{{combin-stub}}