Logarithmically concave sequence

In mathematics, a sequence a = (a0, a1, ..., an) of nonnegative real numbers is called a logarithmically concave sequence, or a log-concave sequence for short, if ai2ai−1ai+1 holds for 0 < i < n .

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

• a is non-negative
• a has no internal zeros; in other words, the support of a is an interval of Z.

These hypotheses mirror the ones required for log-concave functions.

Sequences that fulfill the three conditions are also called Pòlya Frequency sequences of order 2 (PF2 sequences). Refer to chapter 2 of [1] for a discussion on the two notions. For instance, the sequence (1,1,0,0,1) checks the concavity inequalities but not the internal zeros condition.

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

References

1. ^ Brenti, F. (1989). Unimodal Log-Concave and Pòlya Frequency Sequences in Combinatorics. American Mathematical Society.