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Infinitely Logarithmically Concave Sequence

An infinitely logarithmically concave sequence is a nonnegative sequence that remains nonnegative after every iteration of the log-concavity transform. For a sequence a=(a_0,a_1,...), extend it by a_(-1)=0 and define

(La)_n=a_n^2-a_(n-1)a_(n+1).

The condition La>=0 means that a is a logarithmically concave sequence, while infinite logarithmic concavity requires L^ma>=0 for every nonnegative integer m.

Liu and Mao (2026) reported that the Taylor series coefficients a_(2n) in xi(1/2+z)=sum_(n>=0)a_(2n)z^(2n) form an infinitely logarithmically concave sequence. More precisely, every entry of every iterate L^m(a_0,a_2,...) is strictly positive. Here xi is the xi-function. The assertion does not assume the Riemann hypothesis.

Xie et al. (2026) proved that the sequence (k^d)_(k>=0) is infinitely logarithmically concave for a positive integer d iff d!=2. Xie et al. (2026) report using generative AI to develop proof arguments, check estimates, and write computational code, while taking responsibility for the mathematics.

See also

Logarithmically Concave Sequence

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References

Liu, Y. and Mao, J. "Infinite Log-Concavity of the Taylor Coefficients of the Riemann xi-Function." 4 Oct 2026. https://arxiv.org/abs/2610.04972.Xie, M. H. Y.; Zhang, C. X. T.; and Zhang, P. B. "A Unified Approach to Infinite Log-Concavity of Combinatorial Sequences." 8 Oct 2026. https://arxiv.org/abs/2610.12237.

Cite this as:

Weisstein, Eric W. "Infinitely Logarithmically Concave Sequence." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/InfinitelyLogarithmicallyConcaveSequence.html

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