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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. Liu and Mao (2026) report generative AI assistance with approaches, technical checks, and exposition and state that they verified the arguments and computations.

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, write computational code, and revise the exposition, 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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