On the concavity properties of certain arithmetic sequences and polynomials

Given a sequence α = ( a k ) k ≥ 0 of nonnegative numbers, define a new sequence L ( α ) = ( b k ) k ≥ 0 by b k = a k 2 - a k - 1 a k + 1 . The sequence α is called r - log-concave if L i ( α ) = L ( L i - 1 ( α ) ) is a nonnegative sequence for all 1 ≤ i ≤ r . In this paper, we study the r -log-con...

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Bibliographic Details
Published inMathematische Zeitschrift Vol. 305; no. 3
Main Author Zhu, Bao-Xuan
Format Journal Article
LanguageEnglish
Published Berlin/Heidelberg Springer Berlin Heidelberg 01.11.2023
Springer Nature B.V
Subjects
Binomial coefficients
Combinatorial analysis
Concavity
Linear transformations
Mathematical analysis
Mathematics
Mathematics and Statistics
Polynomials
Log-concavity
3
3-Log-concavity
05C30
05A20
Total positivity
30D15
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Summary:Given a sequence α = ( a k ) k ≥ 0 of nonnegative numbers, define a new sequence L ( α ) = ( b k ) k ≥ 0 by b k = a k 2 - a k - 1 a k + 1 . The sequence α is called r - log-concave if L i ( α ) = L ( L i - 1 ( α ) ) is a nonnegative sequence for all 1 ≤ i ≤ r . In this paper, we study the r -log-concavity and its q -analogue for r = 2 , 3 using total positivity of matrices. We show the 6-log-concavity of the Taylor coefficients of the Riemann ξ -function. We give some criteria for r - q -log-concavity for r = 2 , 3 . As applications, we get 3- q -log-concavity of q -binomial coefficients and different q -Stirling numbers of two kinds, which extends results for q -log-concavity. We also present some results for r - q -log-concavity from the linear transformations. Finally, we pose an interesting question.
ISSN:0025-5874
1432-1823
DOI:10.1007/s00209-023-03361-z