Log-concavity and LC-positivity |
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Authors: | Yi Wang Yeong-Nan Yeh |
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Institution: | a Department of Applied Mathematics, Dalian University of Technology, Dalian 116024, China b Institute of Mathematics, Academia Sinica, Taipei 11529, Taiwan |
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Abstract: | A triangle {a(n,k)}0?k?n of nonnegative numbers is LC-positive if for each r, the sequence of polynomials is q-log-concave. It is double LC-positive if both triangles {a(n,k)} and {a(n,n−k)} are LC-positive. We show that if {a(n,k)} is LC-positive then the log-concavity of the sequence {xk} implies that of the sequence {zn} defined by , and if {a(n,k)} is double LC-positive then the log-concavity of sequences {xk} and {yk} implies that of the sequence {zn} defined by . Examples of double LC-positive triangles include the constant triangle and the Pascal triangle. We also give a generalization of a result of Liggett that is used to prove a conjecture of Pemantle on characteristics of negative dependence. |
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Keywords: | Sequences Linear transformations Convolutions Log-concavity q-log-concavity LC-positivity |
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