On the concavity properties of certain arithmetic sequences and polynomials
Tóm tắt
Given a sequence
$$\alpha =(a_k)_{k\ge 0}$$
of nonnegative numbers, define a new sequence
$${\mathcal {L}}(\alpha )=(b_k)_{k\ge 0}$$
by
$$b_{k}=a^2_{k}-a_{k-1}a_{k+1}$$
. The sequence
$$\alpha $$
is called r-log-concave if
$${\mathcal {L}}^{i}(\alpha )=\mathcal {L}({\mathcal {L}}^{i-1}(\alpha ))$$
is a nonnegative sequence for all
$$1\le i\le r$$
. 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
$$\xi $$
-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.
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