Mathematische Zeitschrift

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.

If X is a rational surface without nonzero holomorphic vector field and f is an automorphism of X, we study in several examples the Zariski tangent space of the local deformation space of the pair (X, f).

We provide a comprehensive treatment of the single and double commutation method as a tool for constructing soliton solutions of the Toda and Kac–van Moerbeke hierarchy on arbitrary background. In addition, we present a novel construction based on the single commutation method. As an illustration we compute the N-soliton solution of the Toda and Kac–van Moerbeke hierarchy.