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- W4302304010 abstract "Abstract The $q$ -coloured Delannoy numbers $D_{n,k}(q)$ count the number of lattice paths from $(0,,0)$ to $(n,,k)$ using steps $(0,,1)$ , $(1,,0)$ and $(1,,1)$ , among which the $(1,,1)$ steps are coloured with $q$ colours. The focus of this paper is to study some analytical properties of the polynomial matrix $D(q)=[d_{n,k}(q)]_{n,kgeq 0}=[D_{n-k,k}(q)]_{n,kgeq 0}$ , such as the strong $q$ -log-concavity of polynomial sequences located in a ray or a transversal line of $D(q)$ and the $q$ -total positivity of $D(q)$ . We show that the zeros of all row sums $R_n(q)=sum nolimits _{k=0}^{n}d_{n,k}(q)$ are in $(-infty,, -1)$ and are dense in the corresponding semi-closed interval. We also prove that the zeros of all antidiagonal sums $A_n(q)=sum nolimits _{k=0}^{lfloor n/2 rfloor }d_{n-k,k}(q)$ are in the interval $(-infty,, -1]$ and are dense there." @default.
- W4302304010 created "2022-10-06" @default.
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- W4302304010 date "2022-08-01" @default.
- W4302304010 modified "2023-10-18" @default.
- W4302304010 title "Some analytical properties of the matrix related to <i>q</i>-coloured Delannoy numbers" @default.
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- W4302304010 doi "https://doi.org/10.1017/s0013091522000384" @default.
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