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2016c.00800 Hou, Shui-Hung Hou, Edwin On a recursion formula related to confluent Vandermonde. Am. Math. Mon. 122, No. 8, 766-772 (2015). 2015 Mathematical Association of America (MAA), Washington, DC EN H65 confluent Vandermonde matrix interpolation determinant recursion formula
• doi:10.4169/amer.math.monthly.122.8.766
• Consider the polynomial $p(s):=\prod_{k=1}^{r}(s-\lambda_{k})^{n_{k}}$ where the $\lambda_{1},\dots,\lambda_{r}$ are distinct and the integers $n_{k}\geq1$ with $\sum n_{k}=n$. The confluent Vandermonde matrix $V(p)$ is defined to be the $n\times n$ matrix $[V_{1}|V _{2}|\dots |V_{r}]$ where $V_{k}$ is an $n\times n_{k}$ block whose $(i,j)$th entry equals $\binom{i-1}{j-1}\lambda_{k}^{i-j}$ (the case where each $n_{k}=1$ is the usual Vandermonde matrix). The matrix $V(p)$ arises in various generalized interpolation problems. The present paper offers a new proof of a recursion formula for $v(p):=\det V(p)$: $$v(p)=v(q)\prod_{k=1}^{r-1}(\lambda_{r}-\lambda_{k})^{n_{k}n_{r}}$$ where $q(s):=p(s)/(s-\lambda_{r})^{n_{r}}$. This leads immediately to a classical identity of L. Schendel, namely, \$v(p)=\prod_{i John D. Dixon (Ottawa)