@article {MATHEDUC.06528720,
author = {Hou, Shui-Hung and Hou, Edwin},
title = {On a recursion formula related to confluent Vandermonde.},
year = {2015},
journal = {American Mathematical Monthly},
volume = {122},
number = {8},
issn = {0002-9890},
pages = {766-772},
publisher = {Mathematical Association of America (MAA), Washington, DC},
doi = {10.4169/amer.math.monthly.122.8.766},
abstract = {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