Let $\{x\_1,\dots,x\_n\}$ be pairwise distinct complex numbers and $l\_1+l\_2+\dots+l\_n=N$. The $N\times N$ confluent Vandermonde matrix is defined as
$$V=
\begin{bmatrix}
v\_{1,0}&v\_{2,0}&\dots&v\_{n,0}\\\\
v\_{1,1}&v\_{2,1}&\dots&v\_{n,1}\\\\
\vdots\\\\
v\_{1,N-1}&v\_{2,N-1}&\dots&v\_{n,N-1}
\end{bmatrix}$$ where $v\_{j,k}=\begin{bmatrix}x\_j^k,&kx\_j^{k-1},&\dots&k(k-1)\times\dots\times (k-l\_j+1) x\_j^{k-l\_j+1}\end{bmatrix}$. Let $\|\cdot\|$ denote the row sum matrix norm. In some applications (e.g. interpolation, signal processing) one would like to estimate the quantity $\|V^{-1}\|$.


Gautschi [1] has shown that for $l\_1=\dots=l\_n=2$ one has
$$ \|V^{-1}\| \leq \max\_{1\leq \lambda\leq n} \beta\_{\lambda} \prod\_{\nu=1,\nu\neq\lambda}^n \biggl(\frac{1+|x\_{\lambda}|}{|x\_{\nu}-x\_{\lambda}|}\biggr)^2$$ where $\beta\_{\lambda}=\max\biggl(1+|x\_{\lambda}|,1+2(1+|x\_{\lambda}|)\sum\_{\nu\neq\lambda}{1\over |x\_\nu-x\_\lambda|}\biggr)$.


I am interested in a somewhat cruder estimates, as follows: if $|x\_j|\leq 1$ and $|x\_i-x\_j|\geq \delta$, then for the above case we have $$\|V^{-1}\| \leq C n 2^N\delta^{-N+1}\qquad (\*)$$ for some absolute constant $C$.


**Is it true that something like $(\*)$ holds for the general configuration
$\{l\_1,\dots,l\_n\}$?**


EDIT: using [2], this seems to boil down to the following. Let
$$ h\_j(x)=\prod\_{i \neq j}(x-x\_i)^{-l\_i}.$$ **For $t=0,1,\dots,l\_j$ evaluate $h\_j^{(t)}(x\_j).$**


[1] W.Gautschi, "On Inverses of Vandermonde and Confluent Vandermonde matrices II", Numerische Mathematik 5, 425-430, 1963.


[2] R.Schapelle, "The Inverse of the Confluent Vandermonde Matrix", IEEE Trans. on Automatic Control, October 1972, pp.724-725.



Norm of inverse confluent Vandermonde matrix
For $j=1,\dots,n$ and $k=0,1,\dots,l\_j-1$ denote by $u\_{j,k}$ the row with index $l\_1+\dots +l\_{j-1}+k$ of the matrix $V^{-1}$. By using a generalization of the Hermite interpolation formula (see [3]), in [2] it is shown that the elements of $u\_{j,k}$ are the coefficients of the polynomial
$$ {1\over k!} \sum\_{t=0}^{l\_j-1-k} {1\over t!} h\_j^{(t)}(x\_j) (x-x\_j)^{k+t} \prod\_{i\neq j} (x-x\_i)^{l\_i} $$


Now thanks to the answer to [this MSE question](https://math.stackexchange.com/q/248537/32051), one has
$$
|h\_j^{(t)}(x\_j)|\leq N(N+1)\cdots (N+t-1)\delta^{-N-t}.
$$


The sum of absolute values of the coefficients of the polynomials $(x-x\_j)^{k+t} \prod\_{i\neq j} (x-x\_i)^{l\_i}$ is at most (see [4, Lemma])
$$
(1+|x\_j|)^{k+t} \prod\_{i\neq j}(1+|x\_i|)^{l\_i} \leq 2^{N-(l\_j-k-t)}.
$$


So now
$$
\|u\_{j,k}\| \leq {1\over k!}\sum\_{t=0}^{l\_j-1-k} {1\over t!} {N(N+1)\cdots (N+t-1) \over {\delta^{N+t}}}2^{N-l\_j+k+t}\\\\
\leq \biggl({2\over \delta}\biggr)^N {1\over {2^{l\_j-k}k!}}\sum\_{t=0}^{l\_j-1-k} {l\_j-1-k \choose t} {N(N+1)\cdots(N+t-1)\over (l\_j-k-t)\cdots(l\_j-k-2)(l\_j-k-1)} \biggl({2\over \delta}\biggr)^t\\\\
\leq \biggl({2\over \delta}\biggr)^N {1\over {2^{l\_j-k}k!}} \biggl(1+{2N\over \delta}\biggr)^{l\_j-1-k}\\\\
=\biggl({2\over \delta}\biggr)^N {2\over k!} \biggl({1\over 2}+{N\over\delta}\biggr)^{l\_j-1-k}.
$$


[3] A.Spitzbart, "A Generalization of Hermite's Interpolation Formula", The American Mathematical Monthly, Vol.67 No.1, p.42-46, 1960.


[4] W.Gautschi, "On Inverses of Vandermonde and Confluent Vandermonde matrices", Numerische Mathematik 4, p.117-123, 1962.


