1 Introduction
Let be a list of real numbers. The classical Vandermonde matrix is defined as follows:
As is wellknown, the Vandermonde matrix is invertible if and only if the are pairwise distinct. See, for example, [3].
In what follows, is a positive integer and is a fixed real number. Let us define the entries by
and let , the maximum of the absolute values of the entries of . The size of the entries of inverses of Vandermonde matrices have been studied for a long time (e.g., [1]). Recently, in a paper by the first two authors and Daniel Kane [2]
, we needed to estimate
, and we proved that . In fact, even more is true: the limit exists and equals . In this paper, we generalize this result, replacing with any real number greater than .Our main results are as follows:
Theorem 1.
Let and . Then for . Hence .
Theorem 2.
Let and . Then .
Theorem 3.
For all real the limit exists.
2 Preliminaries
For every real number , and for all integers , let us define the power sum
The following lemma will be useful in later arguments.
Lemma 4.
Let be integers with , , and let be a positive real number.

[(a)]

If , then .

If , then .
Proof.
We have
where
and
Now there is a bijection given by
where is the unique integer such that . Hence, it follows easily that for , and for . ∎
Recall the following formula for the entries of the inverse of a Vandermonde matrix (see, e.g., [4, §1.2.3, Exercise 40]).
Lemma 5.
Let be pairwise distinct real numbers. If then
For define
We now obtain a relationship between the entries of and and .
Lemma 6.
Let . Then
(1) 
for .
Proof.
By Lemma 5, we have
which in turn, by Vieta’s formulas, gives
(2) 
for . The result now follows by the definitions of and . ∎
Next, we obtain some inequalities for .
Lemma 7.
Define . Then
Proof.
For , we have
A quick computation shows that the inequality
is equivalent to
Let be the minimum positive integer such that . Then . Hence, for , we have
so that
(3) 
∎
Finally, we have the easy
Lemma 8.
For we have .
Proof.
is a symmetric matrix, so its inverse is also. ∎
3 Proof of Theorem 1
4 Proof of Theorem 2
Proof.
Since , it follows that . Hence in Theorem 1 we can take , and this gives . However, by explicit calculation, we have
so that
(7) 
Hence
and the result follows. ∎
5 Proof of Theorem 3
Proof.
We have
where the equality
arises from the onetoone correspondence between the subsets of of cardinality and those of cardinality .
For define
Hence the limits
(8) 
exist and are finite.
From this theorem we can explicitly compute for .
Corollary 9.
Let be the real zero of the polynomial .

[(a)]

If , then .

If , then .
Proof.
Remark 10.
The quantity converges rather slowly to its limit when is close to . The following table gives some numerical estimates for .
3  1.785312341998534190367486 

2.4862447382651613433  
2  5.194119929182595417 
26.788216012030303413  
1.5  67.3672156 
1.4  282.398 
1.3  3069.44 
1.2  422349.8 
6 Final remarks
We close with a conjecture we have been unable to prove.
Conjecture 11.
Let and . Then, for all sufficiently large , we have for some , .
References
 [1] W. Gautschi. On inverses of Vandermonde and confluent Vandermonde matrix. Numer. Mathematik 4 (1962), 117–123.
 [2] D. M. Kane, C. Sanna, and J. Shallit. Waring’s theorem for binary powers. Combinatorica 39 (2019), 1335–1350.
 [3] A. Klinger. The Vandermonde matrix. Amer. Math. Monthly 74 (1967), 571–574.
 [4] D. E. Knuth. The Art of Computer Programming, Vol. 1, Fundamental Algorithms. AddisonWesley, third edition, 1997.
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