On the characterization of the degree of interpolation polynomials in terms of certain combinatorical matrices

In this note, we show that the degree of the interpolation polynomial for equidistant base points is characterized by the regularity of matrices of combinatorical type.

Furthermore, denote by q a the interpolation polynomial of degree at most defined by the base points x 0 , . . . , x and the value vector a: In this text, we show the following remarkable characterization of the degree of the interpolation polynomial for equidistant base points in terms of the matrices A s,a : When we were preparing this text, we learned much about the relation between combinatorics and binomial identities and the evaluation of special determinants. We cordially refer the reader to the outstanding collections [8,9] of determinants evaluation. We also refer to the book [11] which presents special combinatorical identities.

The proof of the main Theorem
Theorem 1.1 relies on some properties of the matrix A s,a and the polynomial q a that can be obtained independently from each other. In this section, we state these results and show how they prove Theorem 1.1.
The first result is on the determinant of the matrix (1): There is a constant σ only depending on the size of the matrix A s,a defined by (1), such that its determinant is given by The second result deals with the derivatives of the interpolation polynomial (2)  In the next sections, we prove Propositions 2.1 and 2.2.

Preliminaries
Some submatrices of A s,a (1) are of special type and were studied in our previous work [7]. They will be important in the reasoning of Proposition 2.1.

The matrix A s,a and its determinant
Let us consider the matrix A s,a ∈ M +1 R defined by (1). Its components are given by 2 Furthermore, denote by A [κ] ∈ M R the matrix obtained from A s,a by removing the last row and the κth column. Of course, it does not depend on s and a and is given by The matrices A [κ] are of the form (5) and have the following property: Proposition 3.1 There exist a number σ depending only on such that An immediate corollary is Proposition 2.1: Proof (Proposition 2.1) Expanding the determinant of A s,a with respect to the last row yields In the sequel, we prove Proposition 3.1: Proof (Proposition 3.1) We use (6') for k = and note that only one summand is left, namely, the one with μ = μ = (0, 1, . . . − 1). In this case, we have V ,μ = V ,μ = V , and therefore, The proof is completed if we show that the quotient of 1≤i 1 Here, γ p denotes the number of factors with value p in the third product, that is Rearranging the factors, we obtain We write and finally achieve Remark 3.1 The proof of Proposition 3.1 can also be performed by reducing the determinant at hand to a matrix with polynomial entries: Then, applying the determinant evaluation as known from the Vandermonde matrix, one can show that the determinant is given by ⎛ see [8,Propostion 1]. However, in both cases, the explicit calculations above remain to be done.

On the interpolant q a
In this section, we study the interpolation polynomial q a (2) associated with equidistant base points and a value vector a, see, e.g., [3,4]. Explicitly, we have where L j denotes the jth Lagrange polynomial associated with the base points x 0 = ξ, For j = 0, . . . , , the latter are given by The proof of Proposition 2.2 will be presented at the end of this section and it makes use of a combinatorical fact on the polynomial: To formulate this fact, we introduce the symbols: Furthermore, for j > 0 and m = , the symbols τ ,m, j obey Proof The first part is obtained by expanding of The second part is obviously true for Using this recurrence, (12) holds by induction.

A final remark on the general interpolation problem
Remark 5.1 In the case of arbitrary base points x 0 , . . . , x , one can introduce modified symbols τ ,m, j = 1≤i 1 <...<i m ≤ i 1 ,...,i m = j x i 1 · · · x i m to get an analogous of (11).
In [2], these symbols are denoted by e −m, j and are used to present representations of the Schur functions. In our setting, they can be used to get an expansion like (12), too, and a formula of the form (13) can also be obtained: