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.

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 : Theorem 1. The interpolation problem q a (x i ) = a i associated to equidistant base points x i = ξ + ih, i = 0, 1, . . . , ℓ and the value vector a = (a 0 , a 1 , . . . , a ℓ ) yields a polynomial of degree ℓ − m if and only if det A s,a = 0, for s = 0, . . . , m − 1, and det A m,a = 0.
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 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.
The first result is on the determinant of the matrix (1): There is a constant σ ℓ 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): Proposition 3. For integers s ≥ 0 and 0 ≤ k ≤ s there are constants σ ℓ,s,k such that (ℓ − s)-th derivative of the interpolation polynomial (2) for equidistant base points is given by Theorem 1 is a corollary of Propositions 2 and 3: In the next sections we prove Proposition 2 and Proposition 3.
3 On the determinant of A s,a

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.
A consequence of this discussion is Theorem 4 ([7, Theorem 3]). Let α, β be sequences of complex numbers with 1 α i β i ∈ Ê + and α i β j − β i α j = 0 for all 1 ≤ i, j ≤ k, and r be an positive injective sequence. Then the matrix (5) is regular if and only if k ≤ ℓ.

The matrix A s,a and its determinant
Let us consider the matrix A s,a ∈ M ℓ+1 Ê defined by (1). Its components are given by 1 Furthermore, denote by A [κ] ∈ M ℓ Ê the matrix obtained from A s,a by removing the last row and the κ-th 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 5. There exists a number σ ℓ depending only on ℓ such that An immediate corollary is Proposition 2: Expanding the determinant of A s,a with respect to the last row yields 1 In [7] we show that the regularity result remains valid when one of the αi or βi vanishes. 1 We use the convention 0 0 = 1 to cover all cases by this formula.
In the sequel we prove Proposition 5: Proposition 5. We use (6') for k = ℓ and note that only one summand is left, namely the one with µ = µ ∁ = (0, 1, . . . ℓ − 1). In this case we have V ℓ,µ = V ℓ,µ ∁ = V ℓ and, therefore, The proof is completed if we show that the quotient of i 1 ) and ℓ κ−1 only depends on ℓ. For j > i we have and, therefore, Here γ p denotes the number of factors with value p in the third product, i.e.
Rearranging the factors, we obtain We write and finally achieve Remark 6. The proof of Proposition 3 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 to equidistant base points and a value vector a, see e.g. [4,3].
Proposition 7. For 0 ≤ j ≤ ℓ the polynomial K(t) of degree ℓ + 1 obeys Furthermore, for j > 0 and m = ℓ the symbols τ ℓ,m,j obey Proof. The first part is obtained by expanding of K(t) t−j = Using this recurrence, (12) holds by induction.

Proposition 3.
We define x = ξ + th,L j (t) := L j (ξ + th), andq a (t) := q a (ξ + th), that isL (10). By Proposition 7 we havê and In particular, we obtain at t = 0 For instance, for the first values of s the right hand side of this expression is: (t) we obtain (4).
In [2] these symbols are denoted by e ℓ−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: