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A divided difference identity for a class of multiple integrals

Michael S. Floater

Abstract

We derive an identity that relates a class of multiple integrals involving Vandermonde polynomials to divided differences. Alternatively the identity can be viewed as an integral formula for divided differences. As part of the derivation we show that both sums of pure partial derivatives and mixed partial derivatives of Vandermonde polynomials are zero, which might be of independent interest.

A divided difference identity for a class of multiple integrals

Abstract

We derive an identity that relates a class of multiple integrals involving Vandermonde polynomials to divided differences. Alternatively the identity can be viewed as an integral formula for divided differences. As part of the derivation we show that both sums of pure partial derivatives and mixed partial derivatives of Vandermonde polynomials are zero, which might be of independent interest.
Paper Structure (10 sections, 11 theorems, 50 equations)

This paper contains 10 sections, 11 theorems, 50 equations.

Table of Contents

  1. Introduction
  2. A divided difference viewpoint
  3. Partial derivatives of $V(\mathbf{t})$
  4. Derivatives of $\omega$
  5. Pure partial derivatives of $V$
  6. Sums of pure derivatives
  7. Sums of mixed derivatives
  8. The sum function
  9. Integration on a rectangle
  10. Proof of Theorem \ref{['thm:ddid']}

Key Result

Theorem 1

For $f \in C^n[y_1,y_{n+1}]$, where $[y_1,y_2,\ldots,y_{n+1}]f$ denotes the divided difference of $f$ at the points $y_1,y_2,\ldots,y_{n+1}$.

Theorems & Definitions (11)

  • Theorem 1
  • Corollary 1
  • Lemma 1
  • Lemma 2
  • Lemma 3
  • Lemma 4
  • Lemma 5
  • Lemma 6
  • Lemma 7
  • Lemma 8
  • ...and 1 more