Explicit Hermite Interpolation Polynomials via the Cycle Index with Applications

Abstract

The cycle index polynomial of a symmetric group is a basic tool in combinatorics and especially in Pόlya enumeration theory. It seems irrelevant to numerical analysis. Through Faá di Bruno's formula, cycle index is connected with numerical analysis. In this work, the Hermite interpolation polynomial is explicitly expressed in terms of cycle index. Applications in Gauss-Turán quadrature formula are also considered.