A Set of Rational Functions Related to the Euler–Fobenius Polynomials

Abstract

The asymptotic behavior of the fundamental cardinal spline (Schoenberg’s terminology) of odd degree n is governed by that particular root of the Euler–Frobenius polynomial $\Pi _n $ which lies inside of and nearest to the unit circle. Rigorous bounds for this dominant root and an asymptotic formula for the remaining roots are given. These results are based on the partial fraction expansion of $q_n ( - e^{\pi w} )$, where $q_n (z) = z\Pi _n (z)(1 - z)^{ - n - 1} $ Various other properties of the functions $q_n$ are discussed, as is their use in the representation of the fundamental cardinal splines of odd degree.