A Generalization of Classical Formulas in Numerical Integration and Series Convergence Acceleration

TL;DR

This paper unifies classical summation formulas like Euler-Maclaurin and Gregory's quadrature into a single series expansion, introducing a family of polynomials that generalize these formulas and have applications in numerical analysis.

Contribution

It presents a new unified summation formula that encompasses classical formulas and introduces a polynomial family with useful properties and applications.

Findings

Derived a generalized series expansion unifying classical summation formulas.

Established properties and symmetries of the new polynomial family.

Applied the expansion to finite impulse response (FIR) filter analysis.

Abstract

Summation formulas, such as the Euler-Maclaurin expansion or Gregory's quadrature, have found many applications in mathematics, ranging from accelerating series, to evaluating fractional sums and analyzing asymptotics, among others. We show that these summation formulas actually arise as particular instances of a single series expansion, including Euler's method for alternating series. This new summation formula gives rise to a family of polynomials, which contain both the Bernoulli and Gregory numbers in their coefficients. We prove some properties of those polynomials, such as recurrence identities and symmetries. Lastly, we present one case study, which illustrates one potential application of the new expansion for finite impulse response (FIR) filters.