  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.
