Simplified formula for summing discrete values of some functions | Prikladnaya Diskretnaya Matematika - Applied Discrete Mathematics. 2024. № 64. DOI: 10.17223/20710410/64/7

A simplified version of the Euler-Maclaurin summation formula is obtained. The formula includes an integral estimate of the sum of discrete samples of a function and an amendment to it in the form of the sum of a number of weighted boundary values of its odd derivatives. Simplification is the exclusion of the half-sum of the boundary values of the function from the summation result and is achieved by shifting hr samples inside the integration segments. It is proved that the optimal shift of each sample to the middle of the segment r = 1/2. This shift sets the limits of the integral estimate yo, ym and the values of the weighting coefficients of the derivatives of the correction series. An analytical expression for these coefficients and their generating function are found. Examples show the validity of the resulting formula and the generating function of its coefficients. The formula was used to obtain approximate expressions for the Riemann zeta function, psi function, polygamma functions, as well as the sums of infinite inverse power series and harmonic series. Based on an analysis of the error of these expressions, the advantages of the simplified formula over the Euler-Maclaurin formula in accuracy and brevity are shown.