Simplified formula for summing discrete values of~some~functions

V. R. Osipov

Prikladnaâ diskretnaâ matematika, no. 2 (2024), pp. 79-98.

Résumé

The simplified variant of Euler — Maclaurin summation formula is obtained: \vskip -2ex $${\textstyle\sum\limits_{k=0}^{m}}f(a+kh)=\frac{1}{h}\int\limits_{y_0}^{y_m}f(y) dy-{\textstyle\sum\limits_k} h^{2k-1}b_k\big(f^{(2k-1)}(y_m)-f^{(2k-1)}(y_0)\big),$$ \vskip -2ex where $y_0=a-h/2$, $y_m=a+(m+1/2)h$. The formula includes an integrated estimation of the sum of function discrete samples and the correction to it in the form of the series sum of weight boundary values of its odd derivatives. Simplification is the exception a half-sum of boundary function values from summing result and is reached by shiftihg $hr$ samples inside the integrated segments. The optimal value of this shift for each sample to the middle of the segment $r=1/2$ is proven. This shift specifies integrated estimation limits $y_0$, $y_m$, and values of correction series weighting coefficients. The analytical expression of these coefficients and their generating function have been found: $b_k=\dfrac{1-2^{1-2k}}{(2k)!}B_{2k}$, $\Psi_b(t)=1-\dfrac{t}{2}\mathrm{cosech}\,\dfrac{t}{2}=\textstyle\sum\limits_{k=1}^{\infty}b_k t^{2k}$, where $B_{2k}$ are Bernoulli numbers. Using examples of obtaining exact expressions for the sums $\sum\limits_{k=1} ^{m} k^n$, $\sum\limits_{k=k_0} ^{m} a^ { h k }$, where $m, n\in\mathbb{N}$, the validity of the resulting formula and the generating function of its coefficients is shown. The formula was used to obtain approximate expressions for the Riemann zeta function, psi function, polygamma function, as well as 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 terms of accuracy and brevity are shown.

Détail

Keywords: sum, row, generating function, Bernoulli's number, error.
Mots-clés : coefficient, correction

Bibliographie

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[4] Whittaker E. T. and Watson G. N., A Course of Modern Analysis, 4th Ed., Cambridge University Press, Cambridge, 1927 | MR