Geodesic
A Discrete Analog of Euler's Summation Formula
Matematičeskie zametki, Tome 71 (2002) no. 6, pp. 931-936
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In this paper, we prove a discrete analog of Euler's summation formula. The difference from the classical Euler formula is in that the derivatives are replaced by finite differences and the integrals by finite sums. Instead of Bernoulli numbers and Bernoulli polynomials, special numbers $P_n$ and special polynomials $P_n(x)$ introduced by Korobov in 1996 appear in the formula. 
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A. V. Ustinov. A Discrete Analog of Euler's Summation Formula. Matematičeskie zametki, Tome 71 (2002) no. 6, pp. 931-936. http://geodesic.mathdoc.fr/item/MZM_2002_71_6_a12/

[1] Korobov N. M., “Spetsialnye polinomy i ikh prilozheniya”, Diofantovy priblizheniya. Matem. zapiski, no. 2, MGU, M., 1996, 77–89 | MR

[2] Korobov N. M., Teoretikochislovye metody v priblizhennom analize, Fizmatgiz, M., 1963

[3] Gelfond A. O., Ischislenie konechnykh raznostei, Nauka, M., 1967