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General Euler-Boole's and dual Euler-Boole's formulas

The aim of this talk is to give general Euler-Boole's and dual Euler-Boole's formulae. More precisely, we derive formulae of Boole type where the integral is approximated not only with the values of the function in certain points but also with values of its derivatives up to (2n-1)-th order in end points of the interval. Our method produces formulae of arbitrary degree of exactness. Dual Euler-Boole's formulae are derived by analogy with Simpson's and dual Simpson's rule, and Simpson's 3/8 and Maclaurin rule. Finally, by analogy with Bullen-Simpson's and Bullen-Simpson's 3/8 inequality, general Bullen-Boole's inequality for a class of (2k)-convex functions is derived.

Extended Euler formulae; multiplication theorem for Bernoulli polynomials; Boole's quadrature formula; dual Boole's quadrature formula; sharp estimates of the quadrature formulae; Bullen-Boole's inequality; (2k)-convex functions

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