Euler-boole Summation Revisited
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چکیده
is a well-known formula from classical analysis giving a relation between the finite sum of values of a function f , whose first m derivatives are absolutely integrable on [a, n], and its integral, for a,m, n ∈ N, a < n. This elementary formula appears often in introductory texts [2, 17], usually in reference to a particular application—Stirling’s asymptotic formula. However, general approaches to such formulae are not often mentioned in the same context. In the formula above, the Bl are the Bernoulli numbers and the B̃l(x) are the periodic Bernoulli polynomials. The Bernoulli polynomials are most succinctly characterized by a generating function [1, 23.1.1]:
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Euler-Boole Summation Revisited
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