Euler-Maclaurin Method for Linear Differential Equations with Piecewise Constant Arguments with One Delay: Stability and Oscillations

and Applied Analysis 3 Proposition 8 (see [33]). The Bernoulli’s polynomial Bj(x) satisfies the following properties: (i) B 0 (x) = 1, B 1 (x) = x − 1/2, B k (x) = ∑ k j=0 C j k B j x , (ii) B 2k+1 (1) = B 2k+1 (0) = B 2k+1 = 0, (iii) B 2k (1) = B 2k (0) = B 2k , (iv) B k (x) = (1/(k + 1))B 󸀠 k+1 (x), k = 1, 2, . . . . 3.2. The Euler-Maclaurin Method and Discretization. Let q > 0 be an integer, and assume that the function f(t) is at least 2q + 2 times continuously differentiable on [c, d]. Further assume that h evenly divides c and d; then Atkinson’s version of the Euler-Maclaurin formula [34] is as follows: