4 1. Preliminary Notions Similarly, I3 = (y y )f(y) f( y + 1) y y +1 f(t) dt = ( y− y)f(y) y +1 y f(t) dt + f( y + 1). Since I1 = I I2 I3, it follows that b n=a+2 f(n) = b a+1 f(t) dt + x y (t t )f (t) dt (x x )f(x) + x b f(t) dt f( y + 1) ( y− y)f(y) + a+1 y f(t) dt. From this, we get that b n=a+1 f(n) = x y f(t) dt+ x y (t−t )f (t) dt+( x−x)f(x)−( y−y)f(y), which proves (1.3). The Euler-MacLaurin formula can be considerably generalized. For in- stance, one can show the following. Proposition 1.3. Let ν be a positive integer. Let f be a function such that the derivatives f , f , . . . , f (2ν) are all continuous on the interval [M, N]. Then, N n=M f(n) = N M f(t) dt + 1 2 [f(M) + f(N)] + B2 2! f (x) N M + B4 4! f (x) N M + · · · + B2ν (2ν)! f (2ν−1) (x) N M 1 (2ν)! N M B2ν(t t )f (2ν) (t) dt, where the Bi’s are the Bernoulli numbers defined implicitly by i=0 Bi yi i! = y ey 1 and where Bi(x) stands for the i-th Bernoulli polynomial which is defined as the unique polynomial of degree i satisfying to x+1 x Bi(u) du = xi. Thus, B1(u) = u 1 2 , B2(u) = u2 u + 1 6 , B3(u) = u3 3 2 u2 + 1 2 u, . . . .
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