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24 Bernoulli and Euler PolynomialsProperties

§24.8 Series Expansions

Contents

§24.8(i) Fourier Series

If n=1,2, and 0x1, then

24.8.1 B2n(x) =(-1)n+12(2n)!(2π)2nk=1cos(2πkx)k2n,
24.8.2 B2n+1(x) =(-1)n+12(2n+1)!(2π)2n+1k=1sin(2πkx)k2n+1.

The second expansion holds also for n=0 and 0<x<1.

If n=1 with 0<x<1, or n=2,3, with 0x1, then

24.8.3 Bn(x)=-n!(2πi)nk=-k0e2πikxkn.

If n=1,2, and 0x1, then

24.8.4 E2n(x) =(-1)n4(2n)!π2n+1k=0sin((2k+1)πx)(2k+1)2n+1,
24.8.5 E2n-1(x) =(-1)n4(2n-1)!π2nk=0cos((2k+1)πx)(2k+1)2n.

§24.8(ii) Other Series

24.8.6 B4n+2 =(8n+4)k=1k4n+1e2πk-1,
n=1,2,,
24.8.7 B2n =(-1)n+14n22n-1k=1k2n-1eπk+(-1)k+n,
n=2,3,.

Let αβ=π2. Then

24.8.8 B2n4n(αn-(-β)n)=αnk=1k2n-1e2αk-1-(-β)nk=1k2n-1e2βk-1,
n=2,3,.
24.8.9 E2n=(-1)nk=1k2ncosh(12πk)-4k=0(-1)k(2k+1)2ne2π(2k+1)-1,
n=1,2,.