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24 Bernoulli and Euler PolynomialsProperties

§24.12 Zeros

Contents

§24.12(i) Bernoulli Polynomials: Real Zeros

In the interval 0x1 the only zeros of B2n+1(x), n=1,2,, are 0,12,1, and the only zeros of B2n(x)-B2n, n=1,2,, are 0,1.

For the interval 12x< denote the zeros of Bn(x) by xj(n), j=1,2,, with

24.12.1 12x1(n)x2(n).

Then the zeros in the interval -<x12 are 1-xj(n).

When n(2) is even

24.12.2 34+12n+2π <x1(n)
<34+12n+1π,
24.12.3 x1(n)-34 12n+1π,
n,

and as n with m(1) fixed,

24.12.4 x2m-1(n) m-14,
x2m(n) m+14.

When n is odd x1(n)=12, x2(n)=1 (n3), and as n with m(1) fixed,

24.12.5 x2m-1(n) m-12,
x2m(n) m.

Let R(n) be the total number of real zeros of Bn(x). Then R(n)=n when 1n5, and

24.12.6 R(n)2n/(π),
n.

§24.12(ii) Euler Polynomials: Real Zeros

For the interval 12x< denote the zeros of En(x) by yj(n), j=1,2,, with

24.12.7 12y1(n)y2(n).

Then the zeros in the interval -<x12 are 1-yj(n).

When n(2) is even y1(n)=1, and as n with m(1) fixed,

24.12.8 ym(n)m.

When n is odd y1(n)=12,

24.12.9 32-πn+13(n!)<y2(n)<32,
n=3,7,11,,
24.12.10 32<y2(n)<32+πn+13(n!),
n=5,9,13,,

and as n with m(1) fixed,

24.12.11 y2m(n)m-12.

§24.12(iii) Complex Zeros

For complex zeros of Bernoulli and Euler polynomials, see Delange (1987) and Dilcher (1988). A related topic is the irreducibility of Bernoulli and Euler polynomials. For details and references, see Dilcher (1987b), Kimura (1988), or Adelberg (1992).

§24.12(iv) Multiple Zeros

Bn(x), n=1,2,, has no multiple zeros. The only polynomial En(x) with multiple zeros is E5(x)=(x-12)(x2-x-1)2.