§24.12 Zeros

In the interval $0\le x\le 1$ the only zeros of $B_{2n+1}\left(x\right)$, $n=1,2,\mathrm{\dots }$, are $0,\frac{1}{2},1$, and the only zeros of $B_{2n}\left(x\right)-B_{2n}$, $n=1,2,\mathrm{\dots }$, are $0,1$.

For the interval denote the zeros of $B_n\left(x\right)$ by $x_j^{(n)}$, $j=1,2,\mathrm{\dots }$, with

Then the zeros in the interval are $1-x_j^{(n)}$.

When $n(\ge 2)$ is even

and as $n\to \mathrm{\infty }$ with $m(\ge 1)$ fixed,

When $n$ is odd $x_1^{(n)}=\frac{1}{2}$, $x_2^{(n)}=1$ $(n\ge 3)$, and as $n\to \mathrm{\infty }$ with $m(\ge 1)$ fixed,

Let $R(n)$ be the total number of real zeros of $B_n\left(x\right)$. Then $R(n)=n$ when $1\le n\le 5$, and

For the interval denote the zeros of $E_n\left(x\right)$ by $y_j^{(n)}$, $j=1,2,\mathrm{\dots }$, with

Then the zeros in the interval are $1-y_j^{(n)}$.

When $n(\ge 2)$ is even $y_1^{(n)}=1$, and as $n\to \mathrm{\infty }$ with $m(\ge 1)$ fixed,

When $n$ is odd $y_1^{(n)}=\frac{1}{2}$,

and as $n\to \mathrm{\infty }$ with $m(\ge 1)$ fixed,

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).

$B_n\left(x\right)$, $n=1,2,\mathrm{\dots }$, has no multiple zeros. The only polynomial $E_n\left(x\right)$ with multiple zeros is $E_5\left(x\right)=(x-\frac{1}{2}){(x^2-x-1)}^2$.