# §24.12 Zeros

## §24.12(i) Bernoulli Polynomials: Real Zeros

In the interval $0\leq x\leq 1$ the only zeros of $B_{2n+1}\left(x\right)$, $n=1,2,\ldots$, are $0,\tfrac{1}{2},1$, and the only zeros of $B_{2n}\left(x\right)-B_{2n}$, $n=1,2,\ldots$, are $0,1$.

For the interval $\tfrac{1}{2}\leq x<\infty$ denote the zeros of $B_{n}\left(x\right)$ by $x_{j}^{(n)}$, $j=1,2,\ldots$, with

 24.12.1 $\tfrac{1}{2}\leq x_{1}^{(n)}\leq x_{2}^{(n)}\leq\cdots.$ ⓘ Symbols: $n$: integer and $x$: real or complex Permalink: http://dlmf.nist.gov/24.12.E1 Encodings: TeX, pMML, png See also: Annotations for 24.12(i), 24.12 and 24

Then the zeros in the interval $-\infty are $1-x_{j}^{(n)}$.

When $n(\geq 2)$ is even

 24.12.2 $\displaystyle\frac{3}{4}+\frac{1}{2^{n+2}\pi}$ $\displaystyle ⓘ Symbols: $\pi$: the ratio of the circumference of a circle to its diameter, $n$: integer and $x$: real or complex Referenced by: §24.12(i) Permalink: http://dlmf.nist.gov/24.12.E2 Encodings: TeX, pMML, png See also: Annotations for 24.12(i), 24.12 and 24 24.12.3 $\displaystyle x^{(n)}_{1}-\frac{3}{4}$ $\displaystyle\sim\frac{1}{2^{n+1}\pi},$ $n\to\infty$, ⓘ Symbols: $\sim$: asymptotic equality, $\pi$: the ratio of the circumference of a circle to its diameter, $n$: integer and $x$: real or complex Permalink: http://dlmf.nist.gov/24.12.E3 Encodings: TeX, pMML, png See also: Annotations for 24.12(i), 24.12 and 24

and as $n\to\infty$ with $m(\geq 1)$ fixed,

 24.12.4 $\displaystyle x^{(n)}_{2m-1}$ $\displaystyle\to m-\tfrac{1}{4},$ $\displaystyle x^{(n)}_{2m}$ $\displaystyle\to m+\tfrac{1}{4}.$ ⓘ Symbols: $m$: integer, $n$: integer and $x$: real or complex Permalink: http://dlmf.nist.gov/24.12.E4 Encodings: TeX, TeX, pMML, pMML, png, png See also: Annotations for 24.12(i), 24.12 and 24

When $n$ is odd $x^{(n)}_{1}=\frac{1}{2}$, $x^{(n)}_{2}=1$ $(n\geq 3)$, and as $n\to\infty$ with $m(\geq 1)$ fixed,

 24.12.5 $\displaystyle x^{(n)}_{2m-1}$ $\displaystyle\to m-\tfrac{1}{2},$ $\displaystyle x^{(n)}_{2m}$ $\displaystyle\to m.$ ⓘ Symbols: $m$: integer, $n$: integer and $x$: real or complex Permalink: http://dlmf.nist.gov/24.12.E5 Encodings: TeX, TeX, pMML, pMML, png, png See also: Annotations for 24.12(i), 24.12 and 24

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

 24.12.6 $R(n)\sim 2n/(\pi e),$ $n\to\infty$.

## §24.12(ii) Euler Polynomials: Real Zeros

For the interval $\frac{1}{2}\leq x<\infty$ denote the zeros of $E_{n}\left(x\right)$ by $y^{(n)}_{j}$, $j=1,2,\ldots$, with

 24.12.7 $\tfrac{1}{2}\leq y^{(n)}_{1}\leq y^{(n)}_{2}\leq\cdots.$ ⓘ Symbols: $n$: integer and $y^{(n)}_{j}$: zeros Permalink: http://dlmf.nist.gov/24.12.E7 Encodings: TeX, pMML, png See also: Annotations for 24.12(ii), 24.12 and 24

Then the zeros in the interval $-\infty are $1-y^{(n)}_{j}$.

When $n(\geq 2)$ is even $y^{(n)}_{1}=1$, and as $n\to\infty$ with $m(\geq 1)$ fixed,

 24.12.8 $y^{(n)}_{m}\to m.$ ⓘ Symbols: $m$: integer, $n$: integer and $y^{(n)}_{j}$: zeros Permalink: http://dlmf.nist.gov/24.12.E8 Encodings: TeX, pMML, png See also: Annotations for 24.12(ii), 24.12 and 24

When $n$ is odd $y^{(n)}_{1}=\tfrac{1}{2}$,

 24.12.9 $\frac{3}{2}-\frac{\pi^{n+1}}{3(n!)} $n=3,7,11,\dots$,
 24.12.10 $\frac{3}{2} $n=5,9,13,\dots$,

and as $n\to\infty$ with $m(\geq 1)$ fixed,

 24.12.11 $y^{(n)}_{2m}\to m-\tfrac{1}{2}.$ ⓘ Symbols: $m$: integer, $n$: integer and $y^{(n)}_{j}$: zeros Permalink: http://dlmf.nist.gov/24.12.E11 Encodings: TeX, pMML, png See also: Annotations for 24.12(ii), 24.12 and 24

## §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

$B_{n}\left(x\right)$, $n=1,2,\ldots$, has no multiple zeros. The only polynomial $E_{n}\left(x\right)$ with multiple zeros is $E_{5}\left(x\right)=(x-\frac{1}{2})(x^{2}-x-1)^{2}$.