24 Bernoulli and Euler Polynomials Properties 24.11 Asymptotic Approximations 24.13 Integrals

§24.12 Zeros

Permalink:: http://dlmf.nist.gov/24.12

§24.12(i) Bernoulli Polynomials: Real Zeros
§24.12(ii) Euler Polynomials: Real Zeros
§24.12(iii) Complex Zeros
§24.12(iv) Multiple Zeros

§24.12(i) Bernoulli Polynomials: Real Zeros

Notes:: See Inkeri (1959), Leeming (1989), and Delange (1991). For the first paragraph see Olver (1997b, p. 283). (24.12.2) follows from a result in Lehmer (1940); see also Dilcher (1988, p. 77).
Keywords:: Bernoulli polynomials
Referenced by:: ¶ ‣ §2.10(i)
Permalink:: http://dlmf.nist.gov/24.12.i

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

For the interval $\tfrac{1}{2}\leq x<\infty$ denote the zeros of $\mathop{B_{{n}}\/}\nolimits\!\left(x\right)$ 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:: : integer and : real or complex
Permalink:: http://dlmf.nist.gov/24.12.E1
Encodings:: TeX, pMML, png

Then the zeros in the interval $-\infty<x\leq\frac{1}{2}$ are $1-x_{j}^{{(n)}}$ .

When $n(\geq 2)$ is even

24.12.2 $\frac{3}{4}+\frac{1}{2^{{n+2}}\pi}<x^{{(n)}}_{1}<\frac{3}{4}+\frac{1}{2^{{n+1}% }\pi},$

Symbols:: : integer and : real or complex
Referenced by:: §24.12(i)
Permalink:: http://dlmf.nist.gov/24.12.E2
Encodings:: TeX, pMML, png

24.12.3 $x^{{(n)}}_{1}-\frac{3}{4}\sim\frac{1}{2^{{n+1}}\pi},$ $n\to\infty$ ,

Symbols:: $\sim$ : asymptotic equality, : integer and : real or complex
Permalink:: http://dlmf.nist.gov/24.12.E3
Encodings:: TeX, pMML, png

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

24.12.4

$x^{{(n)}}_{{2m-1}}\to m-\tfrac{1}{4},$

$x^{{(n)}}_{{2m}}\to m+\tfrac{1}{4}.$

Symbols:: : integer, : integer and : real or complex
Permalink:: http://dlmf.nist.gov/24.12.E4
Encodings:: TeX, TeX, pMML, pMML, png, png

When 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

$x^{{(n)}}_{{2m-1}}\to m-\tfrac{1}{2},$

$x^{{(n)}}_{{2m}}\to m.$

Symbols:: : integer, : integer and : real or complex
Permalink:: http://dlmf.nist.gov/24.12.E5
Encodings:: TeX, TeX, pMML, pMML, png, png

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

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

Symbols:: : base of exponential function, $\sim$ : asymptotic equality, : integer and : number of zeros
Permalink:: http://dlmf.nist.gov/24.12.E6
Encodings:: TeX, pMML, png

§24.12(ii) Euler Polynomials: Real Zeros

Notes:: See Howard (1976) and Delange (1988).
Keywords:: Euler polynomials
Permalink:: http://dlmf.nist.gov/24.12.ii

For the interval $\frac{1}{2}\leq x<\infty$ denote the zeros of $\mathop{E_{{n}}\/}\nolimits\!\left(x\right)$ 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:: : integer and $y^{{(n)}}_{j}$ : zeros
Permalink:: http://dlmf.nist.gov/24.12.E7
Encodings:: TeX, pMML, png

Then the zeros in the interval $-\infty<x\leq\frac{1}{2}$ 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:: : integer, : integer and $y^{{(n)}}_{j}$ : zeros
Permalink:: http://dlmf.nist.gov/24.12.E8
Encodings:: TeX, pMML, png

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

24.12.9 $\frac{3}{2}-\frac{\pi^{{n+1}}}{3(n!)}<y^{{(n)}}_{2}<\frac{3}{2},$ $n=3,7,11,\dots$ ,

Symbols:: : : factorial, : integer and $y^{{(n)}}_{j}$ : zeros
Permalink:: http://dlmf.nist.gov/24.12.E9
Encodings:: TeX, pMML, png

24.12.10 $\frac{3}{2}<y^{{(n)}}_{2}<\frac{3}{2}+\frac{\pi^{{n+1}}}{3(n!)},$ $n=5,9,13,\dots$ ,

Symbols:: : : factorial, : integer and $y^{{(n)}}_{j}$ : zeros
Permalink:: http://dlmf.nist.gov/24.12.E10
Encodings:: TeX, pMML, png

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

24.12.11 $y^{{(n)}}_{{2m}}\to m-\tfrac{1}{2}.$

Symbols:: : integer, : integer and $y^{{(n)}}_{j}$ : zeros
Permalink:: http://dlmf.nist.gov/24.12.E11
Encodings:: TeX, pMML, png

§24.12(iii) Complex Zeros

Keywords:: Bernoulli polynomials, Euler polynomials
Permalink:: http://dlmf.nist.gov/24.12.iii

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

Notes:: See Dilcher (2008) and Brillhart (1969).
Keywords:: Bernoulli polynomials, Euler polynomials
Permalink:: http://dlmf.nist.gov/24.12.iv

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

© 2010–2013 NIST / Privacy Policy / Disclaimer / Feedback; Version 1.0.6; Release date 2013-05-06 Math-Image version (See MathML) . A Printed Companionis available. 24.11 Asymptotic Approximations 24.13 Integrals

§24.12 Zeros

Contents

§24.12(i) Bernoulli Polynomials: Real Zeros

§24.12(ii) Euler Polynomials: Real Zeros

§24.12(iii) Complex Zeros

§24.12(iv) Multiple Zeros