24 Bernoulli and Euler Polynomials Properties 24.11 Asymptotic Approximations 24.13 Integrals

§24.12 Zeros

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Permalink:: http://dlmf.nist.gov/24.12
See also:: Annotations for Ch.24

§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

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Keywords:: 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).
Referenced by:: §2.10(i)
Permalink:: http://dlmf.nist.gov/24.12.i
See also:: Annotations for §24.12 and Ch.24

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

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

24.12.1 $\tfrac{1}{2}\leq x_{1}^{(n)}\leq x_{2}^{(n)}\leq\cdots.$

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Symbols:: : integer and : 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 Ch.24

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

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Symbols:: $\pi$ : the ratio of the circumference of a circle to its diameter, : integer and : 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 Ch.24

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

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Symbols:: $\sim$ : asymptotic equality, $\pi$ : the ratio of the circumference of a circle to its diameter, : integer and : 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 Ch.24

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

24.12.4

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

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

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Symbols:: : integer, : integer and : 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 Ch.24

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

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

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

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Symbols:: : integer, : integer and : 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 Ch.24

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

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

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Symbols:: $\sim$ : asymptotic equality, $\pi$ : the ratio of the circumference of a circle to its diameter, $\mathrm{e}$ : base of natural logarithm, : integer and : number of zeros
Permalink:: http://dlmf.nist.gov/24.12.E6
Encodings:: TeX, pMML, png
See also:: Annotations for §24.12(i), §24.12 and Ch.24

§24.12(ii) Euler Polynomials: Real Zeros

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Keywords:: Euler polynomials, real, zeros
Notes:: See Howard (1976) and Delange (1988).
Permalink:: http://dlmf.nist.gov/24.12.ii
See also:: Annotations for §24.12 and Ch.24

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

24.12.7 $\tfrac{1}{2}\leq y^{(n)}_{1}\leq y^{(n)}_{2}\leq\cdots.$

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Symbols:: : 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 Ch.24

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

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Symbols:: : integer, : 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 Ch.24

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

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

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Symbols:: $\pi$ : the ratio of the circumference of a circle to its diameter, : factorial (as in ), : integer and $y^{(n)}_{j}$ : zeros
Permalink:: http://dlmf.nist.gov/24.12.E9
Encodings:: TeX, pMML, png
See also:: Annotations for §24.12(ii), §24.12 and Ch.24

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

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Symbols:: $\pi$ : the ratio of the circumference of a circle to its diameter, : factorial (as in ), : integer and $y^{(n)}_{j}$ : zeros
Permalink:: http://dlmf.nist.gov/24.12.E10
Encodings:: TeX, pMML, png
See also:: Annotations for §24.12(ii), §24.12 and Ch.24

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

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

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Symbols:: : integer, : 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 Ch.24

§24.12(iii) Complex Zeros

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Keywords:: Bernoulli polynomials, Euler polynomials, complex, zeros
Permalink:: http://dlmf.nist.gov/24.12.iii
See also:: Annotations for §24.12 and Ch.24

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

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Keywords:: Bernoulli polynomials, Euler polynomials, multiple, zeros
Notes:: See Dilcher (2008) and Brillhart (1969).
Permalink:: http://dlmf.nist.gov/24.12.iv
See also:: Annotations for §24.12 and Ch.24

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

24.11 Asymptotic Approximations 24.13 Integrals

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§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