Polynomial of Bernoulli

In Mathematical, the polynomials of Bernoulli appear in the study of many special functions and in particular, the Fonction Zeta of Riemann.

Definition

The polynomials of Bernoulli are the single continuation of polynomials $\backslash \; left\; (B\_n\; \backslash \; right)\; \_\; \{N\; \backslash \; in\; \backslash \; mathbb\; \{NR\}\}$ such as:

• $B\_0\; =\; 1$
• $\backslash \; forall\; N\; \backslash \; in\; \backslash \; mathbb\; \{NR\},\; B\text{'}\_\; \{n+1\}\; =\; (n+1)\; B\_n$
• $\backslash \; forall\; N\; \backslash \; in\; \backslash \; mathbb\; \{NR\},\; \backslash \; int\_0\; ^1\; B\_n\; (X)\; dx\; =\; 0$

Generating functions

The generating Fonction for the polynomials of Bernoulli is

$\backslash \; frac\; \{T\; e^\; \{xt\}\}\; \{e^t-1\}\; =\; \backslash \; sum\_\; \{n=0\}\; ^\; \backslash \; infty\; B\_n\; (X)\; \backslash \; frac\; \{t^n\}\; \{N!\}\backslash ,$.

The generating function for the polynomials of Euler is

$\backslash \; frac\; \{2e^\; \{xt\}\}\; \{e^t+1\}\; =\; \backslash \; sum\_\; \{n=0\}\; ^\; \backslash \; infty\; E\_n\; (X)\; \backslash \; frac\; \{t^n\}\; \{N!\}\backslash ,$.

Numbers of Euler and Bernoulli

The numbers of Bernoulli are given by $B\_n=B\_n\; (0)\; \backslash ,$.

The numbers of Euler are given by $E\_n=2^nE\_n\; (1/2)\; \backslash ,$.

Explicit expressions for the small orders

The few first polynomials of Bernoulli are:

$B\_0\; (X)\; =1\; \backslash ,$

$B\_1\; (X)\; =x-1/2\; \backslash ,$

$B\_2\; (X)\; =x^2-x+1/6\; \backslash ,$

$B\_3\; (X)\; =x^3-\; \backslash \; frac\; \{3\}\; \{2\}\; x^2+\; \backslash \; frac\; \{1\}\; \{2\}\; X\; \backslash ,$

$B\_4\; (X)\; =x^4-2x^3+x^2-\; \backslash \; frac\; \{1\}\; \{30\}\; \backslash ,$

$B\_5\; (X)\; =x^5-\; \backslash \; frac\; \{5\}\; \{2\}\; x^4+\; \backslash \; frac\; \{5\}\; \{3\}\; x^3-\; \backslash \; frac\; \{1\}\; \{6\}\; X\; \backslash ,$

$B\_6\; (X)\; =x^6-3x^5+\; \backslash \; frac\; \{5\}\; \{2\}\; x^4-\; \backslash \; frac\; \{1\}\; \{2\}\; x^2+\; \backslash \; frac\; \{1\}\; \{42\}\; \backslash ,$

The few first polynomials of Euler are:

$E\_0\; (X)\; =1\; \backslash ,$

$E\_1\; (X)\; =x-1/2\; \backslash ,$

$E\_2\; (X)\; =x^2-x\; \backslash ,$

$E\_3\; (X)\; =x^3-\; \backslash \; frac\; \{3\}\; \{2\}\; x^2+\; \backslash \; frac\; \{1\}\; \{4\}\; \backslash ,$

$E\_4\; (X)\; =x^4-2x^3+x\; \backslash ,$

$E\_5\; (X)\; =x^5-\; \backslash \; frac\; \{5\}\; \{2\}\; x^4+\; \backslash \; frac\; \{5\}\; \{2\}\; x^2-\; \backslash \; frac\; \{1\}\; \{2\}\; \backslash ,$

$E\_6\; (X)\; =x^6-3x^5+5x^3-3x\; \backslash ,$

Differences

The polynomials of Bernoulli and Euler obey many relations of the Calcul symbolic system used by Edouard Lucas, for example.

$B\_n\; (x+1)\; -\; B\_n\; (X)\; =nx^\; \{n-1\}\; \backslash ,$

$E\_n\; (x+1)\; +E\_n\; (X)\; =2x^\; \{N\}\; \backslash ,$

Derived

$B\_n\text{'}\; (X)\; =nB\_\; \{n-1\}\; (X)\; \backslash ,$

$E\_n\text{'}\; (X)\; =nE\_\; \{n-1\}\; (X)\; \backslash ,$

Translations

$B\_n\; (x+y)\; =\; \backslash \; sum\_\; \{k=0\}\; ^n\; \{N\; \backslash \; choose\; K\}\; B\_k\; (X)\; y^\; \{n-k\}\; \backslash ,$

$E\_n\; (x+y)\; =\; \backslash \; sum\_\; \{k=0\}\; ^n\; \{N\; \backslash \; choose\; K\}\; E\_k\; (X)\; y^\; \{n-k\}\; \backslash ,$

Symmetries

$B\_n\; (1-x)\; =\; (-\; 1)\; ^n\; B\_n\; (X)\; \backslash ,$

$E\_n\; (1-x)\; =\; (-\; 1)\; ^n\; E\_n\; (X)\; \backslash ,$

$(-\; 1)\; ^n\; B\_n\; (-\; X)\; =\; B\_n\; (X)\; +\; nx^\; \{n-1\}\; \backslash ,$

$(-\; 1)\; ^n\; E\_n\; (-\; X)\; =\; -\; E\_n\; (X)\; +\; 2x^n\; \backslash ,$

Other properties of the polynomials of Bernoulli

$\backslash \; forall\; N\; \backslash \; in\; \backslash \; mathbb\; \{NR\}\; B\_n\; (X)\; =2^\; \{n-1\}\; \backslash \; left\; (B\_n\; \backslash \; left\; (\backslash \; frac\; \{X\}\; \{2\}\; \backslash \; right)\; +\; B\_n\; \backslash \; left\; (\backslash \; frac\; \{x+1\}\; \{2\}\; \backslash \; right)\; \backslash \; right)$

Particular values

$\backslash \; forall\; N\; \backslash \; in\; \backslash \; mathbb\; \{NR\},\; B\_n\; (0)\; =B\_n\; (1)$

$\backslash \; forall\; p\; \backslash \; in\; \backslash \; mathbb\; \{NR\}\; ^\; \{*\},\; B\_\; \{2p+1\}\; (0)\; =\; B\_\; \{2p+1\}\; (1)\; =0$

$\backslash \; forall\; p\; \backslash \; in\; \backslash \; mathbb\; \{NR\}\; ^\; \{*\},\; B\_\; \{2p\}\; \backslash \; left\; (\backslash \; frac\; \{1\}\; \{2\}\; \backslash \; right)\; =\; \backslash \; left\; (\backslash \; frac\; \{1\}\; \{2^\; \{2p-1\}\}\; -1\; \backslash \; right)\; B\_\; \{2p\}\; (0),\; B\_\; \{2p+1\}\; \backslash \; left\; (\backslash \; frac\; \{1\}\; \{2\}\; \backslash \; right)\; =0$

Fourier series

The Fourier series of the polynomials of Bernoulli is also a Série of Dirichlet and is a particular case of the Fonction zeta of Hurwitz

$B\_n\; (X)\; =\; -\; \backslash \; Gamma\; (n+1)\; \backslash \; sum\_\; \{k=1\}\; ^\; \backslash \; infty$

\ frac {e^ {(2 \ pi ikx)}+ e^ {(2 \ pi ik (1-x))}} {(2 \ pi ik) ^n} \,

References

• Mr. Abramowitz and I.A. Stegun, eds. Handbook off Mathematical Functions with Formulated, Graphs, and Mathematical Tables , (1972) Dover, New York. (See Chapter 23.) ; wiki: Abramowitz and Stegun .

• Tom Mr. Apostol Introduction to Analytic Number Theory , (1976) Springer-Verlag, New York. (See Chapter 12.11)

• Linas Vepstas, The Bernoulli Operator, the Operator Gauss-Kuzmin-Wirsing, and the Riemann Zeta

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