Difference between revisions of "Bernoulli B"

From specialfunctionswiki
Jump to: navigation, search
 
(3 intermediate revisions by the same user not shown)
Line 10: Line 10:
  
 
=Properties=
 
=Properties=
<div class="toccolours mw-collapsible mw-collapsed">
+
[[Bernoulli polynomial and Hurwitz zeta]]<br />
<strong>Theorem:</strong> The following formula holds:
 
$$\dfrac{te^{xt}}{e^t-1} = \displaystyle\sum_{k=0}^{\infty} B_k(x)\dfrac{t^k}{k!}.$$
 
<div class="mw-collapsible-content">
 
<strong>Proof:</strong> █
 
</div>
 
</div>
 
  
<div class="toccolours mw-collapsible mw-collapsed">
+
=See Also=
<strong>Theorem:</strong> The following formula holds:
+
[[Bernoulli numbers]]<br />
$$\displaystyle\int_a^x B_n(t) dt = \dfrac{B_{n+1}(x)-B_{n+1}(a)}{n+1}.$$
+
{{:Orthogonal polynomials footer}}
<div class="mw-collapsible-content">
 
<strong>Proof:</strong> █
 
</div>
 
</div>
 
 
 
<div class="toccolours mw-collapsible mw-collapsed">
 
<strong>Theorem:</strong> The following formula holds:
 
$$B_n(mx)=m^{n-1}\displaystyle\sum_{k=0}^{m-1} B_n \left( x + \dfrac{k}{m} \right).$$
 
<div class="mw-collapsible-content">
 
<strong>Proof:</strong> █
 
</div>
 
</div>
 
 
 
{{:Bernoulli polynomial and Hurwitz zeta}}
 
  
{{:Orthogonal polynomials footer}}
+
[[Category:SpecialFunction]]

Latest revision as of 22:46, 20 June 2016

Bernoulli polynomials $B_n$ are orthogonal polynomials defined by the formula $$B_n(x)=\displaystyle\sum_{k=0}^n {n \choose k} b_{n-k}x^k,$$ where $b_k$ are Bernoulli numbers.

$$B_0(x)=1$$ $$B_1(x)=x-\dfrac{1}{2}$$ $$B_2(x)=x^2-x+\dfrac{1}{6}$$ $$B_3(x)=x^3-\dfrac{3x^2}{2}+\dfrac{x}{2}$$ $$B_4(x)=x^4-2x^3+x^2-\dfrac{1}{30}$$

Properties

Bernoulli polynomial and Hurwitz zeta

See Also

Bernoulli numbers

Orthogonal polynomials