Polinomios de Bernoulli

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Los primeros Polinomios de Bernoulli son:

${\displaystyle B_{0}(x)=1\,}$

${\displaystyle B_{1}(x)=x-{\frac {1}{2}}\,}$

${\displaystyle B_{2}(x)=x^{2}-x+{\frac {1}{6}}\,}$

${\displaystyle B_{3}(x)=x^{3}-{\frac {3}{2}}x^{2}+{\frac {1}{2}}x\,}$

${\displaystyle B_{4}(x)=x^{4}-2x^{3}+x^{2}-{\frac {1}{30}}\,}$

${\displaystyle B_{5}(x)=x^{5}-{\frac {5}{2}}x^{4}+{\frac {5}{3}}x^{3}-{\frac {1}{6}}x\,}$

${\displaystyle B_{6}(x)=x^{6}-3x^{5}+{\frac {5}{2}}x^{4}-{\frac {1}{2}}x^{2}+{\frac {1}{42}}\,}$ .

${\displaystyle B_{7}(x)=x^{7}-{\frac {7}{2}}x^{6}+{\frac {7}{2}}x^{5}-{\frac {7}{6}}x^{3}+{\frac {1}{6}}x\,}$ .

${\displaystyle B_{8}(x)=x^{8}-4x^{7}+{\frac {14}{3}}x^{6}-{\frac {7}{3}}x^{4}+{\frac {2}{3}}x^{2}-{\frac {1}{30}}\,}$ .

${\displaystyle B_{9}(x)=x^{9}-{\frac {9}{2}}x^{8}+6x^{7}-{\frac {21}{5}}x^{5}+2x^{3}-{\frac {3}{10}}x\,}$ .

En,^[2][3] se deduce y demuestra que los polinomios de Bernoulli se pueden obtener mediante la siguiente recurrencia integral

${\displaystyle B_{m}(x)=m\int _{0}^{x}B_{m-1}(t)dt-m\int _{0}^{1}\int _{0}^{t}B_{m-1}(s)dsdt.}$

• Número de Bernoulli.
• Polinomios para sumas de potencias de progresiones aritméticas