| $${\mathbf n}$$ | Bernoulli Numbers | Bernoulli Polynomials | Euler Numbers | Euler Polynomials |
|---|---|---|---|---|
| 0 | $${1}$$ | $${1}$$ | $${1}$$ | $${1}$$ |
| 1 | $${- \frac{1}{2}}$$ | $${x - \frac{1}{2}}$$ | $${0}$$ | $${x - \frac{1}{2}}$$ |
| 2 | $${\frac{1}{6}}$$ | $${x^2 - x + \frac{1}{6}}$$ | $${-1}$$ | $${x^2 - x}$$ |
| 3 | $${0}$$ | $${x^3 - \frac{3}{2} x^2 + \frac{1}{2} x}$$ | $${0}$$ | $${x^3 - \frac{3}{2} x^2 + \frac{1}{4} }$$ |
| 4 | $${-\frac{1}{30}}$$ | $${x^4 - 2x^3 + x^2 -\frac{1}{30}}$$ | $${5}$$ | $${x^4 - 2x^3 + x }$$ |
| 5 | $${0}$$ | $${ x^5 - \frac{5}{2} x^4 + \frac{5}{3}x^3 - \frac{1}{6} x }$$ | $${0}$$ | $${ x^5 - \frac{5}{2} x^4 + \frac{5}{2}x^2 - \frac{1}{2} }$$ |
| 6 | $${\frac{1}{42}}$$ | $${ x^6 - 3x^5 + \frac{5}{2} x^4 - \frac{1}{2} x^2 + \frac{1}{42}}$$ | $${-61}$$ | $${ x^6 - 3x^5 + 5 x^3 - 3 x}$$ |
Bermoulli Numbers and Polynomials
\({\displaystyle \frac{x}{e^x - 1} = \sum_{k = 0}^{\infty} \frac{B_k}{k!} x^k}\)
\({\displaystyle x = (e^x - 1) \sum_{k = 0}^{\infty} \frac{B_k}{k!} x^k}\)
\({\displaystyle B_m = - \frac{1}{m+1} \sum_{n=0}^{m-1} \binom{m+1}{n} B_n }\)
\({B_{2n+1} = 0, \ n \in \mathbb{Z}^+ }\)
\({B_{n}'(x) = n B_{n-1}(x) }\)
\({ \displaystyle B_{n}(x) = \int{n B_{n-1}(x) \ dx } }\)
\({ \displaystyle \int_{0}^{1}{ B_{n}(x) \ dx = 0} }\)
Euler Numbers and Polynomials
\({\displaystyle \sum_{n=0}^{\infty} E_n \frac{x^n}{n!} = \text{sech}(x) }\)
\({\displaystyle \sum_{n=0}^{\infty} E_n(x) \frac{z^n}{n!} = \frac{2e^{xz}}{e^z + 1} }\)
\({E_{n}'(x) = n E_{n-1}(x) }\)
\({E_{n}(x + 1) + E_{n}(x) = 2 x^n }\)
\({\displaystyle E_n = 2^n E_n \left( \frac{1}{2} \right) }\)
\({E_{2n} = 0, \ n \in \mathbb{Z}^+ }\)
More information:
Polygamma Cheat Sheet