Bernoulli B
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
Theorem: The following formula holds: $$\dfrac{te^{xt}}{e^t-1} = \displaystyle\sum_{k=0}^{\infty} B_k(x)\dfrac{t^k}{k!}.$$
Proof: █
Theorem: The following formula holds: $$\displaystyle\int_a^x B_n(t) dt = \dfrac{B_{n+1}(x)-B_{n+1}(a)}{n+1}.$$
Proof: █
Theorem: The following formula holds: $$B_n(mx)=m^{n-1}\displaystyle\sum_{k=0}^{m-1} B_n \left( x + \dfrac{k}{m} \right).$$
Proof: █
Bernoulli polynomial and Hurwitz zeta
