mathlib documentation

number_theory.bernoulli_polynomials

Bernoulli polynomials #

The Bernoulli polynomials (defined here : https://en.wikipedia.org/wiki/Bernoulli_polynomials) are an important tool obtained from Bernoulli numbers.

Mathematical overview #

The $n$-th Bernoulli polynomial is defined as $$ B_n(X) = ∑_{k = 0}^n {n \choose k} (-1)^k * B_k * X^{n - k} $$ where $B_k$ is the $k$-th Bernoulli number. The Bernoulli polynomials are generating functions, $$ t * e^{tX} / (e^t - 1) = ∑_{n = 0}^{\infty} B_n(X) * \frac{t^n}{n!} $$

Implementation detail #

Bernoulli polynomials are defined using bernoulli, the Bernoulli numbers.

Main theorems #

TODO #

noncomputable def polynomial.bernoulli (n : ) :

The Bernoulli polynomials are defined in terms of the negative Bernoulli numbers.

Equations
@[simp]
theorem polynomial.sum_bernoulli (n : ) :

The theorem that ∑ Bₙ(t)X^n/n!)(e^X-1)=Xe^{tX}