# mathlibdocumentation

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 #

• sum_bernoulli: The sum of the $k^\mathrm{th}$ Bernoulli polynomial with binomial coefficients up to n is (n + 1) * X^n.
• bernoulli_generating_function: The Bernoulli polynomials act as generating functions for the exponential.

## TODO #

• bernoulli_eval_one_neg : $$B_n(1 - x) = (-1)^n*B_n(x)$$
noncomputable def polynomial.bernoulli (n : ) :

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

Equations
theorem polynomial.bernoulli_def (n : ) :
= (finset.range (n + 1)).sum (λ (i : ), (bernoulli (n - i) * (n.choose i)))
@[simp]
@[simp]
@[simp]
@[simp]
theorem polynomial.sum_bernoulli (n : ) :
(finset.range (n + 1)).sum (λ (k : ), ((n + 1).choose k) = (n + 1)
theorem polynomial.bernoulli_generating_function {A : Type u_1} [comm_ring A] [ A] (t : A) :
power_series.mk (λ (n : ), ((1 / (n.factorial)) * - 1) =

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