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 #

source
noncomputable def polynomial.bernoulli (n : ℕ) :
polynomial ℚ

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

Equations
source
theorem polynomial.bernoulli_def (n : ℕ) :
polynomial.bernoulli n = (finset.range (n + 1)).sum (λ (i : ℕ), ⇑(polynomial.monomial i) (bernoulli (n - i) * ↑(n.choose i)))
source
@[simp]
theorem polynomial.bernoulli_zero  :
polynomial.bernoulli 0 = 1
source
@[simp]
theorem polynomial.bernoulli_eval_zero (n : ℕ) :
polynomial.eval 0 (polynomial.bernoulli n) = bernoulli n
source
@[simp]
theorem polynomial.bernoulli_eval_one (n : ℕ) :
polynomial.eval 1 (polynomial.bernoulli n) = bernoulli' n
source
@[simp]
theorem polynomial.sum_bernoulli (n : ℕ) :
(finset.range (n + 1)).sum (λ (k : ℕ), ↑((n + 1).choose k) • polynomial.bernoulli k) = ⇑(polynomial.monomial n) (↑n + 1)
source
theorem polynomial.bernoulli_generating_function {A : Type u_1} [comm_ring A] [algebra ℚ A] (t : A) :
power_series.mk (λ (n : ℕ), ⇑(polynomial.aeval t) ((1 / ↑(n.factorial)) • polynomial.bernoulli n)) * (power_series.exp A - 1) = power_series.X * ⇑(power_series.rescale t) (power_series.exp A)

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