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 #

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noncomputable def polynomial.bernoulli (n : ℕ) :
ℚ[X]

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

Equations
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theorem polynomial.bernoulli_def (n : ℕ) :
polynomial.bernoulli n = ∑ (i : ℕ) in finset.range (n + 1), ⇑(polynomial.monomial i) ((bernoulli (n - i)) * ↑(n.choose i))
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@[simp]
theorem polynomial.bernoulli_zero  :
polynomial.bernoulli 0 = 1
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@[simp]
theorem polynomial.bernoulli_eval_zero (n : ℕ) :
polynomial.eval 0 (polynomial.bernoulli n) = bernoulli n
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@[simp]
theorem polynomial.bernoulli_eval_one (n : ℕ) :
polynomial.eval 1 (polynomial.bernoulli n) = bernoulli' n
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@[simp]
theorem polynomial.sum_bernoulli (n : ℕ) :
∑ (k : ℕ) in finset.range (n + 1), ↑((n + 1).choose k) • polynomial.bernoulli k = ⇑(polynomial.monomial n) (↑n + 1)
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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!) • 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}