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number_theory.bernoulli_polynomials

Bernoulli polynomials #

The 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, $$ \frac{t e^{tX} }{ e^t - 1} = ∑_{n = 0}^{\infty} B_n(X) \frac{t^n}{n!} $$

Implementation detail #

Bernoulli polynomials are defined using bernoulli, the Bernoulli numbers.

Main theorems #

TODO #

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

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 = (finset.range (n + 1)).sum (λ (i : ℕ), ⇑(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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theorem polynomial.derivative_bernoulli_add_one (k : ℕ) :
⇑polynomial.derivative (polynomial.bernoulli (k + 1)) = (↑k + 1) * polynomial.bernoulli k
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theorem polynomial.derivative_bernoulli (k : ℕ) :
⇑polynomial.derivative (polynomial.bernoulli k) = ↑k * polynomial.bernoulli (k - 1)
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@[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)
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theorem polynomial.bernoulli_eq_sub_sum (n : ℕ) :
↑(n.succ) • polynomial.bernoulli n = ⇑(polynomial.monomial n) ↑(n.succ) - (finset.range n).sum (λ (k : ℕ), ↑((n + 1).choose k) • polynomial.bernoulli k)

Another version of polynomial.sum_bernoulli.

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theorem polynomial.sum_range_pow_eq_bernoulli_sub (n p : ℕ) :
(↑p + 1) * (finset.range n).sum (λ (k : ℕ), ↑k ^ p) = polynomial.eval ↑n (polynomial.bernoulli p.succ) - bernoulli p.succ

Another version of bernoulli.sum_range_pow.

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theorem polynomial.bernoulli_succ_eval (n p : ℕ) :
polynomial.eval ↑n (polynomial.bernoulli p.succ) = bernoulli p.succ + (↑p + 1) * (finset.range n).sum (λ (k : ℕ), ↑k ^ p)

Rearrangement of polynomial.sum_range_pow_eq_bernoulli_sub.

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theorem polynomial.bernoulli_eval_one_add (n : ℕ) (x : ℚ) :
polynomial.eval (1 + x) (polynomial.bernoulli n) = polynomial.eval x (polynomial.bernoulli n) + ↑n * x ^ (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.factorial)) • polynomial.bernoulli n)) * (power_series.exp A - 1) = power_series.X * ⇑(power_series.rescale t) (power_series.exp A)

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