Theorems · Theorem · difference equations
fwdDiff_iter_eq_sum_shift
∀ {M : Type u_1} {G : Type u_2} [inst : AddCommMonoid M] [inst_1 : AddCommGroup G] (h : M) (f : M → G) (n : ℕ) (y : M),
(fwdDiff h)^[n] f y = ∑ k ∈ Finset.range (n + 1), ((-1) ^ (n - k) * ↑(n.choose k)) • f (y + k • h)Express the n-th forward difference of f at y in terms of the values f (y + k), for
0 ≤ k ≤ n.
- Defined in
- Mathlib.Algebra.Group.ForwardDiff
- Cited by
- 5 results in Mathlib
- Foundations
- Depth 72 from the axioms · uses propext, Classical.choice, Quot.sound
- Assumes
- AddCommMonoidAddCommGroup
Around this declaration
Dashed lines are statement dependencies; solid lines are citations in proofs.
Cites36
Mathlib declarations this one mentions in its statement or cites explicitly in its proof. Plumbing is filtered out.
- DFunLike.coeproof · cited by 62,936
- Finsetproof · cited by 13,712
- AddCommGroupstatement and proof · cited by 12,871
- AddCommMonoidstatement and proof · cited by 12,281
- Finset.sumstatement and proof · cited by 5,195
- Nat.cast_oneproof · cited by 2,501
- mul_assocproof · cited by 1,667
- Finset.rangestatement and proof · cited by 1,341
- sub_eq_add_negproof · cited by 1,023
- Module.Endproof · cited by 774
- Nat.iteratestatement and proof · cited by 740
- Commuteproof · cited by 639
Cited by5
Results whose statement or proof uses this declaration.
- IsUltrametricDist.norm_fwdDiff_iter_apply_leproof · cited by 1
- Polynomial.fwdDiff_iter_eq_zero_of_degree_ltproof · cited by 1
- fwdDiff_iter_choose_zeroproof · cited by 1
- fwdDiff_iter_comp_addproof · cited by 1
- PadicInt.fwdDiff_mahlerSeriesproof · cited by 0