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pepamontero merged 2 commits into from
Jul 10, 2026
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Generalize the sum to (n : R) ^ k * x ^ n from (n : R) ^ 2 * x ^ n #1

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87aeaf7
Add theorem `pow_eq_sum_stirlingSecond_mul_descFactorial` in `Stirlin…
Mal-Pat Jul 10, 2026
517f05b
Generalize the sum to `(n : R) ^ k * x ^ n` from `(n : R) ^ 2 * x ^ n`
Mal-Pat Jul 10, 2026
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111 changes: 72 additions & 39 deletions Mathlib/Analysis/SpecificLimits/Normed.lean
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Expand Up @@ -13,6 +13,7 @@ public import Mathlib.Analysis.Normed.Ring.InfiniteSum
public import Mathlib.Analysis.Normed.Module.Basic
public import Mathlib.Analysis.Normed.Order.Lattice
public import Mathlib.Analysis.SpecificLimits.Basic
public import Mathlib.Combinatorics.Enumerative.Stirling
public import Mathlib.Data.List.TFAE
public import Mathlib.Data.Nat.Choose.Bounds
public import Mathlib.Data.Nat.Choose.Cast
Expand Down Expand Up @@ -549,47 +550,79 @@ theorem tsum_coe_mul_geometric_of_norm_lt_one {r : 𝕜} (hr : ‖r‖ < 1) :
(∑' n : ℕ, n * r ^ n : 𝕜) = r / (1 - r) ^ 2 :=
(hasSum_coe_mul_geometric_of_norm_lt_one hr).tsum_eq

/-- If `‖r‖ < 1`, then `∑' n : ℕ, n ^ 2 * r ^ n = r * (1 + r) / (1 - r) ^ 3`, `HasSum` version in
a general ring with summable geometric series. For a version in a field, using division instead
of `Ring.inverse`, see `hasSum_sq_mul_geometric_of_norm_lt_one`. -/
theorem hasSum_sq_mul_geometric_of_norm_lt_one' {x : R} (h : ‖x‖ < 1) :
HasSum (fun n : ℕ ↦ (n : R) ^ 2 * x ^ n) (x * (1 + x) * ((1 - x)⁻¹ʳ) ^ 3) := by
have A : HasSum (fun n : ℕ ↦ ((n + 2).choose 2 : R) * x ^ n) ((1 - x)⁻¹ʳ ^ 3) :=
hasSum_choose_mul_geometric_of_norm_lt_one' 2 h
have B : HasSum (fun (n : ℕ) ↦ (n + 1) * x ^ n) ((1 - x)⁻¹ʳ ^ 2) := by
convert! hasSum_choose_mul_geometric_of_norm_lt_one' 1 h with n
simp
have C : HasSum (fun n ↦ n * x ^ n : ℕ → R) (x * (1 - x)⁻¹ʳ ^ 2) :=
hasSum_coe_mul_geometric_of_norm_lt_one' h
convert! ((A.mul_left 2).sub (B.mul_left 2)).sub C using 1
· ext n
rw [← mul_assoc, Nat.two_mul_cast_choose_two, pow_two, cast_add, cast_ofNat]
noncomm_ring
· symm
calc 2 * (1 - x)⁻¹ʳ ^ 3 - 2 * (1 - x)⁻¹ʳ ^ 2 - x * (1 - x)⁻¹ʳ ^ 2
_ = 2 * (1 - x)⁻¹ʳ ^ 3 - 2 * (((1 - x) * (1 - x)⁻¹ʳ) * (1 - x)⁻¹ʳ ^ 2)
- x * (((1 - x) * (1 - x)⁻¹ʳ) * (1 - x)⁻¹ʳ ^ 2) := by
simp [Ring.mul_inverse_cancel (1 - x) (isUnit_one_sub_of_norm_lt_one h)]
_ = x * (1 + x) * ((1 - x)⁻¹ʳ) ^ 3 := by noncomm_ring

/-- If `‖r‖ < 1`, then `∑' n : ℕ, n ^ 2 * r ^ n = r * (1 + r) / (1 - r) ^ 3`, version in a general
ring with summable geometric series. For a version in a field, using division instead of
`Ring.inverse`, see `tsum_sq_mul_geometric_of_norm_lt_one`. -/
theorem tsum_sq_mul_geometric_of_norm_lt_one' {x : R} (h : ‖x‖ < 1) :
∑' n : ℕ, (n : R) ^ 2 * x ^ n = x * (1 + x) * ((1 - x)⁻¹ʳ) ^ 3 :=
(hasSum_sq_mul_geometric_of_norm_lt_one' h).tsum_eq

/-- If `‖r‖ < 1`, then `∑' n : ℕ, n ^ 2 * r ^ n = r * (1 + r) / (1 - r) ^ 3`,
`HasSum` version. -/
theorem hasSum_sq_mul_geometric_of_norm_lt_one {r : 𝕜} (hr : ‖r‖ < 1) :
HasSum (fun n : ℕ ↦ (n : 𝕜) ^ 2 * r ^ n) (r * (1 + r) / (1 - r) ^ 3) := by
convert! hasSum_sq_mul_geometric_of_norm_lt_one' hr using 1
theorem hasSum_descFactorial_mul_geometric_of_norm_lt_one' (j : ℕ) {x : R} (h : ‖x‖ < 1) :
HasSum (fun n : ℕ ↦ (n.descFactorial j : R) * x ^ n)
((j.factorial : R) * x ^ j * ((1 - x)⁻¹ʳ) ^ (j + 1)) := by
have hx : Commute (x ^ j) (((1 - x)⁻¹ʳ) ^ (j + 1)) := by
obtain ⟨u, hu⟩ := isUnit_one_sub_of_norm_lt_one h
have hxu : Commute x (u : R) := hu.symm ▸ (Commute.one_right x).sub_right (Commute.refl x)
simp [← hu, hxu.units_inv_right.pow_pow]
have A := ((hasSum_choose_mul_geometric_of_norm_lt_one' j h).mul_left
(j.factorial : R)).mul_right (x ^ j)
have Afun : (fun m : ℕ ↦ (j.factorial : R) * (((m + j).choose j : R) * x ^ m) * x ^ j)
= fun m : ℕ ↦ ((m + j).descFactorial j : R) * x ^ (m + j) := by
funext m
push_cast [Nat.descFactorial_eq_factorial_mul_choose]
simp only [pow_add, mul_assoc]
rw [Afun] at A
have B := (hasSum_nat_add_iff (f := fun n : ℕ ↦ (n.descFactorial j : R) * x ^ n) j).mp A
have hzero : ∑ i ∈ Finset.range j, (i.descFactorial j : R) * x ^ i = 0 :=
Finset.sum_eq_zero fun i hi ↦ by
simp [descFactorial_eq_zero_iff_lt.2 (Finset.mem_range.1 hi)]
simpa only [hzero, add_zero, mul_assoc, hx.symm.eq] using B

/-- If `‖x‖ < 1`, then `∑' n : ℕ, n ^ k * x ^ n` is given by the finite sum
`∑ j ∈ range (k + 1), S(k, j) * j ! * x ^ j * ((1 - x)⁻¹ʳ) ^ (j + 1)`,
where `S(k, j)` denotes the Stirling numbers of the second kind. -/
theorem hasSum_pow_mul_geometric_of_norm_lt_one' (k : ℕ) {x : R} (h : ‖x‖ < 1) :
HasSum (fun n : ℕ ↦ (n : R) ^ k * x ^ n)
(∑ j ∈ Finset.range (k + 1),
(stirlingSecond k j : R) * j.factorial * x ^ j * ((1 - x)⁻¹ʳ) ^ (j + 1)) := by
have hfun : (fun n : ℕ ↦ (n : R) ^ k * x ^ n) = fun n : ℕ ↦ ∑ j ∈ Finset.range (k + 1),
(stirlingSecond k j : R) * ((n.descFactorial j : R) * x ^ n) := by
funext n
rw [← Nat.cast_pow, Nat.pow_eq_sum_stirlingSecond_mul_descFactorial n k, Nat.cast_sum,
Finset.sum_mul]
simp only [Nat.cast_mul, mul_assoc]
have hval : (∑ j ∈ Finset.range (k + 1),
(stirlingSecond k j : R) * j.factorial * x ^ j * ((1 - x)⁻¹ʳ) ^ (j + 1))
= ∑ j ∈ Finset.range (k + 1), (stirlingSecond k j : R)
* ((j.factorial : R) * x ^ j * ((1 - x)⁻¹ʳ) ^ (j + 1)) := by
simp only [mul_assoc]
rw [hfun, hval]
exact hasSum_sum fun j _ ↦
(hasSum_descFactorial_mul_geometric_of_norm_lt_one' j h).mul_left _

theorem tsum_pow_mul_geometric_of_norm_lt_one' (k : ℕ) {x : R} (h : ‖x‖ < 1) :
∑' n : ℕ, (n : R) ^ k * x ^ n = (∑ j ∈ Finset.range (k + 1),
(stirlingSecond k j : R) * j.factorial * x ^ j * ((1 - x)⁻¹ʳ) ^ (j + 1)) :=
(hasSum_pow_mul_geometric_of_norm_lt_one' k h).tsum_eq

theorem hasSum_pow_mul_geometric_of_norm_lt_one (k : ℕ) {r : 𝕜} (hr : ‖r‖ < 1) :
HasSum (fun n : ℕ ↦ (n : 𝕜) ^ k * r ^ n)
(∑ j ∈ Finset.range (k + 1),
stirlingSecond k j * j.factorial * r ^ j / (1 - r) ^ (j + 1)) := by
convert! hasSum_pow_mul_geometric_of_norm_lt_one' k hr using 1
simp [div_eq_mul_inv]

/-- If `‖r‖ < 1`, then `∑' n : ℕ, n ^ 2 * r ^ n = r * (1 + r) / (1 - r) ^ 3`. -/
theorem tsum_sq_mul_geometric_of_norm_lt_one {r : 𝕜} (hr : ‖r‖ < 1) :
∑' n : ℕ, (n : 𝕜) ^ 2 * r ^ n = r * (1 + r) / (1 - r) ^ 3 :=
(hasSum_sq_mul_geometric_of_norm_lt_one hr).tsum_eq
theorem tsum_pow_mul_geometric_of_norm_lt_one (k : ℕ) {r : 𝕜} (hr : ‖r‖ < 1) :
∑' n : ℕ, (n : 𝕜) ^ k * r ^ n = (∑ j ∈ Finset.range (k + 1),
stirlingSecond k j * j.factorial * r ^ j / (1 - r) ^ (j + 1)) :=
(hasSum_pow_mul_geometric_of_norm_lt_one k hr).tsum_eq

theorem hasSum_sq_mul_geometric_of_norm_lt_one' {x : R} (h : ‖x‖ < 1) :
HasSum (fun n : ℕ ↦ (n : R) ^ 2 * x ^ n) (x * (1 + x) * ((1 - x)⁻¹ʳ) ^ 3) := by
have h1 : (∑ j ∈ Finset.range (2 + 1),
(stirlingSecond 2 j : R) * j.factorial * x ^ j * ((1 - x)⁻¹ʳ) ^ (j + 1))
= x * ((1 - x)⁻¹ʳ) ^ 2 + 2 * (x ^ 2 * ((1 - x)⁻¹ʳ) ^ 3) := by
simp [Finset.sum_range_succ, stirlingSecond, Nat.factorial, mul_assoc]
have h2 : x * ((1 - x)⁻¹ʳ) ^ 2 + 2 * (x ^ 2 * ((1 - x)⁻¹ʳ) ^ 3)
= x * (1 + x) * ((1 - x)⁻¹ʳ) ^ 3 := by
calc x * ((1 - x)⁻¹ʳ) ^ 2 + 2 * (x ^ 2 * ((1 - x)⁻¹ʳ) ^ 3)
= x * (((1 - x) * (1 - x)⁻¹ʳ) * ((1 - x)⁻¹ʳ) ^ 2) + 2 * (x ^ 2 * ((1 - x)⁻¹ʳ) ^ 3) := by
grind [Ring.mul_inverse_cancel, isUnit_one_sub_of_norm_lt_one]
_ = x * (1 + x) * ((1 - x)⁻¹ʳ) ^ 3 := by grind
grind [hasSum_pow_mul_geometric_of_norm_lt_one']

end MulGeometric

Expand Down
23 changes: 23 additions & 0 deletions Mathlib/Combinatorics/Enumerative/Stirling.lean
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Original file line number Diff line number Diff line change
Expand Up @@ -5,6 +5,7 @@ Authors: Beibei Xiong, Yu Shao, Weijie Jiang, Zhengfeng Yang
-/
module

public import Mathlib.Algebra.BigOperators.Ring.Finset
public import Mathlib.Data.Nat.Factorial.Basic
public import Mathlib.Data.Nat.Choose.Basic
public import Mathlib.Tactic.NormNum.Inv
Expand Down Expand Up @@ -169,4 +170,26 @@ theorem stirlingSecond_succ_self_left (n : ℕ) :
rw [stirlingSecond_succ_succ, ih, stirlingSecond_self, mul_one,
Nat.choose_succ_succ (n + 1), Nat.choose_one_right]

theorem descFactorial_mul_self (n j : ℕ) :
n.descFactorial j * n = n.descFactorial (j + 1) + j * n.descFactorial j := by
rcases le_or_gt j n with h | h <;>
grind [descFactorial_succ, add_mul, descFactorial_eq_zero_iff_lt]

theorem pow_eq_sum_stirlingSecond_mul_descFactorial (n k : ℕ) :
n ^ k = ∑ j ∈ Finset.range (k + 1), stirlingSecond k j * n.descFactorial j := by
induction k with
| zero => simp
| succ k ih =>
have : ∑ j ∈ Finset.range (k + 1), stirlingSecond k j * (j * n.descFactorial j)
= ∑ j ∈ Finset.range (k + 1),
(j + 1) * (stirlingSecond k (j + 1) * n.descFactorial (j + 1)) := by
grind [Finset.sum_range_succ' (fun j ↦ stirlingSecond k j * (j * n.descFactorial j)) k,
Finset.sum_range_succ (fun j ↦ (j + 1) * (stirlingSecond k (j + 1) *
n.descFactorial (j + 1))) k, stirlingSecond_eq_zero_of_lt k.lt_add_one]
rw [pow_succ, ih, Finset.sum_mul,
Finset.sum_range_succ' (fun j ↦ stirlingSecond (k + 1) j * n.descFactorial j) (k + 1)]
simp only [mul_assoc, descFactorial_mul_self, mul_add, Finset.sum_add_distrib, this,
stirlingSecond_succ_succ, add_mul, stirlingSecond_succ_zero, zero_mul, add_zero]
grind

end Nat
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