feat(Analysis): the generalized hypergeometric function

[mcdoll]

We define the (regularized) generalized hypergeometric function and prove convergence in the cases p < q + 1 and p = q + 1. This includes the Gaussian hypergeometric function (p = 2 and q = 1) and the Bessel functions (p = 0 and q = 1).

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[github-actions]

PR summary 33c10bef48

Import changes for modified files

No significant changes to the import graph

Import changes for all files

Files	Import difference
Mathlib.Analysis.SpecialFunctions.Hypergeometric (new file)	2624

---

Declarations diff (regex)

+ Gamma_nat_add_div_Gamma_eq
+ eventually_atTop_regularizedHGFunCoeff_ne_zero
+ finsetProd_div_finsetProd_mul
+ prod_eq_pow_mul_prod
+ radius_regularizedGaussHGFunSeries_eq_one
+ radius_regularizedHGFunSeries_eq_one
+ radius_regularizedHGFunSeries_eq_top
+ radius_regularizedHGFunSeries_eq_top_of_finite
+ regularizedGaussHGFun
+ regularizedGaussHGFunSeries
+ regularizedHGFun
+ regularizedHGFunCoeff
+ regularizedHGFunCoeff_add_one
+ regularizedHGFunCoeff_add_one_div_self
+ regularizedHGFunCoeff_eq_zero_iff
+ regularizedHGFunCoeff_eq_zero_left
+ regularizedHGFunCoeff_eq_zero_left'
+ regularizedHGFunCoeff_eq_zero_right
+ regularizedHGFunSeries
+ regularizedHGFunSeries_coeff
+ regularizedHGFunSeries_eq_zero
+ tendsto_finsetProd_div_add_one
+ tendsto_finsetProd_div_finsetProd_mul

You can run this locally as follows

## from your `mathlib4` directory:
git clone https://github.com/leanprover-community/mathlib-ci.git ../mathlib-ci

## summary with just the declaration names:
../mathlib-ci/scripts/pr_summary/declarations_diff.sh <optional_commit>

## more verbose report:
../mathlib-ci/scripts/pr_summary/declarations_diff.sh long <optional_commit>

The doc-module for scripts/pr_summary/declarations_diff.sh in the mathlib-ci repository contains some details about this script.

Declarations diff (Lean)

✅ Lean-aware diff — post-build, computed from the Lean environment (commit 33c10be).

• +19 new declarations
• −0 removed declarations

+Complex.Gamma_nat_add_div_Gamma_eq
+Complex.eventually_atTop_regularizedHGFunCoeff_ne_zero
+Complex.radius_regularizedGaussHGFunSeries_eq_one
+Complex.radius_regularizedHGFunSeries_eq_one
+Complex.radius_regularizedHGFunSeries_eq_top
+Complex.radius_regularizedHGFunSeries_eq_top_of_finite
+Complex.regularizedGaussHGFun
+Complex.regularizedGaussHGFunSeries
+Complex.regularizedHGFun
+Complex.regularizedHGFunCoeff
+Complex.regularizedHGFunCoeff_add_one
+Complex.regularizedHGFunCoeff_add_one_div_self
+Complex.regularizedHGFunCoeff_eq_zero_iff
+Complex.regularizedHGFunCoeff_eq_zero_left
+Complex.regularizedHGFunCoeff_eq_zero_left'
+Complex.regularizedHGFunCoeff_eq_zero_right
+Complex.regularizedHGFunSeries
+Complex.regularizedHGFunSeries_coeff
+Complex.regularizedHGFunSeries_eq_zero

---

No changes to strong technical debt.

Increase in weak tech debt: (relative, absolute) = (1.00, 0.00)

Current number	Change	Type (weak)
5015	1	exposed public sections

Current commit 33c10bef48
Reference commit d0060d7bbb

This script lives in the mathlib-ci repository. To run it locally, from your mathlib4 directory:

git clone https://github.com/leanprover-community/mathlib-ci.git ../mathlib-ci
../mathlib-ci/scripts/reporting/technical-debt-metrics.sh pr_summary

• The relative value is the weighted sum of the differences with weight given by the inverse of the current value of the statistic.
• The absolute value is the relative value divided by the total sum of the inverses of the current values (i.e. the weighted average of the differences).

[wwylele]

When an Exists appears in the hypothesis, I think we usually expand it into multiple parameters

Suggested change

	variable (a b n) in
	theorem regularizedHGFunCoeff_eq_zero_right
	(hb : ∃ (j : Fin q) (m : ℕ), b j = -n - m) : regularizedHGFunCoeff a b n = 0 := by grind
	variable (a b n) in
	theorem regularizedHGFunCoeff_eq_zero_left
	(ha : ∃ (j : Fin p), ∃ k < n, k = -a j) : regularizedHGFunCoeff a b n = 0 := by grind
	variable (a b n) in
	theorem regularizedHGFunCoeff_eq_zero_right {j : Fin q} {m : ℕ} (h : b j = -n - m) :
	regularizedHGFunCoeff a b n = 0 := by grind
	variable (a b n) in
	theorem regularizedHGFunCoeff_eq_zero_left {j : Fin p} {k : ℕ} (hk : k < n) (h : k = -a j) :
	regularizedHGFunCoeff a b n = 0 := by grind

[wwylele]

Suggested change

	theorem radius_regularizedHGFunSeries_eq_top (a : Fin p → ℂ) (b : Fin q → ℂ) (h : p < q + 1) :
	theorem radius_regularizedHGFunSeries_eq_top (a : Fin p → ℂ) (b : Fin q → ℂ) (h : p ≤ q) :

[wwylele]

Should a and b be Multiset ℂ instead? Advantages are

• Reflects the fact that order doesn't matter
• If one wants to introduce a cancelling lemma when there is a common factor among a and b, Multiset allows writing it naturally in insert/erase, while Fin needs reindexing.