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feat(Analysis/SpecialFunctions): generating function of besselJ - #43978

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wwylele:bessel-sum

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@wwylele wwylele commented Sep 19, 2026

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This proves $e^{\frac{1}{2}z(t-t^{-1})}=\sum_{m=-\infty}^{\infty}t^{m}J_{m}\left(z\right)$


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PR summary 6f37e001ee

Import changes for modified files

Dependency changes

File Base Count Head Count Change
Mathlib.Analysis.SpecialFunctions.Bessel 2782 2797 +15 (+0.54%)
Mathlib.Analysis.SpecialFunctions.RegularizedHypergeometric 2780 2781 +1 (+0.04%)
Import changes for all files
Files Import difference
Mathlib.Analysis.SpecialFunctions.RegularizedHypergeometric 1
Mathlib.Analysis.SpecialFunctions.Bessel 15

Declarations diff (regex)

+ Multiset.nnnorm_prod_map
+ Multiset.norm_prod_map
+ analyticOnNhd_tsum_besselJ_mul_pow
+ besselJ_mul_pow_le
+ deriv_regularizedHGFun
+ hasSum_besselJ_mul_pow
+ mul_deriv_besselJ_eq_besselJ_add_one
+ mul_deriv_besselJ_eq_besselJ_add_one_int
+ mul_deriv_besselJ_eq_besselJ_sub_one
+ mul_deriv_besselJ_eq_besselJ_sub_one_int
+ mul_regularizedHGFunCoeff
+ mul_regularizedHGFunCoeff_of_mem_right
+ norm_Gamma_le_norm_Gamma_add_of_one_le_re
+ norm_Gamma_le_norm_Gamma_add_one
+ norm_besselJ_le_exp
+ norm_besselJ_le_exp_int
+ regularizedHGFun_le_exp_of_one_le_re
+ regularizedHGFun_zero_singleton_eq_mul_add_mul
+ regularizedHGFun_zero_singleton_eq_mul_add_mul_aux
+ summable_besselJ_mul_pow
+ summable_bound
+ summable_deriv_HGF
+ tsum_besselJ_mul_pow
+ two_mul_deriv_besselJ
+ two_mul_deriv_besselJ_int
+ two_mul_self_mul_besselJ

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 6f37e00).

  • +24 new declarations
  • −0 removed declarations
+Complex.analyticOnNhd_tsum_besselJ_mul_pow
+Complex.deriv_regularizedHGFun
+Complex.hasSum_besselJ_mul_pow
+Complex.mul_deriv_besselJ_eq_besselJ_add_one
+Complex.mul_deriv_besselJ_eq_besselJ_add_one_int
+Complex.mul_deriv_besselJ_eq_besselJ_sub_one
+Complex.mul_deriv_besselJ_eq_besselJ_sub_one_int
+Complex.mul_regularizedHGFunCoeff
+Complex.mul_regularizedHGFunCoeff_of_mem_right
+Complex.norm_Gamma_le_norm_Gamma_add_of_one_le_re
+Complex.norm_Gamma_le_norm_Gamma_add_one
+Complex.norm_besselJ_le_exp
+Complex.norm_besselJ_le_exp_int
+Complex.regularizedHGFun_le_exp_of_one_le_re
+Complex.regularizedHGFun_zero_singleton_eq_mul_add_mul
+Complex.summable_besselJ_mul_pow
+Complex.tsum_besselJ_mul_pow
+Complex.two_mul_deriv_besselJ
+Complex.two_mul_deriv_besselJ_int
+Complex.two_mul_self_mul_besselJ
+Multiset.nnnorm_prod_map
+Multiset.norm_prod_map
+nnnormHom.congr_simp
+normHom.congr_simp

No changes to strong technical debt.
No changes to weak technical debt.

Current commit 6f37e001ee
Reference commit fd634dd297

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.py 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).

@github-actions github-actions Bot added the t-analysis Analysis (normed *, calculus) label Sep 19, 2026
@mathlib-dependent-issues mathlib-dependent-issues Bot added the blocked-by-other-PR This PR depends on another PR (this label is automatically managed by a bot) label Sep 19, 2026
@mathlib-dependent-issues

mathlib-dependent-issues Bot commented Sep 19, 2026

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@wwylele
wwylele marked this pull request as draft September 19, 2026 21:40
@mathlib-bors

mathlib-bors Bot commented Sep 19, 2026

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This pull request is now in draft mode. No active bors state needed cleanup.

While this PR remains draft, bors will ignore commands on this PR. Mark it ready for review before using commands like bors r+ or bors try.

@wwylele wwylele changed the title feat(Analysis/SpecialFunctions): sum of besselJ over the first parameter feat(Analysis/SpecialFunctions): generating function of besselJ Sep 20, 2026
@wwylele
wwylele force-pushed the bessel-sum branch 2 times, most recently from 3723751 to c693d60 Compare September 20, 2026 12:56
@wwylele
wwylele requested a balanced review from Copilot September 20, 2026 13:58

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Copilot review overview

🟡 Changes recommended

Two derivative recurrences have incorrect DLMF references, alongside minor naming and spelling issues.

Get a fresh assessment by requesting another Copilot review.

Review effort: Balanced
Findings: 3 Low severity

Open (3)
What changed in this PR

Adds the generating function for integer-order Bessel functions, supported by recurrence, derivative, convergence, and norm-bound lemmas.

Changes:

  • Adds Bessel recurrence and derivative identities.
  • Proves summability, analyticity, and the Bessel generating function.
  • Adds supporting hypergeometric, Gamma, and multiset norm lemmas.
File Description
Mathlib/​Analysis/​SpecialFunctions/​Bessel.lean Proves recurrences, bounds, and generating function.
Mathlib/​Analysis/​SpecialFunctions/​RegularizedHypergeometric.lean Adds coefficient, derivative, and norm bounds.
Mathlib/​Analysis/​SpecialFunctions/​Gamma/​Basic.lean Adds Gamma norm-growth bounds.
Mathlib/​Analysis/​Normed/​Ring/​Basic.lean Adds multiset product norm lemmas.
Mathlib/​Analysis/​Calculus/​FDeriv/​Analytic.lean Registers analytic derivatives for fun_prop.

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Comment thread Mathlib/Analysis/SpecialFunctions/Bessel.lean
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blocked-by-other-PR This PR depends on another PR (this label is automatically managed by a bot) t-analysis Analysis (normed *, calculus)

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