From b497531d55e3bc371ff15abbeb4b134badaa37a9 Mon Sep 17 00:00:00 2001
From: Karan Anand <anandkarancompsci@gmail.com>
Date: Thu, 27 Aug 2026 11:04:24 +0530
Subject: [PATCH] docs: add equations to gamma and beta family READMEs

Add defining equations to the intro sections of gammaln, polygamma,
bernoulli, betainc, betaincinv, gamma-delta-ratio, gamma1pm1,
falling-factorial, rising-factorial, and absgammalnf.

Ref: https://github.com/stdlib-js/metr-issue-tracker/issues/1257
---
 .../math/base/special/absgammalnf/README.md    | 18 ++++++++++++++++++
 .../math/base/special/bernoulli/README.md      | 10 ++++++++++
 .../math/base/special/bernoullif/README.md     | 10 ++++++++++
 .../math/base/special/betainc/README.md        | 10 ++++++++++
 .../math/base/special/betaincinv/README.md     | 10 ++++++++++
 .../base/special/falling-factorial/README.md   | 10 ++++++++++
 .../base/special/gamma-delta-ratio/README.md   | 18 ++++++++++++++++++
 .../math/base/special/gamma1pm1/README.md      | 16 ++++++++++++++++
 .../math/base/special/gammaln/README.md        | 18 ++++++++++++++++++
 .../math/base/special/polygamma/README.md      | 12 ++++++++++++
 .../base/special/rising-factorial/README.md    | 10 ++++++++++
 11 files changed, 142 insertions(+)

diff --git a/lib/node_modules/@stdlib/math/base/special/absgammalnf/README.md b/lib/node_modules/@stdlib/math/base/special/absgammalnf/README.md
index 9bf1f6093c12..d624e6f1ae40 100644
--- a/lib/node_modules/@stdlib/math/base/special/absgammalnf/README.md
+++ b/lib/node_modules/@stdlib/math/base/special/absgammalnf/README.md
@@ -22,6 +22,22 @@ limitations under the License.
 
 > [Natural logarithm][@stdlib/math/base/special/lnf] of the [absolute][@stdlib/math/base/special/absf] value of the [gamma][@stdlib/math/base/special/gamma] function.
 
+<section class="intro">
+
+The function computes the natural logarithm of the absolute value of the [gamma function][gamma-function]
+
+<!-- <equation class="equation" label="eq:absgammaln_function" align="center" raw="y = \ln |\Gamma(x)|" alt="Natural logarithm of the absolute value of the gamma function"> -->
+
+```math
+y = \ln |\Gamma(x)|
+```
+
+<!-- </equation> -->
+
+</section>
+
+<!-- /.intro -->
+
 <section class="usage">
 
 ## Usage
@@ -184,6 +200,8 @@ int main( void ) {
 
 <section class="links">
 
+[gamma-function]: https://en.wikipedia.org/wiki/Gamma_function
+
 [@stdlib/math/base/special/lnf]: https://github.com/stdlib-js/stdlib/tree/develop/lib/node_modules/%40stdlib/math/base/special/lnf
 
 [@stdlib/math/base/special/absf]: https://github.com/stdlib-js/stdlib/tree/develop/lib/node_modules/%40stdlib/math/base/special/absf
diff --git a/lib/node_modules/@stdlib/math/base/special/bernoulli/README.md b/lib/node_modules/@stdlib/math/base/special/bernoulli/README.md
index e9bff8c79eb8..95ad0a71e13b 100644
--- a/lib/node_modules/@stdlib/math/base/special/bernoulli/README.md
+++ b/lib/node_modules/@stdlib/math/base/special/bernoulli/README.md
@@ -24,6 +24,16 @@ limitations under the License.
 
 <section class="intro">
 
+The [Bernoulli numbers][bernoulli-number] are a sequence of rational numbers which are of importance in number theory and analysis
+
+<!-- <equation class="equation" label="eq:bernoulli_numbers" align="center" raw="\frac{t}{e^t - 1} = \sum_{n=0}^{\infty} B_n \frac{t^n}{n!}" alt="Bernoulli numbers generating function"> -->
+
+```math
+\frac{t}{e^t - 1} = \sum_{n=0}^{\infty} B_n \frac{t^n}{n!}
+```
+
+<!-- </equation> -->
+
 </section>
 
 <!-- /.intro -->
diff --git a/lib/node_modules/@stdlib/math/base/special/bernoullif/README.md b/lib/node_modules/@stdlib/math/base/special/bernoullif/README.md
index 6528deb8470d..e0791db86ae0 100644
--- a/lib/node_modules/@stdlib/math/base/special/bernoullif/README.md
+++ b/lib/node_modules/@stdlib/math/base/special/bernoullif/README.md
@@ -24,6 +24,16 @@ limitations under the License.
 
 <section class="intro">
 
+The [Bernoulli numbers][bernoulli-number] are a sequence of rational numbers which are of importance in number theory and analysis
+
+<!-- <equation class="equation" label="eq:bernoulli_numbers" align="center" raw="\frac{t}{e^t - 1} = \sum_{n=0}^{\infty} B_n \frac{t^n}{n!}" alt="Bernoulli numbers generating function"> -->
+
+```math
+\frac{t}{e^t - 1} = \sum_{n=0}^{\infty} B_n \frac{t^n}{n!}
+```
+
+<!-- </equation> -->
+
 </section>
 
 <!-- /.intro -->
diff --git a/lib/node_modules/@stdlib/math/base/special/betainc/README.md b/lib/node_modules/@stdlib/math/base/special/betainc/README.md
index 5d3a7fcb828c..0d48b5699bcb 100644
--- a/lib/node_modules/@stdlib/math/base/special/betainc/README.md
+++ b/lib/node_modules/@stdlib/math/base/special/betainc/README.md
@@ -24,6 +24,16 @@ limitations under the License.
 
 <section class="intro">
 
+The regularized [incomplete beta function][incomplete-beta-function] is defined as
+
+<!-- <equation class="equation" label="eq:incomplete_beta_function" align="center" raw="I_x(a, b) = \frac{B(x;\, a, b)}{B(a, b)} = \frac{1}{B(a, b)} \int_0^x t^{a-1}(1-t)^{b-1}\,dt" alt="Regularized incomplete beta function"> -->
+
+```math
+I_x(a, b) = \frac{B(x;\, a, b)}{B(a, b)} = \frac{1}{B(a, b)} \int_0^x t^{a-1}(1-t)^{b-1}\,dt
+```
+
+<!-- </equation> -->
+
 </section>
 
 <!-- /.intro -->
diff --git a/lib/node_modules/@stdlib/math/base/special/betaincinv/README.md b/lib/node_modules/@stdlib/math/base/special/betaincinv/README.md
index f30a71e53471..5710d804c752 100644
--- a/lib/node_modules/@stdlib/math/base/special/betaincinv/README.md
+++ b/lib/node_modules/@stdlib/math/base/special/betaincinv/README.md
@@ -24,6 +24,16 @@ limitations under the License.
 
 <section class="intro">
 
+The inverse of the regularized [incomplete beta function][incomplete-beta-function] is defined as
+
+<!-- <equation class="equation" label="eq:betaincinv_function" align="center" raw="x = I^{-1}(p;\, a, b) \quad \text{such that} \quad I_x(a, b) = p" alt="Inverse of the regularized incomplete beta function"> -->
+
+```math
+x = I^{-1}(p;\, a, b) \quad \text{such that} \quad I_x(a, b) = p
+```
+
+<!-- </equation> -->
+
 </section>
 
 <!-- /.intro -->
diff --git a/lib/node_modules/@stdlib/math/base/special/falling-factorial/README.md b/lib/node_modules/@stdlib/math/base/special/falling-factorial/README.md
index 1e89ce9bbe33..edce46ac4d79 100644
--- a/lib/node_modules/@stdlib/math/base/special/falling-factorial/README.md
+++ b/lib/node_modules/@stdlib/math/base/special/falling-factorial/README.md
@@ -24,6 +24,16 @@ limitations under the License.
 
 <section class="intro">
 
+The [falling factorial][falling-and-rising-factorials] (or Pochhammer's symbol for the descending factorial) is defined as
+
+<!-- <equation class="equation" label="eq:falling_factorial" align="center" raw="(x)_n = x(x-1)(x-2) \cdots (x-n+1) = \frac{x!}{(x-n)!}" alt="Falling factorial"> -->
+
+```math
+(x)_n = x(x-1)(x-2) \cdots (x-n+1) = \frac{x!}{(x-n)!}
+```
+
+<!-- </equation> -->
+
 </section>
 
 <!-- /.intro -->
diff --git a/lib/node_modules/@stdlib/math/base/special/gamma-delta-ratio/README.md b/lib/node_modules/@stdlib/math/base/special/gamma-delta-ratio/README.md
index 4eef66ce6c5d..3881f878a4c7 100644
--- a/lib/node_modules/@stdlib/math/base/special/gamma-delta-ratio/README.md
+++ b/lib/node_modules/@stdlib/math/base/special/gamma-delta-ratio/README.md
@@ -22,6 +22,22 @@ limitations under the License.
 
 > Compute the ratio of two [gamma][@stdlib/math/base/special/gamma] functions.
 
+<section class="intro">
+
+The function computes the ratio of two [gamma functions][gamma-function]
+
+<!-- <equation class="equation" label="eq:gamma_delta_ratio" align="center" raw="y = \frac{\Gamma(x)}{\Gamma(x + \delta)}" alt="Ratio of two gamma functions"> -->
+
+```math
+y = \frac{\Gamma(x)}{\Gamma(x + \delta)}
+```
+
+<!-- </equation> -->
+
+</section>
+
+<!-- /.intro -->
+
 <section class="usage">
 
 ## Usage
@@ -184,6 +200,8 @@ int main( void ) {
 
 <section class="links">
 
+[gamma-function]: https://en.wikipedia.org/wiki/Gamma_function
+
 <!-- <related-links> -->
 
 [@stdlib/math/base/special/gamma]: https://github.com/stdlib-js/stdlib/tree/develop/lib/node_modules/%40stdlib/math/base/special/gamma
diff --git a/lib/node_modules/@stdlib/math/base/special/gamma1pm1/README.md b/lib/node_modules/@stdlib/math/base/special/gamma1pm1/README.md
index 824b823d9d5d..5f98555f6608 100644
--- a/lib/node_modules/@stdlib/math/base/special/gamma1pm1/README.md
+++ b/lib/node_modules/@stdlib/math/base/special/gamma1pm1/README.md
@@ -22,6 +22,22 @@ limitations under the License.
 
 > Compute `gamma(x+1) - 1`.
 
+<section class="intro">
+
+The function computes `Γ(1+x) - 1` accurately for small `x`
+
+<!-- <equation class="equation" label="eq:gamma1pm1_function" align="center" raw="y = \Gamma(1 + x) - 1" alt="Gamma of one plus x minus one"> -->
+
+```math
+y = \Gamma(1 + x) - 1
+```
+
+<!-- </equation> -->
+
+</section>
+
+<!-- /.intro -->
+
 <section class="usage">
 
 ## Usage
diff --git a/lib/node_modules/@stdlib/math/base/special/gammaln/README.md b/lib/node_modules/@stdlib/math/base/special/gammaln/README.md
index 72401ee74be9..1ceefaeaa926 100644
--- a/lib/node_modules/@stdlib/math/base/special/gammaln/README.md
+++ b/lib/node_modules/@stdlib/math/base/special/gammaln/README.md
@@ -22,6 +22,22 @@ limitations under the License.
 
 > [Natural logarithm][@stdlib/math/base/special/ln] of the [gamma][@stdlib/math/base/special/gamma] function.
 
+<section class="intro">
+
+The natural logarithm of the [gamma function][gamma-function] is defined as
+
+<!-- <equation class="equation" label="eq:gammaln_function" align="center" raw="y = \ln \Gamma(x)" alt="Natural logarithm of the gamma function"> -->
+
+```math
+y = \ln \Gamma(x)
+```
+
+<!-- </equation> -->
+
+</section>
+
+<!-- /.intro -->
+
 <section class="usage">
 
 ## Usage
@@ -193,6 +209,8 @@ int main( void ) {
 
 <section class="links">
 
+[gamma-function]: https://en.wikipedia.org/wiki/Gamma_function
+
 [@stdlib/math/base/special/ln]: https://github.com/stdlib-js/stdlib/tree/develop/lib/node_modules/%40stdlib/math/base/special/ln
 
 <!-- <related-links> -->
diff --git a/lib/node_modules/@stdlib/math/base/special/polygamma/README.md b/lib/node_modules/@stdlib/math/base/special/polygamma/README.md
index 8163af4edca0..866da6c0f679 100644
--- a/lib/node_modules/@stdlib/math/base/special/polygamma/README.md
+++ b/lib/node_modules/@stdlib/math/base/special/polygamma/README.md
@@ -24,6 +24,16 @@ limitations under the License.
 
 <section class="intro">
 
+The [polygamma][polygamma-function] function of order `n` is the `(n+1)`th derivative of the natural logarithm of the [gamma function][gamma-function]
+
+<!-- <equation class="equation" label="eq:polygamma_function" align="center" raw="\psi^{(n)}(x) = \frac{d^{n+1}}{dx^{n+1}} \ln{\Gamma(x)}" alt="Polygamma function"> -->
+
+```math
+\psi^{(n)}(x) = \frac{d^{n+1}}{dx^{n+1}} \ln{\Gamma(x)}
+```
+
+<!-- </equation> -->
+
 </section>
 
 <!-- /.intro -->
@@ -143,6 +153,8 @@ logEachMap( 'ψ^(%d)(x = %0.4f): %0.4f', n, x, polygamma );
 
 <section class="links">
 
+[gamma-function]: https://en.wikipedia.org/wiki/Gamma_function
+
 [polygamma-function]: https://en.wikipedia.org/wiki/Polygamma_function
 
 [@stdlib/math/base/special/ln]: https://github.com/stdlib-js/stdlib/tree/develop/lib/node_modules/%40stdlib/math/base/special/ln
diff --git a/lib/node_modules/@stdlib/math/base/special/rising-factorial/README.md b/lib/node_modules/@stdlib/math/base/special/rising-factorial/README.md
index 9d17dde0e452..23d79e3ae452 100644
--- a/lib/node_modules/@stdlib/math/base/special/rising-factorial/README.md
+++ b/lib/node_modules/@stdlib/math/base/special/rising-factorial/README.md
@@ -24,6 +24,16 @@ limitations under the License.
 
 <section class="intro">
 
+The [rising factorial][falling-and-rising-factorials] (or Pochhammer function) is defined as
+
+<!-- <equation class="equation" label="eq:rising_factorial" align="center" raw="x^{(n)} = x(x+1)(x+2) \cdots (x+n-1) = \frac{(x+n-1)!}{(x-1)!}" alt="Rising factorial"> -->
+
+```math
+x^{(n)} = x(x+1)(x+2) \cdots (x+n-1) = \frac{(x+n-1)!}{(x-1)!}
+```
+
+<!-- </equation> -->
+
 </section>
 
 <!-- /.intro -->