Add docs for new hypergeometric functions

src/functions-reference/real-valued_basic_functions.qmd

@@ -1698,3 +1698,69 @@ Implementation of the $W_0$ branch of the Lambert W function, i.e., solution to
 `R` **`lambert_wm1`**`(T x)`<br>\newline
 Implementation of the $W_{-1}$ branch of the Lambert W function, i.e., solution to the function $W_{-1}(x) \exp^{W_{-1}(x)} = x$
 {{< since 2.25 >}}
+
+## Hypergeometric Functions {#hypergeometric-functions}
+
+Hypergeometric functions refer to a power series of the form
+\begin{equation*}
+_pF_q(a_1,...,a_p;b_1,...,b_q;z) = \sum_{n=0}^\infty \frac{(a_1)_n\cdot\cdot\cdot(a_p)_n}{(b_1)_n\cdot\cdot\cdot(b_q)_n} \frac{x^n}{n!}
+\end{equation*}
+where $(a)_n$ is the Pochhammer symbol defined as $(a)_n = \frac{\Gamma(a+n)}{\Gamma(a)}$.
+
+The gradients of the hypergeometric function are given by:
+\begin{equation*}
+ \frac{\partial }{\partial a_1} =
+  \sum_{k=0}^{\infty}{
+    \frac
+      {\psi\left(k+a_1\right)\left(\prod_{j=1}^p\left(a_j\right)_k\right)z^k}
+      {k!\prod_{j=1}^q\left(b_j\right)_k}}
+    - \psi\left(a_1\right){}_pF_q(a_1,...,a_p;b_1,...,b_q;z)
+\end{equation*}
+\begin{equation*}
+ \frac{\partial }{\partial b_1} =
+  \psi\left(b_1\right){}_pF_q(a_1,...,a_p;b_1,...,b_q;z) -
+  \sum_{k=0}^{\infty}{
+    \frac
+      {\psi\left(k+b_1\right)\left(\prod_{j=1}^p\left(a_j\right)_k\right)z^k}
+      {k!\prod_{j=1}^q\left(b_j\right)_k}}
+\end{equation*}
+\begin{equation*}
+  \frac{\partial }{\partial z} =
+  \frac{\prod_{j=1}^{p}a_j}{\prod_{j=1}^{q} b_j}{}_pF_q(a_1+1,...,a_p+1;b_1+1,...,b_q+1;z)
+\end{equation*}
+
+Stan provides both the generalized hypergeometric function as well as several
+special cases for particular values of p and q.
+
+<!-- real; hypergeometric_1F0; (real a, real z); -->
+\index{{\tt \bfseries hypergeometric\_1F0 }!{\tt (real a, real z): real}|hyperpage}
+
+`real` **`hypergeometric_1F0`**`(real a, real z)`<br>\newline
+Special case of the hypergeometric function with $p=1$ and $q=0$.
+{{< since 2.37 >}}
+
+<!-- real; hypergeometric_2F1; (real a1, real a2, real b1, real z); -->
+\index{{\tt \bfseries hypergeometric\_2F1 }!{\tt (real a1, real a2, real b1, real z): real}|hyperpage}
+
+`real` **`hypergeometric_2F1`**`(real a1, real a2, real b1, real z)`<br>\newline
+Special case of the hypergeometric function with $p=2$ and $q=1$. If the function does not
+meet convergence criteria for given inputs, the function will attempt to apply [Euler's transformation](https://mathworld.wolfram.com/EulersHypergeometricTransformations.html)
+to improve convergence:
+\begin{equation*}
+{}_2F_1(a_1,a_2, b_1, z)={}_2F_1(b_1 - a_1,a_2, b_1, \frac{z}{z-1})\cdot(1-z)^{-a_2}
+\end{equation*}
+{{< since 2.37 >}}
+
+<!-- real; hypergeometric_3F2; (T1 a, T2 b, real z); -->
+\index{{\tt \bfseries hypergeometric\_3F2 }!{\tt (T1 a, T2 b, real z): real}|hyperpage}
+
+`real` **`hypergeometric_3F2`**`(T1 a, T2 b, real z)`<br>\newline
+Special case of the hypergeometric function with $p=3$ and $q=2$, where a and b are vectors of length 3 and 2, respectively.
+{{< since 2.37 >}}
+
+<!-- real; hypergeometric_pFq; (T1 a, T2 b, real z); -->
+\index{{\tt \bfseries hypergeometric\_pFq }!{\tt (T1 a, T2 b, real z): real}|hyperpage}
+
+`real` **`hypergeometric_pFq`**`(T1 a, T2 b, real z)`<br>\newline
+Generalized hypergeometric function, where a and b are vectors of length p and q, respectively.
+{{< since 2.37 >}}