Support for hypergeometric functions

mathics/builtin/specialfns/bessel.py

@@ -566,39 +566,6 @@ class HankelH2(_Bessel):
     sympy_name = "hankel2"
 
 
-class HypergeometricU(MPMathFunction):
-    """
-    <url>
-    :Confluent hypergeometric function: https://en.wikipedia.org/wiki/Confluent_hypergeometric_function</url> (<url>
-    :mpmath: https://mpmath.org/doc/current/functions/bessel.html#mpmath.hyperu</url>, <url>
-    :WMA: https://reference.wolfram.com/language/ref/HypergeometricU.html</url>)
-    <dl>
-      <dt>'HypergeometricU'[$a$, $b$, $z$]
-      <dd>returns $U(a, b, z)$.
-    </dl>
-
-    >> HypergeometricU[3, 2, 1.]
-     = 0.105479
-
-    Plot 'U'[3, 2, x] from 0 to 2 in steps of 0.5:
-    >> Plot[HypergeometricU[3, 2, x], {x, 0.5, 2}]
-     = -Graphics-
-
-    We handle this special case:
-    >> HypergeometricU[0, c, z]
-     = 1
-    """
-
-    attributes = A_LISTABLE | A_NUMERIC_FUNCTION | A_PROTECTED | A_READ_PROTECTED
-    mpmath_name = "hyperu"
-    nargs = {3}
-    rules = {
-        "HypergeometricU[0, c_, z_]": "1",
-    }
-    summary_text = "Tricomi confluent hypergeometric function"
-    sympy_name = ""
-
-
 # Kelvin Functions
 
 

mathics/builtin/specialfns/hypergeom.py

@@ -31,8 +31,8 @@ class HypergeometricPFQ(MPMathFunction):
     :sympy: https://docs.sympy.org/latest/modules/functions/special.html#sympy.functions.special.hyper.hyper</url>, <url>
     :WMA: https://reference.wolfram.com/language/ref/HypergeometricPFQ.html</url>)
     <dl>
-      <dt>'HypergeometricPFQ'[${a_1, a_2, a_p}, {b_1, b_2, b_q}, z$]
-      <dd>returns ${}_p F_q({a_1, a_2, a_p}, {b_1, b_2, b_q}, z)$.
+      <dt>'HypergeometricPFQ'[${a_1, ..., a_p}, {b_1, ..., b_q}, z$]
+      <dd>returns ${}_p F_q({a_1, ..., a_p}; {b_1, ..., b_q}; z)$.
     </dl>
     >> HypergeometricPFQ[{2}, {2}, 1]
      = E
@@ -93,7 +93,7 @@ class Hypergeometric1F1(MPMathFunction):
     :WMA: https://reference.wolfram.com/language/ref/Hypergeometric1F1.html</url>)
     <dl>
       <dt>'Hypergeometric1F1'[$a$, $b$, $z$]
-      <dd>returns $M(a, b, z)$.
+      <dd>returns ${}_1 F_1(a; b; z)$.
     </dl>
 
     Result is symbollicaly simplified by default:
@@ -106,7 +106,7 @@ class Hypergeometric1F1(MPMathFunction):
     Plot 'M'[3, 2, x] from 0 to 2 in steps of 0.5:
     >> Plot[Hypergeometric1F1[3, 2, x], {x, 0.5, 2}]
      = -Graphics-
-    Here, plot explicitly requests numerical results.
+    Here, plot explicitly requests a numerical evaluation.
     """
 
     attributes = A_LISTABLE | A_NUMERIC_FUNCTION | A_PROTECTED | A_READ_PROTECTED
@@ -117,3 +117,101 @@ class Hypergeometric1F1(MPMathFunction):
     }
     summary_text = "compute Kummer confluent hypergeometric function"
     sympy_name = ""
+
+
+class MeijerG(MPMathFunction):
+    """
+    <url>
+    :Meijer G-function: https://en.wikipedia.org/wiki/Meijer_G-function</url> (<url>
+    :mpmath: https://mpmath.org/doc/current/functions/hypergeometric.html#meijerg</url>, <url>
+    :sympy: https://docs.sympy.org/latest/modules/functions/special.html#sympy.functions.special.hyper.meijerg</url>, <url>
+    :WMA: https://reference.wolfram.com/language/ref/MeijerG.html</url>)
+    <dl>
+      <dt>'MeijerG'[${{a_1, ..., a_n}, {a_{n+1}, ..., a_p}}, {{b_1, ..., b_m}, {b_{m+1}, ..., a_q}}, z$]
+      <dd>returns $G^{m,n}_{p,q}(z | {a_1, ..., a_p}; {b_1, ..., b_q})$.
+    </dl>
+    Result is symbollicaly simplified by default:
+    >> MeijerG[{{1, 2}, {}}, {{3}, {}}, 1]
+     = MeijerG[{{1, 2}, {}}, {{3}, {}}, 1]
+    unless a numerical evaluation is explicitly requested:
+    >> MeijerG[{{1, 2},{}}, {{3},{}}, 1] // N
+     = 0.210958
+    """
+
+    attributes = A_NUMERIC_FUNCTION | A_PROTECTED | A_READ_PROTECTED
+    mpmath_name = "meijerg"
+    nargs = {3}
+    rules = {}
+    summary_text = "compute the Meijer G-function"
+    sympy_name = "meijerg"
+
+    def eval(self, a, b, z, evaluation: Evaluation):
+        "MeijerG[a_, b_, z_]"
+        try:
+            a_sympy = [e.to_sympy() for e in a]
+            b_sympy = [e.to_sympy() for e in b]
+            return from_sympy(sympy.meijerg(a_sympy, b_sympy, z.to_sympy()))
+        except Exception:
+            pass
+
+    def eval_N(self, a, b, z, evaluation: Evaluation):
+        "N[MeijerG[a_, b_, z_]]"
+        try:
+            return run_mpmath(
+                mpmath.meijerg,
+                tuple([a.to_python(), b.to_python(), z.to_python()]),
+                FP_MANTISA_BINARY_DIGITS,
+            )
+        except Exception:
+            pass
+
+
+class HypergeometricU(MPMathFunction):
+    """
+    <url>
+    :Confluent hypergeometric function: https://en.wikipedia.org/wiki/Confluent_hypergeometric_function</url> (<url>
+    :mpmath: https://mpmath.org/doc/current/functions/bessel.html#mpmath.hyperu</url>, <url>
+    :WMA: https://reference.wolfram.com/language/ref/HypergeometricU.html</url>)
+    <dl>
+      <dt>'HypergeometricU'[$a$, $b$, $z$]
+      <dd>returns $U(a, b, z)$.
+    </dl>
+    Result is symbollicaly simplified by default:
+    >> HypergeometricU[3, 2, 1]
+     = MeijerG[{{1, 2}, {}}, {{3}, {}}, 1]
+    unless a numerical evaluation is explicitly requested:
+    >> MeijerG[{{1, 2},{}}, {{3},{}}, 1] // N
+     = 0.210958
+
+    >> HypergeometricU[3, 2, 1.]
+     = 0.105479
+
+    Plot 'U'[3, 2, x] from 0 to 2 in steps of 0.5:
+    >> Plot[HypergeometricU[3, 2, x], {x, 0.5, 2}]
+     = -Graphics-
+
+    We handle this special case:
+    >> HypergeometricU[0, b, z]
+     = 1
+    """
+
+    attributes = A_LISTABLE | A_NUMERIC_FUNCTION | A_PROTECTED | A_READ_PROTECTED
+    mpmath_name = "hyperu"
+    nargs = {3}
+    rules = {
+        "HypergeometricU[0, c_, z_]": "1",
+        "HypergeometricU[a_, b_, z_]": "MeijerG[{{1-a},{}},{{0,1-b},{}},z]/Gamma[a]/Gamma[a-b+1]",
+    }
+    summary_text = "Tricomi confluent hypergeometric function"
+    sympy_name = ""
+
+    def eval_N(self, a, b, z, evaluation: Evaluation):
+        "N[HypergeometricU[a_, b_, z_]]"
+        try:
+            return run_mpmath(
+                mpmath.hyperu,
+                tuple([a.to_python(), b.to_python(), z.to_python()]),
+                FP_MANTISA_BINARY_DIGITS,
+            )
+        except Exception:
+            pass