Merge branch 'u/rws/interface_expression_conversion_to_gamma___and_no…

…rmalization' of git://trac.sagemath.org/sage into t/22326/jacobi_p_polynomials_without_pexpect_maxima

src/sage/libs/pynac/pynac.pxd

@@ -144,7 +144,9 @@ cdef extern from "sage/libs/pynac/wrap.h":
     object py_object_from_numeric(GEx e)     except +
 
     # Algorithms
-    GEx g_gcd "gcd"(GEx a, GEx b) except +
+    GEx g_gcd "gcd"(GEx a, GEx b)   except +
+    GEx to_gamma(GEx expr)          except +
+    GEx gamma_normalize(GEx expr)   except +
 
     # Pattern matching wildcards
     GEx g_wild "wild"(unsigned int label) except +

src/sage/symbolic/expression.pyx

@@ -9614,6 +9614,53 @@ cdef class Expression(CommutativeRingElement):
 
     factorial_simplify = simplify_factorial
 
+    def to_gamma(self):
+        """
+        Convert factorial, binomial, and Pochhammer symbol
+        expressions to their gamma function equivalents.
+
+        EXAMPLES::
+
+            sage: m,n = var('m n', domain='integer')
+            sage: factorial(n).to_gamma()
+            gamma(n + 1)
+            sage: binomial(m,n).to_gamma()
+            gamma(m + 1)/(gamma(m - n + 1)*gamma(n + 1))
+        """
+        cdef GEx x
+        sig_on()
+        try:
+            x = to_gamma(self._gobj)
+        finally:
+            sig_off()
+        return new_Expression_from_GEx(self._parent, x)
+
+    def gamma_normalize(self):
+        """
+        Return the expression with any gamma functions that have
+        a common base converted to that base.
+
+        Addtionally the expression is normalized so any fractions
+        can be simplified through cancellation.
+
+        EXAMPLES::
+
+            sage: m,n = var('m n', domain='integer')
+            sage: (gamma(n+2)/gamma(n)).gamma_normalize()
+            (n + 1)*n
+            sage: (gamma(n+2)*gamma(n)).gamma_normalize()
+            (n + 1)*n*gamma(n)^2
+            sage: (gamma(n+2)*gamma(m-1)/gamma(n)/gamma(m+1)).gamma_normalize()
+            (n + 1)*n/((m - 1)*m)
+        """
+        cdef GEx x
+        sig_on()
+        try:
+            x = gamma_normalize(self._gobj)
+        finally:
+            sig_off()
+        return new_Expression_from_GEx(self._parent, x)
+
     def expand_sum(self):
         r"""
         For every symbolic sum in the given expression, try to expand it,