laurent series Fibonacci generating fuction

[lukaskiss222]

Hi all,

I play little bit with Fibonacci gf.
fi = (1+sp.sqrt(5))/2 # this is singularity of Fib. gf. (inverse of golden ratio)
(1/(1-x-x**2)).series(x,1/fi,2) and I get

This is clearly not a laurent series in 1/fi and the first therm is /0.

[jksuom]

The implementation of the series expansion starts by computing the leading term of the denominator at 1/fi:

In [1]: f = 1 - x - x**2                                                        

In [2]: fi = (1 + sqrt(5))/2                                                    

In [3]: f.subs(x, y + 1/fi).leadterm(y)                                         
Out[3]: 
⎛                    2   ⎞
⎜-6 - 2⋅√5 + (1 + √5)    ⎟
⎜─────────────────────, 0⎟
⎜              2         ⎟
⎝      (1 + √5)          ⎠

This fails because expand_mul in Add._eval_as_leading_term does not simplify the constant term to zero. It seems that _mexpand could be used instead:

--- a/sympy/core/add.py
+++ b/sympy/core/add.py
@@ -978,11 +978,12 @@ def as_real_imag(self, deep=True, **hints):
         return (self.func(*re_part), self.func(*im_part))
 
     def _eval_as_leading_term(self, x, cdir=0):
-        from sympy import expand_mul, Order
+        from sympy.core.function import _mexpand
+        from sympy.series.order import Order
 
         old = self
 
-        expr = expand_mul(self)
+        expr = _mexpand(self)
         if not expr.is_Add:
             return expr.as_leading_term(x, cdir=cdir)
 
@@ -1004,7 +1005,7 @@ def _eval_as_leading_term(self, x, cdir=0):
         except TypeError:
             return expr
 
-        new_expr=new_expr.together()
+        new_expr = _mexpand(new_expr.together())
         if new_expr.is_Add:
             new_expr = new_expr.simplify()

[jksuom]

Maybe all expand functions can be dispensed with if new_expr is tested with .is_zero (in which case there will be no need to assign to expr).

--- a/sympy/core/add.py
+++ b/sympy/core/add.py
@@ -978,11 +978,10 @@ def as_real_imag(self, deep=True, **hints):
         return (self.func(*re_part), self.func(*im_part))
 
     def _eval_as_leading_term(self, x, cdir=0):
-        from sympy import expand_mul, Order
+        from sympy import Order
 
         old = self
-
-        expr = expand_mul(self)
+        expr = self
         if not expr.is_Add:
             return expr.as_leading_term(x, cdir=cdir)
 
@@ -1003,12 +1002,11 @@ def _eval_as_leading_term(self, x, cdir=0):
 
         except TypeError:
             return expr
 
-        new_expr=new_expr.together()
+        new_expr = new_expr.together()
         if new_expr.is_Add:
             new_expr = new_expr.simplify()
 
-        if not new_expr:
+        if new_expr.is_zero:
             # simple leading term analysis gave us cancelled terms but we have to send
             # back a term, so compute the leading term (via series)
             n0 = min.getn()