Fix conversion from SymbolicSeries to LaurentSeries

src/sage/rings/laurent_series_ring.py

@@ -42,6 +42,7 @@
 from sage.rings.infinity import infinity
 from sage.rings.integer_ring import ZZ
 from sage.rings.laurent_series_ring_element import LaurentSeries
+from sage.structure.element import Expression
 from sage.structure.parent import Parent
 from sage.structure.unique_representation import UniqueRepresentation
 
@@ -542,6 +543,19 @@ def _element_constructor_(self, x, n=0, prec=infinity):
             sage: P.<x> = LaurentSeriesRing(QQ)
             sage: P({-3: 1})
             x^-3
+
+        Check that :issue:`39839` is fixed::
+
+            sage: var("x")
+            x
+            sage: f = (1/x+sqrt(x+1)).series(x, 5); f
+            1*x^(-1) + 1 + 1/2*x + (-1/8)*x^2 + 1/16*x^3 + (-5/128)*x^4 + Order(x^5)
+            sage: LaurentSeriesRing(QQ, "x")(f)
+            x^-1 + 1 + 1/2*x - 1/8*x^2 + 1/16*x^3 - 5/128*x^4 + O(x^5)
+            sage: PowerSeriesRing(QQ, "x")(f)
+            Traceback (most recent call last):
+            ...
+            ValueError: cannot return dense coefficient list with negative valuation
         """
         from sage.rings.fraction_field_element import FractionFieldElement
         from sage.rings.lazy_series import LazyPowerSeries, LazyLaurentSeries
@@ -589,6 +603,18 @@ def _element_constructor_(self, x, n=0, prec=infinity):
                     x = x.add_bigoh(self.default_prec())
             else:
                 x = x.add_bigoh(prec)
+        elif isinstance(x, Expression):
+            from sage.symbolic.expression import SymbolicSeries
+            if isinstance(x, SymbolicSeries):
+                v = x.default_variable()
+                if str(v) == self.variable_name():
+                    R = self.base_ring()
+                    g = self.gen()
+                    return sum(
+                            (R(a)*g**ZZ(e) for a, e in x.coefficients(v, sparse=True)), self.zero()
+                            ).add_bigoh(x.degree(x.default_variable()))
+                else:
+                    raise TypeError("can only convert series into ring with same variable name")
         return self.element_class(self, x, n).add_bigoh(prec)
 
     def random_element(self, algorithm='default'):

src/sage/rings/lazy_series.py

@@ -2371,7 +2371,7 @@ def cot(self):
         TESTS::
 
             sage: L.<z> = LazyLaurentSeriesRing(QQ); x = var("x")                       # needs sage.symbolic
-            sage: cot(z)[0:6] == cot(x).series(x, 6).coefficients(sparse=False)         # needs sage.symbolic
+            sage: cot(z)[0:6] == (cot(x)-x^-1).series(x, 6).coefficients(sparse=False)  # needs sage.symbolic
             True
         """
         return ~self.tan()
@@ -2653,7 +2653,7 @@ def coth(self):
         TESTS::
 
             sage: L.<z> = LazyLaurentSeriesRing(SR); x = var("x")                       # needs sage.symbolic
-            sage: coth(z)[0:6] == coth(x).series(x, 6).coefficients(sparse=False)       # needs sage.symbolic
+            sage: coth(z)[0:6] == (coth(x)-x^-1).series(x, 6).coefficients(sparse=False)# needs sage.symbolic
             True
         """
         from sage.arith.misc import bernoulli
@@ -2716,7 +2716,7 @@ def csch(self):
         TESTS::
 
             sage: L.<z> = LazyLaurentSeriesRing(SR); x = var("x")                       # needs sage.symbolic
-            sage: csch(z)[0:6] == csch(x).series(x, 6).coefficients(sparse=False)       # needs sage.symbolic
+            sage: csch(z)[0:6] == (csch(x)-x^-1).series(x, 6).coefficients(sparse=False) # needs sage.symbolic
             True
         """
         from sage.arith.misc import bernoulli

src/sage/symbolic/series_impl.pxi

@@ -126,6 +126,44 @@ Check that :issue:`22733` is fixed::
 # ****************************************************************************
 
 
+cpdef bint _is_order(Expression expr):
+    """
+    Return whether ``expr`` has the form ``Order(...)``.
+
+    TESTS::
+
+        sage: from sage.symbolic.expression import _is_order
+        sage: s = (x^2).series(x,1)
+        sage: expr = Expression.coefficients(s)[0][0]; expr
+        Order(x)
+        sage: _is_order(expr)
+        True
+        sage: expr = Expression.coefficients(x.series(x, 0))[0][0]; expr
+        Order(1)
+        sage: _is_order(expr)
+        True
+        sage: expr = Expression.coefficients((x^5).series(x, 3))[0][0]; expr
+        Order(x^3)
+        sage: _is_order(expr)
+        True
+        sage: from sage.functions.other import Order; _is_order(Order(x))
+        True
+        sage: _is_order(Order(x) + 1)
+        False
+        sage: _is_order(Order(x) + 1 - 1)
+        True
+        sage: var("y")
+        y
+        sage: _is_order(Order(y))
+        True
+    """
+    from sage.functions.other import Order
+    from sage.symbolic.operators import add_vararg
+    if expr.operator() is add_vararg and len(expr.operands()) == 1:
+        expr = expr.operands()[0]
+    return expr.operator() is Order
+
+
 cdef class SymbolicSeries(Expression):
     def __init__(self, SR):
         """
@@ -201,22 +239,11 @@ cdef class SymbolicSeries(Expression):
 
     def coefficients(self, x=None, sparse=True):
         r"""
-        Return the coefficients of this symbolic series as a list of pairs.
-
-        INPUT:
-
-        - ``x`` -- (optional) variable
-
-        - ``sparse`` -- boolean (default: ``True``); if ``False`` return a list
-          with as much entries as the order of the series
-
-        OUTPUT: depending on the value of ``sparse``,
-
-        - A list of pairs ``(expr, n)``, where ``expr`` is a symbolic
-          expression and ``n`` is a power (``sparse=True``, default)
+        Return the coefficients of this symbolic series. See :meth:`Expression.coefficients` for more details.
 
-        - A list of expressions where the ``n``-th element is the coefficient of
-          ``x^n`` when ``self`` is seen as polynomial in ``x`` (``sparse=False``).
+        When ``sparse=False``, unlike :meth:`Expression.coefficients`, the dense list of coefficients
+        are padded to the degree of the asymptotic term of this series.
+        Also, this correctly handles the case where there is no term below the asymptotic term.
 
         EXAMPLES::
 
@@ -231,24 +258,63 @@ cdef class SymbolicSeries(Expression):
             1 + (y + 1)*x + ((y + 1)^2)*x^2 + Order(x^3)
             sage: s.coefficients(x, sparse=False)
             [1, y + 1, (y + 1)^2]
+
+        TESTS::
+
+            sage: s = (x^-1 + x^2).series(x,6)
+            sage: s.coefficients()
+            [[1, -1], [1, 2]]
+            sage: s.coefficients(sparse=False)
+            Traceback (most recent call last):
+            ...
+            ValueError: cannot return dense coefficient list with negative valuation
+            sage: Expression.coefficients(s)
+            [[1, -1], [1, 2]]
+            sage: Expression.coefficients(s, sparse=False)
+            Traceback (most recent call last):
+            ...
+            ValueError: cannot return dense coefficient list with negative valuation
+
+            sage: s = (x+x^3).series(x,6)
+            sage: s.coefficients()
+            [[1, 1], [1, 3]]
+            sage: s.coefficients(sparse=False)
+            [0, 1, 0, 1, 0, 0]
+            sage: Expression.coefficients(s)
+            [[1, 1], [1, 3]]
+            sage: Expression.coefficients(s, sparse=False)
+            [0, 1, 0, 1]
+
+            sage: s = (x^5).series(x,1)
+            sage: s.coefficients()
+            []
+            sage: s.coefficients(sparse=False)
+            [0]
+            sage: Expression.coefficients(s)  # incorrect result here, but the user will not see this
+            [[Order(x), 0]]
+            sage: Expression.coefficients(s, sparse=False)
+            [Order(x)]
+
+            sage: s = (x^5).series(x,4)
+            sage: s.coefficients()
+            []
+            sage: s.coefficients(sparse=False)
+            [0, 0, 0, 0]
+            sage: Expression.coefficients(s)  # incorrect result here, but the user will not see this
+            [[Order(x^4), 0]]
+            sage: Expression.coefficients(s, sparse=False)
+            [Order(x^4)]
         """
         if x is None:
             x = self.default_variable()
-        l = [[self.coefficient(x, d), d] for d in range(self.degree(x))]
+        result = super().coefficients(x, sparse)
+        if sparse and len(result) == 1 and ZZ(result[0][1]) == 0 and _is_order(result[0][0]):
+            result = []
         if sparse:
-            return l
-
-        from sage.rings.integer_ring import ZZ
-        if any(not c[1] in ZZ for c in l):
-            raise ValueError("cannot return dense coefficient list with noninteger exponents")
-        val = l[0][1]
-        if val < 0:
-            raise ValueError("cannot return dense coefficient list with negative valuation")
-        deg = l[-1][1]
-        ret = [ZZ(0)] * int(deg+1)
-        for c in l:
-            ret[c[1]] = c[0]
-        return ret
+            return result
+        if len(result) == 1 and _is_order(result[0]):
+            result = []
+        return result + [ZZ.zero()] * (self.degree(x) - len(result))
 
     def power_series(self, base_ring):
         """