(f, g, n, polys=True)
return A, B, C
r"""
    Compute the Gosper's normal form of ``f`` and ``g``.

    Given relatively prime univariate polynomials ``f`` and ``g``,
    rewrite their quotient to a normal form defined as follows:

    .. math::
        \frac{f(n)}{g(n)} = Z \cdot \frac{A(n) C(n+1)}{B(n) C(n)}

    where ``Z`` is an arbitrary constant and ``A``, ``B``, ``C`` are
    monic polynomials in ``n`` with the following properties:

    1. `\gcd(A(n), B(n+h)) = 1 \forall h \in \mathbb{N}`
    2. `\gcd(B(n), C(n+1)) = 1`
    3. `\gcd(A(n), C(n)) = 1`

    This normal form, or rational factorization in other words, is a
    crucial step in Gosper's algorithm and in solving of difference
    equations. It can be also used to decide if two hypergeometric
    terms are similar or not.

    This procedure will return a tuple containing elements of this
    factorization in the form ``(Z*A, B, C)``.

    Examples
    ========

    >>> from sympy.concrete.gosper import gosper_normal
    >>> from sympy.abc import n

    >>> gosper_normal(4*n+5, 2*(4*n+1)*(2*n+3), n, polys=False)
    (1/4, n + 3/2, n + 1/4)
r"""
    Compute the Gosper's normal form of ``f`` and ``g``.
r Compute the Gosper s normal form of f and g .
def gosper_normal(f, g, n, polys=True):
    r"""
    Compute the Gosper's normal form of ``f`` and ``g``.

    Given relatively prime univariate polynomials ``f`` and ``g``,
    rewrite their quotient to a normal form defined as follows:

    .. math::
        \frac{f(n)}{g(n)} = Z \cdot \frac{A(n) C(n+1)}{B(n) C(n)}

    where ``Z`` is an arbitrary constant and ``A``, ``B``, ``C`` are
    monic polynomials in ``n`` with the following properties:

    1. `\gcd(A(n), B(n+h)) = 1 \forall h \in \mathbb{N}`
    2. `\gcd(B(n), C(n+1)) = 1`
    3. `\gcd(A(n), C(n)) = 1`

    This normal form, or rational factorization in other words, is a
    crucial step in Gosper's algorithm and in solving of difference
    equations. It can be also used to decide if two hypergeometric
    terms are similar or not.

    This procedure will return a tuple containing elements of this
    factorization in the form ``(Z*A, B, C)``.

    Examples
    ========

    >>> from sympy.concrete.gosper import gosper_normal
    >>> from sympy.abc import n

    >>> gosper_normal(4*n+5, 2*(4*n+1)*(2*n+3), n, polys=False)
    (1/4, n + 3/2, n + 1/4)

    """
    (p, q), opt = parallel_poly_from_expr(
        (f, g), n, field=True, extension=True)

    a, A = p.LC(), p.monic()
    b, B = q.LC(), q.monic()

    C, Z = A.one, a/b
    h = Dummy('h')

    D = Poly(n + h, n, h, domain=opt.domain)

    R = A.resultant(B.compose(D))
    roots = set(R.ground_roots().keys())

    for r in set(roots):
        if not r.is_Integer or r < 0:
            roots.remove(r)

    for i in sorted(roots):
        d = A.gcd(B.shift(+i))

        A = A.quo(d)
        B = B.quo(d.shift(-i))

        for j in xrange(1, i + 1):
            C *= d.shift(-j)

    A = A.mul_ground(Z)

    if not polys:
        A = A.as_expr()
        B = B.as_expr()
        C = C.as_expr()

    return A, B, C
def gosper_normal ( f , g , n , polys = True ) : ( p , q ) , opt = parallel_poly_from_expr ( ( f , g ) , n , field = True , extension = True ) a , A = p . LC ( ) , p . monic ( ) b , B = q . LC ( ) , q . monic ( ) C , Z = A . one , a / b h = Dummy ( 'h' ) D = Poly ( n + h , n , h , domain = opt . domain ) R = A . resultant ( B . compose ( D ) ) roots = set ( R . ground_roots ( ) . keys ( ) ) for r in set ( roots ) : if not r . is_Integer or r < 0 : roots . remove ( r ) for i in sorted ( roots ) : d = A . gcd ( B . shift ( + i ) ) A = A . quo ( d ) B = B . quo ( d . shift ( - i ) ) for j in xrange ( 1 , i + 1 ) : C *= d . shift ( - j ) A = A . mul_ground ( Z ) if not polys : A = A . as_expr ( ) B = B . as_expr ( ) C = C . as_expr ( ) return A , B , C