def dup_spherical_bessel_fn(n, K):
    """ Low-level implementation of fn(n, x) """
    seq = [[K.one], [K.one, K.zero]]

    for i in range(2, n + 1):
        a = dup_mul_ground(dup_lshift(seq[-1], 1, K), K(2*i - 1), K)
        seq.append(dup_sub(a, seq[-2], K))

    return dup_lshift(seq[n], 1, K)

def dup_revert(f, n, K):
    """
    Compute ``f**(-1)`` mod ``x**n`` using Newton iteration.

    This function computes first ``2**n`` terms of a polynomial that
    is a result of inversion of a polynomial modulo ``x**n``. This is
    useful to efficiently compute series expansion of ``1/f``.

    Examples
    ========

    >>> from sympy.polys import ring, QQ
    >>> R, x = ring("x", QQ)

    >>> f = -QQ(1,720)*x**6 + QQ(1,24)*x**4 - QQ(1,2)*x**2 + 1

    >>> R.dup_revert(f, 8)
    61/720*x**6 + 5/24*x**4 + 1/2*x**2 + 1

    """
    g = [K.revert(dup_TC(f, K))]
    h = [K.one, K.zero, K.zero]

    N = int(_ceil(_log(n, 2)))

    for i in range(1, N + 1):
        a = dup_mul_ground(g, K(2), K)
        b = dup_mul(f, dup_sqr(g, K), K)
        g = dup_rem(dup_sub(a, b, K), h, K)
        h = dup_lshift(h, dup_degree(h), K)

    return g

def dup_revert(f, n, K):
    """
    Compute ``f**(-1)`` mod ``x**n`` using Newton iteration.

    This function computes first ``2**n`` terms of a polynomial that
    is a result of inversion of a polynomial modulo ``x**n``. This is
    useful to efficiently compute series expansion of ``1/f``.

    Examples
    ========

    >>> from sympy.polys.domains import QQ
    >>> from sympy.polys.densetools import dup_revert

    >>> f = [-QQ(1,720), QQ(0), QQ(1,24), QQ(0), -QQ(1,2), QQ(0), QQ(1)]

    >>> dup_revert(f, 8, QQ)
    [61/720, 0/1, 5/24, 0/1, 1/2, 0/1, 1/1]

    """
    g = [K.revert(dup_TC(f, K))]
    h = [K.one, K.zero, K.zero]

    N = int(_ceil(_log(n, 2)))

    for i in xrange(1, N + 1):
        a = dup_mul_ground(g, K(2), K)
        b = dup_mul(f, dup_sqr(g, K), K)
        g = dup_rem(dup_sub(a, b, K), h, K)
        h = dup_lshift(h, dup_degree(h), K)

    return g

def dup_chebyshevt(n, K):
    """Low-level implementation of Chebyshev polynomials of the 1st kind. """
    seq = [[K.one], [K.one, K.zero]]

    for i in range(2, n + 1):
        a = dup_mul_ground(dup_lshift(seq[-1], 1, K), K(2), K)
        seq.append(dup_sub(a, seq[-2], K))

    return seq[n]

def dup_legendre(n, K):
    """Low-level implementation of Legendre polynomials. """
    seq = [[K.one], [K.one, K.zero]]

    for i in range(2, n + 1):
        a = dup_mul_ground(dup_lshift(seq[-1], 1, K), K(2*i - 1, i), K)
        b = dup_mul_ground(seq[-2], K(i - 1, i), K)

        seq.append(dup_sub(a, b, K))

    return seq[n]

def dup_gegenbauer(n, a, K):
    """Low-level implementation of Gegenbauer polynomials. """
    seq = [[K.one], [K(2)*a, K.zero]]

    for i in range(2, n + 1):
        f1 = K(2) * (i + a - K.one) / i
        f2 = (i + K(2)*a - K(2)) / i
        p1 = dup_mul_ground(dup_lshift(seq[-1], 1, K), f1, K)
        p2 = dup_mul_ground(seq[-2], f2, K)
        seq.append(dup_sub(p1, p2, K))

    return seq[n]

def dup_hermite(n, K):
    """Low-level implementation of Hermite polynomials. """
    seq = [[K.one], [K(2), K.zero]]

    for i in range(2, n + 1):
        a = dup_lshift(seq[-1], 1, K)
        b = dup_mul_ground(seq[-2], K(i - 1), K)

        c = dup_mul_ground(dup_sub(a, b, K), K(2), K)

        seq.append(c)

    return seq[n]

def dup_jacobi(n, a, b, K):
    """Low-level implementation of Jacobi polynomials. """
    seq = [[K.one], [(a + b + K(2))/K(2), (a - b)/K(2)]]

    for i in range(2, n + 1):
        den = K(i)*(a + b + i)*(a + b + K(2)*i - K(2))
        f0 = (a + b + K(2)*i - K.one) * (a*a - b*b) / (K(2)*den)
        f1 = (a + b + K(2)*i - K.one) * (a + b + K(2)*i - K(2)) * (a + b + K(2)*i) / (K(2)*den)
        f2 = (a + i - K.one)*(b + i - K.one)*(a + b + K(2)*i) / den
        p0 = dup_mul_ground(seq[-1], f0, K)
        p1 = dup_mul_ground(dup_lshift(seq[-1], 1, K), f1, K)
        p2 = dup_mul_ground(seq[-2], f2, K)
        seq.append(dup_sub(dup_add(p0, p1, K), p2, K))

    return seq[n]

def dup_zz_cyclotomic_p(f, K, irreducible=False):
    """
    Efficiently test if ``f`` is a cyclotomic polnomial.

    **Examples**

    >>> from sympy.polys.factortools import dup_zz_cyclotomic_p
    >>> from sympy.polys.domains import ZZ

    >>> f = [1, 0, 1, 0, 0, 0,-1, 0, 1, 0,-1, 0, 0, 0, 1, 0, 1]
    >>> dup_zz_cyclotomic_p(f, ZZ)
    False

    >>> g = [1, 0, 1, 0, 0, 0,-1, 0,-1, 0,-1, 0, 0, 0, 1, 0, 1]
    >>> dup_zz_cyclotomic_p(g, ZZ)
    True

    """
    if K.is_QQ:
        try:
            K0, K = K, K.get_ring()
            f = dup_convert(f, K0, K)
        except CoercionFailed:
            return False
    elif not K.is_ZZ:
        return False

    lc = dup_LC(f, K)
    tc = dup_TC(f, K)

    if lc != 1 or (tc != -1 and tc != 1):
        return False

    if not irreducible:
        coeff, factors = dup_factor_list(f, K)

        if coeff != K.one or factors != [(f, 1)]:
            return False

    n = dup_degree(f)
    g, h = [], []

    for i in xrange(n, -1, -2):
        g.insert(0, f[i])

    for i in xrange(n-1, -1, -2):
        h.insert(0, f[i])

    g = dup_sqr(dup_strip(g), K)
    h = dup_sqr(dup_strip(h), K)

    F = dup_sub(g, dup_lshift(h, 1, K), K)

    if K.is_negative(dup_LC(F, K)):
        F = dup_neg(F, K)

    if F == f:
        return True

    g = dup_mirror(f, K)

    if K.is_negative(dup_LC(g, K)):
        g = dup_neg(g, K)

    if F == g and dup_zz_cyclotomic_p(g, K):
        return True

    G = dup_sqf_part(F, K)

    if dup_sqr(G, K) == F and dup_zz_cyclotomic_p(G, K):
        return True

    return False

def test_dup_lshift():
    assert dup_lshift([], 3, ZZ) == []
    assert dup_lshift([1], 3, ZZ) == [1,0,0,0]