def _eval_sum_hyper(f, i, a):
    """ Returns (res, cond). Sums from a to oo. """
    from sympy.functions import hyper
    from sympy.simplify import hyperexpand, hypersimp, fraction, simplify
    from sympy.polys.polytools import Poly, factor

    if a != 0:
        return _eval_sum_hyper(f.subs(i, i + a), i, 0)

    if f.subs(i, 0) == 0:
        if simplify(f.subs(i, Dummy('i', integer=True, positive=True))) == 0:
            return S(0), True
        return _eval_sum_hyper(f.subs(i, i + 1), i, 0)

    hs = hypersimp(f, i)
    if hs is None:
        return None

    numer, denom = fraction(factor(hs))
    top, topl = numer.as_coeff_mul(i)
    bot, botl = denom.as_coeff_mul(i)
    ab = [top, bot]
    factors = [topl, botl]
    params = [[], []]
    for k in range(2):
        for fac in factors[k]:
            mul = 1
            if fac.is_Pow:
                mul = fac.exp
                fac = fac.base
                if not mul.is_Integer:
                    return None
            p = Poly(fac, i)
            if p.degree() != 1:
                return None
            m, n = p.all_coeffs()
            ab[k] *= m**mul
            params[k] += [n/m]*mul

    # Add "1" to numerator parameters, to account for implicit n! in
    # hypergeometric series.
    ap = params[0] + [1]
    bq = params[1]
    x = ab[0]/ab[1]
    h = hyper(ap, bq, x)

    return f.subs(i, 0)*hyperexpand(h), h.convergence_statement
dataset/ETHPy150Open sympy/sympy/sympy/concrete/summations.py/_eval_sum_hyper/original