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/*
Copyright (C) 2016 Fredrik Johansson
This file is part of FLINT.
FLINT is free software: you can redistribute it and/or modify it under
the terms of the GNU Lesser General Public License (LGPL) as published
by the Free Software Foundation; either version 3 of the License, or
(at your option) any later version. See <https://www.gnu.org/licenses/>.
*/
#include "acb.h"
void
acb_bernoulli_poly_ui(acb_t res, ulong n, const acb_t x, slong prec)
{
acb_t s, x2;
arb_t t, c;
ulong k;
if (n == 0)
{
acb_one(res);
return;
}
if (n == 1)
{
acb_mul_2exp_si(res, x, 1);
acb_sub_ui(res, res, 1, prec);
acb_mul_2exp_si(res, res, -1);
return;
}
if (acb_is_real(x))
{
arb_bernoulli_poly_ui(acb_realref(res), n, acb_realref(x), prec);
arb_zero(acb_imagref(res));
return;
}
/* assuming small n simplifies the code that follows */
if (n >> (FLINT_BITS / 2) || !acb_is_finite(x))
{
acb_indeterminate(res);
return;
}
acb_init(s);
acb_init(x2);
arb_init(t);
arb_init(c);
acb_mul(x2, x, x, prec);
/* s = x^2 - x n / 2 */
acb_mul_ui(s, x, n, prec);
acb_mul_2exp_si(s, s, -1);
acb_sub(s, x2, s, prec);
/* c = n (n-1) / 2; s = s + c / 6 */
arb_set_ui(c, n * (n - 1));
arb_mul_2exp_si(c, c, -1);
arb_div_ui(t, c, 6, prec);
acb_add_arb(s, s, t, prec);
for (k = 4; k <= n; k += 2)
{
/* c = binomial(n,k) */
arb_mul_ui(c, c, (n + 1 - k) * (n + 2 - k), prec);
arb_div_ui(c, c, k * (k - 1), prec);
/* s = s x^2 + b_k c */
acb_mul(s, s, x2, prec);
arb_bernoulli_ui(t, k, prec);
arb_mul(t, t, c, prec);
acb_add_arb(s, s, t, prec);
}
if (n >= 3 && n % 2)
acb_mul(s, s, x, prec);
acb_swap(res, s);
acb_clear(s);
acb_clear(x2);
arb_clear(t);
arb_clear(c);
}