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/*
Copyright (C) 2021 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 "arb.h"
#include "arb_hypgeom.h"
void
_arb_hypgeom_gamma_stirling_term_bounds(slong * bound, const mag_t zinv, slong N)
{
mag_t b, u;
slong n;
mag_init(b);
mag_init(u);
/* bound[0] = WORD_MAX; -- should not be used */
/* first term 1/(12z) */
mag_set(b, zinv);
mag_div_ui(b, b, 12);
bound[1] = MAG_EXP(b);
/* u = 1/(2 pi z)^2 */
mag_const_pi_lower(u);
mag_mul_2exp_si(u, u, 1);
mag_inv(u, u);
mag_mul(u, u, zinv);
mag_mul(u, u, u);
/* zeta(2n) 2 (2n-2)! / (2pi)^(2n) / z^(2n-1) */
/* ratio bounded by (2n-2)(2n-3)/(2 pi z)^2 */
for (n = 2; n < N; n++)
{
mag_mul_ui(b, b, (2*n-2) * (2*n-3));
mag_mul(b, b, u);
bound[n] = MAG_EXP(b);
}
mag_clear(b);
mag_clear(u);
}