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/*
Copyright (C) 2015 Fredrik Johansson
This file is part of Arb.
Arb 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 2.1 of the License, or
(at your option) any later version. See <http://www.gnu.org/licenses/>.
*/
#include "acb_hypgeom.h"
/* this can be improved */
static int
use_recurrence(const acb_t n, const acb_t a, const acb_t b, slong prec)
{
if (!acb_is_int(n) || !arb_is_nonnegative(acb_realref(n)))
return 0;
if (arf_cmpabs_ui(arb_midref(acb_realref(n)), prec) > 0)
return 0;
if (arb_is_nonnegative(acb_realref(a)) ||
arf_get_d(arb_midref(acb_realref(a)), ARF_RND_DOWN) > -0.9)
return 0;
return 1;
}
void
acb_hypgeom_jacobi_p_ui_direct(acb_t res, ulong n,
const acb_t a, const acb_t b, const acb_t z, slong prec)
{
acb_ptr terms;
acb_t t, u, v;
slong k;
terms = _acb_vec_init(n + 1);
acb_init(t);
acb_init(u);
acb_init(v);
acb_one(terms);
acb_add_ui(u, z, 1, prec);
for (k = 1; k <= n; k++)
{
acb_add_ui(t, a, n + 1 - k, prec);
acb_mul(t, t, u, prec);
acb_div_ui(t, t, 2 * k, prec);
acb_mul(terms + k, terms + k - 1, t, prec);
}
acb_sub_ui(u, z, 1, prec);
acb_one(v);
for (k = 1; k <= n; k++)
{
acb_add_ui(t, b, n + 1 - k, prec);
acb_mul(t, t, u, prec);
acb_div_ui(t, t, 2 * k, prec);
acb_mul(v, v, t, prec);
acb_mul(terms + n - k, terms + n - k, v, prec);
}
acb_set(res, terms);
for (k = 1; k <= n; k++)
acb_add(res, res, terms + k, prec);
_acb_vec_clear(terms, n + 1);
acb_clear(t);
acb_clear(u);
acb_clear(v);
}
void
acb_hypgeom_jacobi_p(acb_t res, const acb_t n, const acb_t a, const acb_t b,
const acb_t z, slong prec)
{
acb_t t, u, v, w;
if (use_recurrence(n, a, b, prec))
{
acb_hypgeom_jacobi_p_ui_direct(res,
arf_get_si(arb_midref(acb_realref(n)), ARF_RND_DOWN), a, b, z, prec);
return;
}
acb_init(t);
acb_init(u);
acb_init(v);
acb_init(w);
acb_neg(t, n);
acb_add_ui(v, a, 1, prec);
acb_add(u, n, v, prec);
acb_add(u, u, b, prec);
acb_sub_ui(w, z, 1, prec);
acb_mul_2exp_si(w, w, -1);
acb_neg(w, w);
acb_hypgeom_2f1(w, t, u, v, w, 0, prec);
acb_rising(t, v, n, prec);
acb_mul(w, w, t, prec);
acb_add_ui(t, n, 1, prec);
acb_rgamma(t, t, prec);
acb_mul(w, w, t, prec);
acb_set(res, w);
acb_clear(t);
acb_clear(u);
acb_clear(v);
acb_clear(w);
}
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