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/*
Copyright (C) 2015 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"
#include "acb_hypgeom.h"
void
acb_hypgeom_2f1_direct(acb_t res, const acb_t a, const acb_t b,
const acb_t c, const acb_t z, int regularized, slong prec)
{
/* 2F1R(a,b,-n,z) = (a)_(n+1) * (b)_(n+1) * z^(n+1) / (n+1)!
* 2F1(a+n+1, b+n+1, n+2, z) */
if (regularized && acb_is_int(c)
&& arf_sgn(arb_midref(acb_realref(c))) <= 0)
{
if ((acb_is_int(a) && arb_is_nonpositive(acb_realref(a)) &&
arf_cmp(arb_midref(acb_realref(a)), arb_midref(acb_realref(c))) >= 0) ||
(acb_is_int(b) && arb_is_nonpositive(acb_realref(b)) &&
arf_cmp(arb_midref(acb_realref(b)), arb_midref(acb_realref(c))) >= 0))
{
acb_zero(res);
}
else
{
acb_t n, n1, t, u, v;
acb_ptr aa;
int p, q;
acb_init(n);
acb_init(n1);
acb_init(t);
acb_init(u);
acb_init(v);
aa = _acb_vec_init(4);
acb_neg(n, c);
acb_add_ui(n1, n, 1, prec);
acb_add(aa, a, n1, prec);
acb_add(aa + 1, b, n1, prec);
acb_add_ui(aa + 2, n1, 1, prec);
if (acb_is_one(aa))
{
p = q = 1;
acb_swap(aa, aa + 1);
}
else if (acb_is_one(aa + 1))
{
p = q = 1;
}
else
{
p = q = 2;
acb_one(aa + 3);
}
acb_hypgeom_pfq_direct(t, aa, p, aa + 2, q, z, -1, prec);
/* z^(n+1) */
acb_pow(u, z, n1, prec);
acb_mul(t, t, u, prec);
acb_rising(u, a, n1, prec);
acb_mul(t, t, u, prec);
acb_rising(u, b, n1, prec);
acb_mul(t, t, u, prec);
/* 1/(n+1)! */
acb_rgamma(u, aa + 2, prec);
acb_mul(t, t, u, prec);
acb_set(res, t);
acb_clear(n);
acb_clear(n1);
acb_clear(t);
acb_clear(u);
acb_clear(v);
_acb_vec_clear(aa, 4);
}
}
else
{
acb_ptr aa;
int p, q;
aa = _acb_vec_init(4);
acb_set(aa + 2, c);
if (acb_is_one(a))
{
p = q = 1;
acb_set(aa, b);
}
else if (acb_is_one(b))
{
p = q = 1;
acb_set(aa, a);
}
else
{
p = q = 2;
acb_set(aa, a);
acb_set(aa + 1, b);
acb_one(aa + 3);
}
acb_hypgeom_pfq_direct(res, aa, p, aa + 2, q, z, -1, prec);
if (regularized)
{
acb_rgamma(aa + 2, aa + 2, prec);
acb_mul(res, res, aa + 2, prec);
}
_acb_vec_clear(aa, 4);
}
if (!acb_is_finite(res))
acb_indeterminate(res);
}