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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_legendre_p_uiui_rec(acb_t res, ulong n, ulong m, const acb_t z, slong prec)
{
acb_t t, u, v;
slong k;
if (!acb_is_finite(z))
{
acb_indeterminate(res);
return;
}
if (m > n)
{
acb_zero(res);
return;
}
if ((n - m) / 4 > prec)
{
acb_indeterminate(res);
return;
}
acb_init(t);
acb_init(u);
acb_init(v);
/* t = p(m,m) = (-1)^m (2m-1)!! */
if (m == 0)
arb_one(acb_realref(t));
else
arb_doublefac_ui(acb_realref(t), 2 * m - 1, prec);
if (m % 2)
arb_neg(acb_realref(t), acb_realref(t));
if (n > m)
{
/* t = p(m+1,m) = z(2m+1)p(m,m), u = p(m,m) */
acb_mul_ui(u, t, 2 * m + 1, prec);
acb_mul(u, u, z, prec);
acb_swap(t, u);
for (k = m + 2; k <= n; k++)
{
/* t, u = ((2*k-1)*z*t - (k+m-1)*u) / (k-m), t */
acb_mul(v, t, z, prec);
acb_mul_ui(v, v, 2 * k - 1, prec);
acb_submul_ui(v, u, k + m - 1, prec);
acb_div_ui(u, v, k - m, prec);
acb_swap(t, u);
}
}
acb_set(res, t);
acb_clear(t);
acb_clear(u);
acb_clear(v);
}