void
arb_poly_rising_ui_series(arb_poly_t res, const arb_poly_t f, ulong r, slong trunc, slong prec)
{
slong len;
if ((f->length == 0 && r != 0) || trunc == 0)
{
arb_poly_zero(res);
return;
}
if (r == 0)
{
arb_poly_one(res);
return;
}
len = poly_pow_length(f->length, r, trunc);
if (f == res)
{
arb_poly_t tmp;
arb_poly_init(tmp);
arb_poly_rising_ui_series(tmp, f, r, len, prec);
arb_poly_swap(tmp, res);
arb_poly_clear(tmp);
}
else
{
arb_poly_fit_length(res, len);
_arb_poly_rising_ui_series(res->coeffs, f->coeffs, f->length, r, len, prec);
_arb_poly_set_length(res, len);
_arb_poly_normalise(res);
}
}
static __inline__ void
_log_rising_ui_series(arb_ptr t, const arb_t x, slong r, slong len, slong prec)
{
arb_struct f[2];
slong rflen;
arb_init(f);
arb_init(f + 1);
arb_set(f, x);
arb_one(f + 1);
rflen = FLINT_MIN(len, r + 1);
_arb_poly_rising_ui_series(t, f, FLINT_MIN(2, len), r, rflen, prec);
_arb_poly_log_series(t, t, rflen, len, prec);
arb_clear(f);
arb_clear(f + 1);
}
static void
bound_rfac(arb_ptr F, const acb_t s, ulong n, slong len, slong wp)
{
if (len == 1)
{
acb_rising_ui_get_mag(arb_radref(F), s, n);
arf_set_mag(arb_midref(F), arb_radref(F));
mag_zero(arb_radref(F + 0));
}
else
{
arb_struct sx[2];
arb_init(sx + 0);
arb_init(sx + 1);
acb_abs(sx + 0, s, wp);
arb_one(sx + 1);
_arb_vec_zero(F, len);
_arb_poly_rising_ui_series(F, sx, 2, n, len, wp);
arb_clear(sx + 0);
arb_clear(sx + 1);
}
}
void
_arb_poly_rgamma_series(arb_ptr res, arb_srcptr h, long hlen, long len, long prec)
{
int reflect;
long i, rflen, r, n, wp;
arb_ptr t, u, v;
arb_struct f[2];
hlen = FLINT_MIN(hlen, len);
wp = prec + FLINT_BIT_COUNT(prec);
t = _arb_vec_init(len);
u = _arb_vec_init(len);
v = _arb_vec_init(len);
arb_init(f);
arb_init(f + 1);
/* use zeta values at small integers */
if (arb_is_int(h) && (arf_cmpabs_ui(arb_midref(h), prec / 2) < 0))
{
r = arf_get_si(arb_midref(h), ARF_RND_DOWN);
_arb_poly_lgamma_series_at_one(u, len, wp);
_arb_vec_neg(u, u, len);
_arb_poly_exp_series(t, u, len, len, wp);
if (r == 1)
{
_arb_vec_swap(v, t, len);
}
else if (r <= 0)
{
arb_set(f, h);
arb_one(f + 1);
rflen = FLINT_MIN(len, 2 - r);
_arb_poly_rising_ui_series(u, f, FLINT_MIN(2, len), 1 - r, rflen, wp);
_arb_poly_mullow(v, t, len, u, rflen, len, wp);
}
else
{
arb_one(f);
arb_one(f + 1);
rflen = FLINT_MIN(len, r);
_arb_poly_rising_ui_series(v, f, FLINT_MIN(2, len), r - 1, rflen, wp);
/* TODO: use div_series? */
_arb_poly_inv_series(u, v, rflen, len, wp);
_arb_poly_mullow(v, t, len, u, len, len, wp);
}
}
else
{
/* otherwise use Stirling series */
arb_gamma_stirling_choose_param(&reflect, &r, &n, h, 1, 0, wp);
/* rgamma(h) = (gamma(1-h+r) sin(pi h)) / (rf(1-h, r) * pi), h = h0 + t*/
if (reflect)
{
/* u = gamma(r+1-h) */
arb_sub_ui(f, h, r + 1, wp);
arb_neg(f, f);
_arb_poly_gamma_stirling_eval(t, f, n, len, wp);
_arb_poly_exp_series(u, t, len, len, wp);
for (i = 1; i < len; i += 2)
arb_neg(u + i, u + i);
/* v = sin(pi x) */
arb_const_pi(f + 1, wp);
arb_mul(f, h, f + 1, wp);
_arb_poly_sin_series(v, f, 2, len, wp);
_arb_poly_mullow(t, u, len, v, len, len, wp);
/* rf(1-h,r) * pi */
if (r == 0)
{
arb_const_pi(u, wp);
_arb_vec_scalar_div(v, t, len, u, wp);
}
else
{
arb_sub_ui(f, h, 1, wp);
arb_neg(f, f);
arb_set_si(f + 1, -1);
rflen = FLINT_MIN(len, r + 1);
_arb_poly_rising_ui_series(v, f, FLINT_MIN(2, len), r, rflen, wp);
arb_const_pi(u, wp);
_arb_vec_scalar_mul(v, v, rflen, u, wp);
/* divide by rising factorial */
/* TODO: might better to use div_series, when it has a good basecase */
_arb_poly_inv_series(u, v, rflen, len, wp);
_arb_poly_mullow(v, t, len, u, len, len, wp);
}
}
else
{
/* rgamma(h) = rgamma(h+r) rf(h,r) */
if (r == 0)
{
arb_add_ui(f, h, r, wp);
_arb_poly_gamma_stirling_eval(t, f, n, len, wp);
_arb_vec_neg(t, t, len);
_arb_poly_exp_series(v, t, len, len, wp);
}
else
{
arb_set(f, h);
arb_one(f + 1);
rflen = FLINT_MIN(len, r + 1);
_arb_poly_rising_ui_series(t, f, FLINT_MIN(2, len), r, rflen, wp);
arb_add_ui(f, h, r, wp);
_arb_poly_gamma_stirling_eval(v, f, n, len, wp);
_arb_vec_neg(v, v, len);
_arb_poly_exp_series(u, v, len, len, wp);
_arb_poly_mullow(v, u, len, t, rflen, len, wp);
}
}
}
/* compose with nonconstant part */
arb_zero(t);
_arb_vec_set(t + 1, h + 1, hlen - 1);
_arb_poly_compose_series(res, v, len, t, hlen, len, prec);
arb_clear(f);
arb_clear(f + 1);
_arb_vec_clear(t, len);
_arb_vec_clear(u, len);
_arb_vec_clear(v, len);
}
