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); }
void _arb_poly_lgamma_series(arb_ptr res, arb_srcptr h, slong hlen, slong len, slong prec) { int reflect; slong r, n, wp; arb_t zr; arb_ptr t, u; if (!arb_is_positive(h)) { _arb_vec_indeterminate(res, len); return; } hlen = FLINT_MIN(hlen, len); wp = prec + FLINT_BIT_COUNT(prec); t = _arb_vec_init(len); u = _arb_vec_init(len); arb_init(zr); /* 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); if (r <= 0) { _arb_vec_indeterminate(res, len); goto cleanup; } else { _arb_poly_lgamma_series_at_one(u, len, wp); if (r != 1) { arb_one(zr); _log_rising_ui_series(t, zr, r - 1, len, wp); _arb_vec_add(u, u, t, len, wp); } } } else if (len <= 2) { arb_lgamma(u, h, wp); if (len == 2) arb_digamma(u + 1, h, wp); } else { /* otherwise use Stirling series */ arb_gamma_stirling_choose_param(&reflect, &r, &n, h, 0, 0, wp); arb_add_ui(zr, h, r, wp); _arb_poly_gamma_stirling_eval(u, zr, n, len, wp); if (r != 0) { _log_rising_ui_series(t, h, r, len, wp); _arb_vec_sub(u, u, t, len, wp); } } /* compose with nonconstant part */ arb_zero(t); _arb_vec_set(t + 1, h + 1, hlen - 1); _arb_poly_compose_series(res, u, len, t, hlen, len, prec); cleanup: arb_clear(zr); _arb_vec_clear(t, len); _arb_vec_clear(u, len); }