void
_acb_hypgeom_2f1_transform_limit(acb_t res, const acb_poly_t a, const acb_poly_t b,
const acb_poly_t c, const acb_poly_t z, int which, slong prec)
{
acb_poly_t ba, ca, cb, cab, ac1, bc1, ab1, ba1, w, t, u, v, s;
acb_t tt;
acb_poly_init(ba);
acb_poly_init(ca); acb_poly_init(cb); acb_poly_init(cab);
acb_poly_init(ac1); acb_poly_init(bc1);
acb_poly_init(ab1); acb_poly_init(ba1);
acb_poly_init(w); acb_poly_init(t);
acb_poly_init(u); acb_poly_init(v);
acb_poly_init(s);
acb_init(tt);
acb_poly_add_si(s, z, -1, prec); /* s = 1 - z */
acb_poly_neg(s, s);
acb_poly_sub(ba, b, a, prec); /* ba = b - a */
acb_poly_sub(ca, c, a, prec); /* ca = c - a */
acb_poly_sub(cb, c, b, prec); /* cb = c - b */
acb_poly_sub(cab, ca, b, prec); /* cab = c - a - b */
acb_poly_add_si(ac1, ca, -1, prec); acb_poly_neg(ac1, ac1); /* ac1 = a - c + 1 */
acb_poly_add_si(bc1, cb, -1, prec); acb_poly_neg(bc1, bc1); /* bc1 = b - c + 1 */
acb_poly_add_si(ab1, ba, -1, prec); acb_poly_neg(ab1, ab1); /* ab1 = a - b + 1 */
acb_poly_add_si(ba1, ba, 1, prec); /* ba1 = b - a + 1 */
/* t = left term, u = right term (DLMF 15.8.1 - 15.8.5) */
if (which == 2)
{
acb_poly_inv_series(w, z, 2, prec); /* w = 1/z */
acb_hypgeom_2f1_series_direct(t, a, ac1, ab1, w, 1, 2, prec);
acb_hypgeom_2f1_series_direct(u, b, bc1, ba1, w, 1, 2, prec);
}
else if (which == 3)
{
acb_poly_inv_series(w, s, 2, prec); /* w = 1/(1-z) */
acb_hypgeom_2f1_series_direct(t, a, cb, ab1, w, 1, 2, prec);
acb_hypgeom_2f1_series_direct(u, b, ca, ba1, w, 1, 2, prec);
}
else if (which == 4)
{
acb_poly_set(w, s); /* w = 1-z */
acb_poly_add(v, ac1, b, prec); /* v = a+b-c+1 */
acb_hypgeom_2f1_series_direct(t, a, b, v, w, 1, 2, prec);
acb_poly_add_si(v, cab, 1, prec); /* v = c-a-b+1 */
acb_hypgeom_2f1_series_direct(u, ca, cb, v, w, 1, 2, prec);
}
else if (which == 5)
{
acb_poly_inv_series(w, z, 2, prec); /* w = 1-1/z */
acb_poly_neg(w, w);
acb_poly_add_si(w, w, 1, prec);
acb_poly_add(v, ac1, b, prec); /* v = a+b-c+1 */
acb_hypgeom_2f1_series_direct(t, a, ac1, v, w, 1, 2, prec);
acb_poly_add_si(v, cab, 1, prec); /* v = c-a-b+1 */
acb_poly_add_si(u, a, -1, prec); /* u = 1-a */
acb_poly_neg(u, u);
acb_hypgeom_2f1_series_direct(u, ca, u, v, w, 1, 2, prec);
}
else
{
flint_printf("invalid transformation!\n");
flint_abort();
}
/* gamma factors */
acb_poly_rgamma_series(v, a, 2, prec);
acb_poly_mullow(u, u, v, 2, prec);
acb_poly_rgamma_series(v, ca, 2, prec);
acb_poly_mullow(t, t, v, 2, prec);
acb_poly_rgamma_series(v, b, 2, prec);
if (which == 2 || which == 3)
acb_poly_mullow(t, t, v, 2, prec);
else
acb_poly_mullow(u, u, v, 2, prec);
acb_poly_rgamma_series(v, cb, 2, prec);
if (which == 2 || which == 3)
acb_poly_mullow(u, u, v, 2, prec);
else
acb_poly_mullow(t, t, v, 2, prec);
if (which == 2 || which == 3)
{
if (which == 2)
acb_poly_neg(s, z); /* -z, otherwise 1-z since before */
acb_poly_neg(v, a);
acb_poly_pow_series(v, s, v, 2, prec);
acb_poly_mullow(t, t, v, 2, prec);
acb_poly_neg(v, b);
acb_poly_pow_series(v, s, v, 2, prec);
acb_poly_mullow(u, u, v, 2, prec);
}
else
{
acb_poly_pow_series(v, s, cab, 2, prec);
acb_poly_mullow(u, u, v, 2, prec);
if (which == 5)
{
acb_poly_neg(v, a);
acb_poly_pow_series(v, z, v, 2, prec);
acb_poly_mullow(t, t, v, 2, prec);
acb_poly_neg(v, ca);
acb_poly_pow_series(v, z, v, 2, prec);
acb_poly_mullow(u, u, v, 2, prec);
}
}
acb_poly_sub(t, t, u, prec);
if (which == 2 || which == 3)
acb_poly_sin_pi_series(v, ba, 2, prec);
else
acb_poly_sin_pi_series(v, cab, 2, prec);
acb_poly_get_coeff_acb(tt, t, 1);
acb_poly_get_coeff_acb(res, v, 1);
acb_div(res, tt, res, prec);
acb_const_pi(tt, prec);
acb_mul(res, res, tt, prec);
acb_poly_clear(ba);
acb_poly_clear(ca); acb_poly_clear(cb); acb_poly_clear(cab);
acb_poly_clear(ac1); acb_poly_clear(bc1);
acb_poly_clear(ab1); acb_poly_clear(ba1);
acb_poly_clear(w); acb_poly_clear(t);
acb_poly_clear(u); acb_poly_clear(v);
acb_poly_clear(s);
acb_clear(tt);
}
void
acb_hypgeom_bessel_k_0f1_series(acb_poly_t res,
const acb_poly_t nu, const acb_poly_t z,
slong len, slong prec)
{
acb_poly_t s, u, A, B;
acb_poly_struct b[2];
arb_t c;
slong wlen;
int singular;
acb_poly_init(s);
acb_poly_init(u);
acb_poly_init(A);
acb_poly_init(B);
acb_poly_init(b + 0);
acb_poly_init(b + 1);
arb_init(c);
singular = (nu->length == 0) || acb_is_int(nu->coeffs);
wlen = len + (singular != 0);
/* A = (z/2)^nu, B = 1/A */
acb_poly_scalar_mul_2exp_si(A, z, -1);
acb_poly_pow_series(A, A, nu, wlen, prec);
acb_poly_inv_series(B, A, wlen, prec);
acb_poly_mullow(u, z, z, wlen, prec);
acb_poly_scalar_mul_2exp_si(u, u, -2);
acb_poly_one(b + 1);
acb_poly_add_si(b + 0, nu, 1, prec);
acb_hypgeom_pfq_series_direct(s, NULL, 0, b, 2, u, 1, -1, wlen, prec);
acb_poly_mullow(A, A, s, wlen, prec);
acb_poly_add_si(b + 0, nu, -1, prec);
acb_poly_neg(b + 0, b + 0);
acb_hypgeom_pfq_series_direct(s, NULL, 0, b, 2, u, 1, -1, wlen, prec);
acb_poly_mullow(B, B, s, wlen, prec);
acb_poly_sub(A, B, A, prec);
acb_poly_scalar_mul_2exp_si(A, A, -1);
/* multiply by pi csc(pi nu) */
acb_poly_sin_pi_series(B, nu, wlen, prec);
if (singular)
{
acb_poly_shift_right(A, A, 1);
acb_poly_shift_right(B, B, 1);
}
acb_poly_div_series(res, A, B, len, prec);
arb_const_pi(c, prec);
_acb_vec_scalar_mul_arb(res->coeffs, res->coeffs, res->length, c, prec);
acb_poly_clear(s);
acb_poly_clear(u);
acb_poly_clear(A);
acb_poly_clear(B);
acb_poly_clear(b + 0);
acb_poly_clear(b + 1);
arb_clear(c);
}
void
acb_hypgeom_u_1f1_series(acb_poly_t res,
const acb_poly_t a, const acb_poly_t b, const acb_poly_t z,
long len, long prec)
{
acb_poly_t s, u, A, B;
acb_poly_struct aa[3];
arb_t c;
long wlen;
int singular;
acb_poly_init(s);
acb_poly_init(u);
acb_poly_init(A);
acb_poly_init(B);
acb_poly_init(aa + 0);
acb_poly_init(aa + 1);
acb_poly_init(aa + 2);
arb_init(c);
singular = (b->length == 0) || acb_is_int(b->coeffs);
wlen = len + (singular != 0);
/* A = rgamma(a-b+1) * 1F~1(a,b,z) */
acb_poly_sub(u, a, b, prec);
acb_poly_add_si(u, u, 1, prec);
acb_poly_rgamma_series(A, u, wlen, prec);
/* todo: handle a = 1 efficiently */
acb_poly_set(aa, a);
acb_poly_set(aa + 1, b);
acb_poly_one(aa + 2);
acb_hypgeom_pfq_series_direct(s, aa, 1, aa + 1, 2, z, 1, -1, wlen, prec);
acb_poly_mullow(A, A, s, wlen, prec);
/* B = rgamma(a) * 1F~1(a-b+1,2-b,z) * z^(1-b) */
acb_poly_set(aa, u);
acb_poly_add_si(aa + 1, b, -2, prec);
acb_poly_neg(aa + 1, aa + 1);
acb_hypgeom_pfq_series_direct(s, aa, 1, aa + 1, 2, z, 1, -1, wlen, prec);
acb_poly_rgamma_series(B, a, wlen, prec);
acb_poly_mullow(B, B, s, wlen, prec);
acb_poly_add_si(u, b, -1, prec);
acb_poly_neg(u, u);
acb_poly_pow_series(s, z, u, wlen, prec);
acb_poly_mullow(B, B, s, wlen, prec);
acb_poly_sub(A, A, B, prec);
/* multiply by pi csc(pi b) */
acb_poly_sin_pi_series(B, b, wlen, prec);
if (singular)
{
acb_poly_shift_right(A, A, 1);
acb_poly_shift_right(B, B, 1);
}
acb_poly_div_series(res, A, B, len, prec);
arb_const_pi(c, prec);
_acb_vec_scalar_mul_arb(res->coeffs, res->coeffs, res->length, c, prec);
acb_poly_clear(s);
acb_poly_clear(u);
acb_poly_clear(A);
acb_poly_clear(B);
acb_poly_clear(aa + 0);
acb_poly_clear(aa + 1);
acb_poly_clear(aa + 2);
arb_clear(c);
}
