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);
}
int main()
{
slong iter;
flint_rand_t state;
flint_printf("hardy_z_series....");
fflush(stdout);
flint_randinit(state);
for (iter = 0; iter < 200 * arb_test_multiplier(); iter++)
{
slong m, n1, n2, bits1, bits2, bits3;
acb_poly_t S, A, B, C;
dirichlet_group_t G;
dirichlet_char_t chi;
ulong q;
bits1 = 2 + n_randint(state, 200);
bits2 = 2 + n_randint(state, 200);
bits3 = 2 + n_randint(state, 200);
m = 1 + n_randint(state, 8);
n1 = 1 + n_randint(state, 8);
n2 = 1 + n_randint(state, 8);
do {
q = 1 + n_randint(state, 15);
} while (q % 4 == 2);
dirichlet_group_init(G, q);
dirichlet_char_init(chi, G);
do {
dirichlet_char_index(chi, G, n_randint(state, G->phi_q));
} while (!dirichlet_char_is_primitive(G, chi));
acb_poly_init(S);
acb_poly_init(A);
acb_poly_init(B);
acb_poly_init(C);
acb_poly_randtest(S, state, m, bits1, 3);
acb_dirichlet_hardy_z_series(A, S, G, chi, n1, bits2);
acb_poly_set(B, S); /* aliasing */
if (q == 1 && n_randint(state, 2))
acb_poly_neg(B, B);
acb_dirichlet_hardy_z_series(B, B, G, chi, n2, bits3);
acb_poly_set(C, A);
acb_poly_truncate(C, FLINT_MIN(n1, n2));
acb_poly_truncate(B, FLINT_MIN(n1, n2));
if (!acb_poly_overlaps(B, C))
{
flint_printf("FAIL\n\n");
flint_printf("S = "); acb_poly_printd(S, 15); flint_printf("\n\n");
flint_printf("A = "); acb_poly_printd(A, 15); flint_printf("\n\n");
flint_printf("B = "); acb_poly_printd(B, 15); flint_printf("\n\n");
abort();
}
acb_poly_clear(S);
acb_poly_clear(A);
acb_poly_clear(B);
acb_poly_clear(C);
dirichlet_char_clear(chi);
dirichlet_group_clear(G);
}
flint_randclear(state);
flint_cleanup();
flint_printf("PASS\n");
return EXIT_SUCCESS;
}
int main()
{
slong iter;
flint_rand_t state;
flint_printf("gamma_upper_series....");
fflush(stdout);
flint_randinit(state);
for (iter = 0; iter < 1000 * arb_test_multiplier(); iter++)
{
slong m, n1, n2, bits1, bits2, bits3;
acb_poly_t S, A, B, C;
acb_t s, t, c;
int regularized;
regularized = n_randint(state, 3);
bits1 = 2 + n_randint(state, 200);
bits2 = 2 + n_randint(state, 200);
bits3 = 2 + n_randint(state, 200);
m = 1 + n_randint(state, 10);
n1 = 1 + n_randint(state, 10);
n2 = 1 + n_randint(state, 10);
acb_poly_init(S);
acb_poly_init(A);
acb_poly_init(B);
acb_poly_init(C);
acb_init(s);
acb_init(t);
acb_init(c);
acb_poly_randtest(S, state, m, bits1, 3);
acb_poly_randtest(A, state, m, bits1, 3);
acb_poly_randtest(B, state, m, bits1, 3);
acb_randtest(s, state, bits1, 3);
acb_hypgeom_gamma_upper_series(A, s, S, regularized, n1, bits2);
acb_hypgeom_gamma_upper_series(B, s, S, regularized, n2, bits3);
acb_poly_set(C, A);
acb_poly_truncate(C, FLINT_MIN(n1, n2));
acb_poly_truncate(B, FLINT_MIN(n1, n2));
if (!acb_poly_overlaps(B, C))
{
flint_printf("FAIL (consistency)\n\n");
flint_printf("regularized = %d\n\n", regularized);
flint_printf("S = "); acb_poly_printd(S, 15); flint_printf("\n\n");
flint_printf("A = "); acb_poly_printd(A, 15); flint_printf("\n\n");
flint_printf("B = "); acb_poly_printd(B, 15); flint_printf("\n\n");
flint_printf("C = "); acb_poly_printd(C, 15); flint_printf("\n\n");
flint_abort();
}
/* f(h(x)) = -exp(-h(x)) h(x)^(s-1) */
acb_poly_neg(C, S);
acb_poly_exp_series(C, C, n1, bits2);
acb_sub_ui(t, s, 1, bits2);
acb_poly_pow_acb_series(B, S, t, n1, bits2);
acb_poly_mullow(C, C, B, n1, bits2);
acb_poly_neg(C, C);
if (regularized == 0)
{
/* integral(f(h(x)) h'(x))' = f(h(x)) h'(x) */
acb_poly_derivative(B, S, bits2);
acb_poly_mullow(C, C, B, n1, bits2);
acb_poly_truncate(C, n1 - 1);
}
else if (regularized == 1)
{
/* (integral(f(h(x)) h'(x)) / c)' = (f(h(x)) h'(x)) / c */
acb_poly_derivative(B, S, bits2);
acb_poly_mullow(C, C, B, n1, bits2);
acb_gamma(c, s, bits2);
acb_poly_scalar_div(C, C, c, bits2);
acb_poly_truncate(C, n1 - 1);
}
else if (regularized == 2)
{
/* (h(x)^-s integral(f(h(x)) h'(x)))' =
* h(x)^-(s+1) (h(x) f(h(x)) - s integral(f(h(x)) h'(x))) h'(x) */
acb_poly_t D;
acb_poly_init(D);
acb_poly_derivative(B, S, bits2);
acb_poly_mullow(D, C, B, n1, bits2);
acb_poly_integral(D, D, bits2);
acb_poly_scalar_mul(D, D, s, bits2);
acb_poly_mullow(C, C, S, n1, bits2);
acb_poly_sub(D, C, D, bits2);
acb_add_ui(t, s, 1, bits2);
acb_neg(t, t);
acb_poly_pow_acb_series(B, S, t, n1, bits2);
acb_poly_mullow(C, D, B, n1, bits2);
acb_poly_derivative(B, S, bits2);
acb_poly_mullow(C, C, B, n1, bits2);
acb_poly_truncate(C, n1 - 1);
acb_poly_clear(D);
}
acb_poly_derivative(B, A, bits2);
if (!acb_poly_overlaps(B, C))
{
flint_printf("FAIL (derivative)\n\n");
flint_printf("regularized = %d\n\n", regularized);
flint_printf("S = "); acb_poly_printd(S, 15); flint_printf("\n\n");
flint_printf("A = "); acb_poly_printd(A, 15); flint_printf("\n\n");
flint_printf("B = "); acb_poly_printd(B, 15); flint_printf("\n\n");
flint_printf("C = "); acb_poly_printd(C, 15); flint_printf("\n\n");
flint_abort();
}
acb_hypgeom_gamma_upper_series(S, s, S, regularized, n1, bits2);
if (!acb_poly_overlaps(A, S))
{
flint_printf("FAIL (aliasing)\n\n");
flint_printf("regularized = %d\n\n", regularized);
flint_printf("S = "); acb_poly_printd(S, 15); flint_printf("\n\n");
flint_printf("A = "); acb_poly_printd(A, 15); flint_printf("\n\n");
flint_abort();
}
acb_poly_clear(S);
acb_poly_clear(A);
acb_poly_clear(B);
acb_poly_clear(C);
acb_clear(s);
acb_clear(t);
acb_clear(c);
}
flint_randclear(state);
flint_cleanup();
flint_printf("PASS\n");
return EXIT_SUCCESS;
}
int main()
{
slong iter;
flint_rand_t state;
flint_printf("erfi_series....");
fflush(stdout);
flint_randinit(state);
for (iter = 0; iter < 200 * arb_test_multiplier(); iter++)
{
slong m, n1, n2, bits1, bits2, bits3;
acb_poly_t S, A, B, C;
acb_t I;
bits1 = 2 + n_randint(state, 200);
bits2 = 2 + n_randint(state, 200);
bits3 = 2 + n_randint(state, 200);
m = 1 + n_randint(state, 10);
n1 = 1 + n_randint(state, 10);
n2 = 1 + n_randint(state, 10);
acb_poly_init(S);
acb_poly_init(A);
acb_poly_init(B);
acb_poly_init(C);
acb_init(I);
acb_poly_randtest(S, state, m, bits1, 3);
acb_poly_randtest(A, state, m, bits1, 3);
acb_poly_randtest(B, state, m, bits1, 3);
acb_hypgeom_erfi_series(A, S, n1, bits2);
acb_onei(I);
acb_poly_set_acb(C, I);
acb_poly_mul(B, S, C, bits3);
acb_hypgeom_erf_series(B, B, n2, bits3);
acb_poly_neg(C, C);
acb_poly_mul(B, B, C, bits3);
acb_poly_set(C, A);
acb_poly_truncate(C, FLINT_MIN(n1, n2));
acb_poly_truncate(B, FLINT_MIN(n1, n2));
if (!acb_poly_overlaps(B, C))
{
flint_printf("FAIL\n\n");
flint_printf("S = "); acb_poly_printd(S, 15); flint_printf("\n\n");
flint_printf("A = "); acb_poly_printd(A, 15); flint_printf("\n\n");
flint_printf("B = "); acb_poly_printd(B, 15); flint_printf("\n\n");
abort();
}
acb_hypgeom_erfi_series(S, S, n1, bits2);
if (!acb_poly_overlaps(A, S))
{
flint_printf("FAIL (aliasing)\n\n");
abort();
}
acb_poly_clear(S);
acb_poly_clear(A);
acb_poly_clear(B);
acb_poly_clear(C);
acb_clear(I);
}
flint_randclear(state);
flint_cleanup();
flint_printf("PASS\n");
return EXIT_SUCCESS;
}
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_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);
}
