The generalized hypergeometric function

The orders $(p, q) = (0,0), (1,0), (0,1), (1,1), (2,1)$ are passed to acb_exp, the power $(1-z)^{-a}$, acb_hypgeom_0f1, acb_hypgeom_m and acb_hypgeom_2f1, which implement analytic continuation and asymptotic expansions. All other orders use direct summation of the series with acb_hypgeom_pfq_direct (with a power series variant for regularized functions whose lower parameters may be nonpositive integers), which converges for $p \le q$ and for $p = q+1$, $|z| < 1$, and gives a finite result for $p > q+1$ only when the series terminates. Parameters that occur in both $a$ and $b$ are not removed (except a single upper parameter equal to 1, which cancels the $k!$); removing them first saves time. See The generalized hypergeometric function.

acb_hypgeom_pfq computes

for complex ball parameters and argument, or the regularized function ${}_p\mathbf{F}_q = {}_pF_q/(\Gamma(b_1)\cdots\Gamma(b_q))$ if regularized is nonzero. arb_hypgeom_pfq is the real version. The function is a front end: it recognizes the orders for which this module has an implementation with analytic continuation and asymptotic expansions, and sums the series directly with a rigorous tail bound otherwise.

Dispatch#

For the first five orders, all $z$ are supported (with the branch cut $[1, \infty)$ for $p = q + 1$). For the other orders the function sums the defining series with acb_hypgeom_pfq_direct, choosing the number of terms with acb_hypgeom_pfq_choose_n and bounding the remainder with acb_hypgeom_pfq_bound_factor. This gives

• a finite result for $p \le q$ and all $z$ (the series is entire), although large $|z|$ needs many terms and a high precision to overcome the cancellation, since no asymptotic expansion is implemented for general orders;
• a finite result for $p = q + 1$ and $|z| < 1$; the closer $|z|$ is to 1, the more terms are needed, and no analytic continuation to $|z| \ge 1$ is implemented for general orders;
• a finite result for $p > q + 1$ only if the series terminates, that is, if some $a_i$ is a nonpositive integer; otherwise the series diverges and the result is indeterminate.

If an upper parameter is exactly 1, it is removed together with the $k!$; other parameters that occur among both the $a_i$ and the $b_j$ are not cancelled, which costs time, and the caller can remove them beforehand. With regularized set and a lower parameter that may be a nonpositive integer, the series is summed with power series arithmetic (acb_hypgeom_pfq_series_direct) so that the vanishing factors $1/\Gamma(b_j)$ cancel the poles.

Example#

The dilogarithm is $\operatorname{Li}_2(z) = z\, {}_3F_2(1,1,1;2,2;z)$; a ${}_3F_1$ with an upper parameter $-3$ is a polynomial even though $p > q + 1$; and ${}_1F_2(1; -2, 3; z)$ is undefined while its regularized version is not:

#include <flint/acb.h>
#include <flint/acb_hypgeom.h>

int main(void)
{
    acb_ptr a, b;
    acb_t z, r;

    a = _acb_vec_init(3);
    b = _acb_vec_init(2);
    acb_init(z);
    acb_init(r);

    /* Li_2(1/2) = (1/2) 3F2(1,1,1; 2,2; 1/2) */
    acb_set_ui(a + 0, 1); acb_set_ui(a + 1, 1); acb_set_ui(a + 2, 1);
    acb_set_ui(b + 0, 2); acb_set_ui(b + 1, 2);
    acb_set_d(z, 0.5);
    acb_hypgeom_pfq(r, a, 3, b, 2, z, 0, 64);
    acb_mul(r, r, z, 64);
    flint_printf("z 3F2(1,1,1;2,2;z) = "); acb_printn(r, 18, 0); flint_printf("\n");
    acb_hypgeom_dilog(r, z, 64);
    flint_printf("Li2(z)             = "); acb_printn(r, 18, 0); flint_printf("\n");

    /* a terminating series with p > q + 1 */
    acb_set_si(a + 0, -3); acb_set_ui(a + 1, 2); acb_set_ui(a + 2, 5);
    acb_set_ui(b + 0, 7);
    acb_set_ui(z, 4);
    acb_hypgeom_pfq(r, a, 3, b, 1, z, 0, 64);
    flint_printf("3F1(-3,2,5;7;4)    = "); acb_printn(r, 18, 0); flint_printf("\n");

    /* the same series without termination diverges */
    acb_set_d(a + 0, -2.5);
    acb_set_d(z, 0.125);
    acb_hypgeom_pfq(r, a, 3, b, 1, z, 0, 64);
    flint_printf("3F1(-2.5,2,5;7;z)  = "); acb_printn(r, 18, 0); flint_printf("\n");

    /* regularized: a lower parameter at a pole of the gamma function */
    acb_set_ui(a + 0, 1);
    acb_set_si(b + 0, -2); acb_set_ui(b + 1, 3);
    acb_set_d(z, 0.5);
    acb_hypgeom_pfq(r, a, 1, b, 2, z, 1, 64);
    flint_printf("1F2~(1;-2,3;z)     = "); acb_printn(r, 18, 0); flint_printf("\n");
    acb_hypgeom_pfq(r, a, 1, b, 2, z, 0, 64);
    flint_printf("1F2(1;-2,3;z)      = "); acb_printn(r, 18, 0); flint_printf("\n");

    _acb_vec_clear(a, 3);
    _acb_vec_clear(b, 2);
    acb_clear(z);
    acb_clear(r);
    flint_cleanup();
    return 0;
}

z 3F2(1,1,1;2,2;z) = [0.58224052646501251 +/- 5.21e-18]
Li2(z)             = [0.582240526465012506 +/- 2.06e-19]
3F1(-3,2,5;7;4)    = [-501.857142857142857 +/- 2.91e-16]
3F1(-2.5,2,5;7;z)  = nan + nan*I
1F2~(1;-2,3;z)     = [0.00113163791418884985 +/- 2.03e-21]
1F2(1;-2,3;z)      = nan + nan*I

The polynomial ${}_3F_1(-3,2,5;7;4) = -3513/7$ is summed exactly up to rounding. In the regularized ${}_1\mathbf{F}_2(1;-2,3;z)$ the terms with $k \le 2$ vanish, because $1/\Gamma(-2+k) = 0$, and the remaining terms sum to $\frac{(1)_3\, z^3}{3!\,\Gamma(6)}\,{}_1F_2(4; 6, 4; z) = \frac{z^3}{120}\,{}_0F_1(6; z)$.