Function series_j_nu 

pub fn series_j_nu<P, E, V>(nu: V, x: V, needed: V::Mask) -> V
where
    E: FloatElement,
    V: FloatVector<Element = E> + SpecializedTranscendentalMath<E> + SpecializedSpecialMath<E>,
    P: Policy,

$J_\nu(x)$ by its ascending series, at arbitrary real order, for small x.

J_\nu(x) = \sum_{k\ge 0} \frac{(-1)^k}{k!\,\Gamma(\nu+k+1)}\left(\frac{x}{2}\right)^{\nu+2k}

advanced by the ratio $t_{k+1}/t_k = -\frac{(x/2)^2}{(k+1)(\nu+k+1)}$, so the only per-order quantity is the seed. No table at any order, like the rest of this module.

Domain: roughly $x \lesssim 6$, and the limit is cancellation not convergence

The series converges everywhere, and Boost’s own comment says so (“this series will actually converge rapidly for all small x - say up to x < 20”) before adding “but the first few terms are large and divergent which leads to large errors :-(”. That is the real bound.

The terms peak near $k \approx x$ at a magnitude around $e^{x}/(\pi x)$ while the answer is $O(x^{-1/2})$, so summing them loses about $1.4427x - \log_2\sqrt{2\pi x}$ bits. Measured envelope-relative at $\nu = 1/3$ in binary64: 0.00 eps at x = 8, 158 at x = 10, 6.1e3 at x = 12. So it is usable to about 8 and comfortable to 6, and the (6, 16) hole between here and [hankel_j_nu] is a real gap that neither arm covers (notes/special/tools/model_fractional_arms.py).

Nothing here detects that. The caller gates on x.

The seed sets the accuracy floor

Every term is proportional to $t_0 = (x/2)^\nu / \Gamma(\nu+1)$, so the seed’s relative error passes straight through to the result and nothing later can recover it. That is one powf and one tgamma, so the floor is roughly their combined error (about 2 ulp), and no amount of extra terms improves it.

$\Gamma(\nu+1)$ has poles at negative integer $\nu$, where the seed becomes zero rather than infinite. Negative integer orders never arrive here: they reflect through $J_{-n} = (-1)^n J_n$ before any kernel sees them.