Module spherical 

The spherical Bessel functions $j_n$, $y_n$, $i_n$, $k_n$.

j_n(x) = \sqrt{\tfrac{\pi}{2x}}\,J_{n+1/2}(x), \qquad
y_n(x) = \sqrt{\tfrac{\pi}{2x}}\,Y_{n+1/2}(x)

and likewise for the modified pair. Both references ship these publicly: Boost as sph_bessel / sph_neumann, SciPy as spherical_jn / spherical_yn / spherical_in / spherical_kn. Boost has no modified spherical pair. SciPy has no oscillating primes as separate names. This module is the union.

The sqrt never gets formed, because it would only be cancelled

Boost implements sph_bessel as literally sqrt(pi/(2x)) * cyl_bessel_j(n + 1/2, x). That is two square roots that multiply to $1/x$: the cylindrical kernel builds its answer on $\sqrt{2/\pi x}$ and the wrapper immediately multiplies by $\sqrt{\pi/2x}$.

Here the recurrence is seeded in the spherical normalization directly:

j_{-1} = \frac{\cos x}{x},\quad j_0 = \frac{\sin x}{x}, \qquad
y_{-1} = \frac{\sin x}{x},\quad y_0 = -\frac{\cos x}{x}

The scaling factor between the two conventions does not depend on the order, so the recurrence is unchanged and walk_jy is reused verbatim (seeds in one normalization, values out in the same one). Two sqrts, two divisions and a rounding disappear, and $j_0$ becomes exactly sinc, which is correct at $x = 0$ where the cylindrical route is $0 \cdot \infty$.

Boost needs a small-$z$ series below $x = 1$ for that reason. This module needs none: the downward recurrence already covers small $x$, and is the arm that runs there anyway, since $n < x$ is what selects the forward one.

Order

$n \ge 0$, matching both references (Boost takes unsigned, SciPy documents n >= 0), and spelled usize so the constraint is the type rather than an assertion. That is the one place this family deliberately diverges from the cylindrical entry points, whose i32 exists because negative orders there are meaningful. A caller who wants one here can use $j_{-n-1}(x) = (-1)^{n+1} y_n(x)$.

Negative x

Unlike their cylindrical parents at half-integer order, $j_n$, $y_n$ and $i_n$ are elementary in $\sin x$, $\cos x$, $\sinh x$, $\cosh x$ and powers of $1/x$, so they are real on the whole line and have definite parity:

j_n(-x) = (-1)^n j_n(x), \qquad y_n(-x) = (-1)^{n+1} y_n(x), \qquad i_n(-x) = (-1)^n i_n(x)

which is SciPy’s convention for spherical_jn / spherical_yn / spherical_in. The kernels evaluate on $\lvert x\rvert$ and apply the sign at the end. They must, because the downward walk’s trip count is $a + 24 + c\,x$ and a negative $x$ would shorten it to nothing. $k_n$ has no parity (it is $e^{-x}$ against $e^{x}$) and is NaN off the positive axis, as the cylindrical $K$ and SciPy’s spherical_kn are.

Functions

sph_deriv

The runtime-order twin of sph_deriv_n.

sph_deriv_n

$f_n'(x)$ from the pair the walk returns: $f_n' = \pm f_{n-1} - \frac{n+1}{x} f_n$.

sph_ik_impl

The runtime-order twin of sph_ik_impl_n.

sph_ik_impl_n

$(i_{n-1},\; i_n,\; k_{n-1},\; k_n)$, scaled by $(e^{-|x|}, e^{x})$ when SCALED. $i_n$ is folded by parity onto the whole line. $k_n$ is NaN for $x < 0$.

sph_jy_impl

The runtime-order twin of sph_jy_impl_n.

sph_jy_impl_n

$(j_{n-1},\; j_n,\; y_{n-1},\; y_n)$, on the whole real line.