spherical Bessel functions (Subscript[j, l])(z)

(Subscript[j, 0])(z)

PowerExpand[(Sqrt[(π/(2  z))]) BesselJ[(1/2), z]]

(Subscript[j, 1])(z)

PowerExpand[(Sqrt[(π/(2  z))]) BesselJ[(3/2), z]]

Series[%, {z, 0, 2}]

(Subscript[j, 2])(z)

PowerExpand[(Sqrt[(π/(2  z))]) BesselJ[(5/2), z]]

Series[%, {z, 0, 3}]

(Subscript[j, 3])(z)

PowerExpand[(Sqrt[(π/(2  z))]) BesselJ[(7/2), z]]

Series[%, {z, 0, 4}]

Around the origin, they indeed behave as (Subscript[j, l])(z)∝(z^l).

Asymptotically, they behave as (1/z) sin z, (1/z) (-cos z), (1/z) (-sin z), (1/z) cos z, respectively, consistent with the known behavior (1/z) sin(z - ((l π)/2)).

spherical Bessel functions (Subscript[n, l])(z)

(Subscript[n, 0])(z)

PowerExpand[(Sqrt[(π/(2  z))]) BesselY[(1/2), z]]

(Subscript[n, 1])(z)

PowerExpand[(Sqrt[(π/(2  z))]) BesselY[(3/2), z]]

Series[%, {z, 0, -1}]

(Subscript[n, 2])(z)

PowerExpand[(Sqrt[(π/(2  z))]) BesselY[(5/2), z]]

Series[%, {z, 0, -2}]

(Subscript[n, 3])(z)

PowerExpand[(Sqrt[(π/(2  z))]) BesselY[(7/2), z]]

Series[%, {z, 0, -3}]

Around the origin, they indeed behave as (Subscript[n, l])(z)∝(z^(-l - 1)).

Asymptotically, they behave as (1/z) (-cos z), (1/z) (-sin z), (1/z) cos z, (1/z) sin z, respectively, consistent with the known behavior -(1/z) cos(z - ((l π)/2)).