Finite sum of spherical Bessel functions

Question

In L.G. Afanasyev, A.V. Tarasov, Breakup of relativistic pi+pi- atoms in matter, Phys. At. Nucl. 59 (1996) 2130 the following identity is given for the spherical Bessel functions $j_n(z)=\sqrt{\frac{\pi}{2z}}\,J_{n+1/2}(z)$: $$j_{L-2s}(z)=\sum_{p=0}^sB_{ps}\left(\frac{2}{z}\right)^pj_{L-p}(z), \tag{1}$$ with $$B_{ps}=(-1)^{s-p}\Gamma(p+1)\binom{s}{p}\binom{L-s+1/2}{p}.$$ This result is given with reference to the handbook A.P. Prudnikov, Yu.A. Brychkov, O.I. Marichev, Integrals and Series V1: Special Functions. However the formula in Prudnikov et al. looks like this $$\sum_{k=0}^n\frac{(-1)^k}{(n-k)!}\binom{m-n}{k}\left(\frac{2}{z}\right)^kJ_{m-k}(z)=\frac{(-1)^n}{n!}J_{m-2n}(z),\;\;\;[m>n], \tag{2}$$ and as I understand assumes that $m$ is an integer. To get from this result (1), one should be sure that (2) remains valid when $m=L+1/2$ is a half integer. How can this be justified rigorously?