Digital Library of Mathematical Functions
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10 Bessel FunctionsSpherical Bessel Functions

§10.53 Power Series

10.53.1 jn(z)=znk=0(-12z2)kk!(2n+2k+1)!!,
10.53.2 yn(z)=-1zn+1k=0n(2n-2k-1)!!(12z2)kk!+(-1)n+1zn+1k=n+1(-12z2)kk!(2k-2n-1)!!.
10.53.3 in(1)(z) =znk=0(12z2)kk!(2n+2k+1)!!,
10.53.4 in(2)(z) =(-1)nzn+1k=0n(2n-2k-1)!!(-12z2)kk!+1zn+1k=n+1(12z2)kk!(2k-2n-1)!!.

For hn(1)(z) and hn(2)(z) combine (10.47.10), (10.53.1), and (10.53.2). For kn(z) combine (10.47.11), (10.53.3), and (10.53.4).