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10 Bessel FunctionsSpherical Bessel Functions

§10.51 Recurrence Relations and Derivatives

Contents
  1. §10.51(i) Unmodified Functions
  2. §10.51(ii) Modified Functions

§10.51(i) Unmodified Functions

Let fn(z) denote any of jn(z), yn(z), hn(1)(z), or hn(2)(z). Then

10.51.1 fn1(z)+fn+1(z) =((2n+1)/z)fn(z),
nfn1(z)(n+1)fn+1(z) =(2n+1)fn(z),
n=1,2,,
10.51.2 fn(z) =fn1(z)((n+1)/z)fn(z),
n=1,2,,
fn(z) =fn+1(z)+(n/z)fn(z),
n=0,1,.
10.51.3 (1zddz)m(zn+1fn(z)) =znm+1fnm(z),
m=0,1,,n,
(1zddz)m(znfn(z)) =(1)mznmfn+m(z),
m=0,1,.

§10.51(ii) Modified Functions

Let gn(z) denote in(1)(z), in(2)(z), or (1)n kn(z). Then

10.51.4 gn1(z)gn+1(z) =((2n+1)/z)gn(z)
ngn1(z)+(n+1)gn+1(z) =(2n+1)gn(z),
n=1,2,,
10.51.5 gn(z) =gn1(z)((n+1)/z)gn(z),
n=1,2,,
gn(z) =gn+1(z)+(n/z)gn(z),
n=0,1,.
10.51.6 (1zddz)m(zn+1gn(z)) =znm+1gnm(z),
m=0,1,,n,
(1zddz)m(zngn(z)) =znmgn+m(z),
m=0,1,.