Digital Library of Mathematical Functions
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NIST
10 Bessel FunctionsSpherical Bessel Functions

§10.51 Recurrence Relations and Derivatives

Contents

§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 fn-1(z)+fn+1(z) =((2n+1)/z)fn(z),
nfn-1(z)-(n+1)fn+1(z) =(2n+1)fn(z),
n=1,2,,
10.51.2 fn(z) =fn-1(z)-((n+1)/z)fn(z),
n=1,2,,
fn(z) =-fn+1(z)+(n/z)fn(z),
n=0,1,.
10.51.3 (1zz)m(zn+1fn(z)) =zn-m+1fn-m(z),
m=0,1,,n,
(1zz)m(z-nfn(z)) =(-1)mz-n-mfn+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 gn-1(z)-gn+1(z) =((2n+1)/z)gn(z)
ngn-1(z)+(n+1)gn+1(z) =(2n+1)gn(z),
n=1,2,,
10.51.5 gn(z) =gn-1(z)-((n+1)/z)gn(z),
n=1,2,,
gn(z) =gn+1(z)+(n/z)gn(z),
n=0,1,.
10.51.6 (1zz)m(zn+1gn(z)) =zn-m+1gn-m(z),
m=0,1,,n,
(1zz)m(z-ngn(z)) =z-n-mgn+m(z),
m=0,1,.