Digital Library of Mathematical Functions
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10 Bessel FunctionsModified Bessel Functions

§10.29 Recurrence Relations and Derivatives

Contents

§10.29(i) Recurrence Relations

With 𝒵ν(z) defined as in §10.25(ii),

10.29.1 𝒵ν-1(z)-𝒵ν+1(z) =(2ν/z)𝒵ν(z),
𝒵ν-1(z)+𝒵ν+1(z) =2𝒵ν(z).
10.29.2 𝒵ν(z) =𝒵ν-1(z)-(ν/z)𝒵ν(z),
𝒵ν(z) =𝒵ν+1(z)+(ν/z)𝒵ν(z).
10.29.3 I0(z) =I1(z),
K0(z) =-K1(z).

§10.29(ii) Derivatives

For k=0,1,2,,

10.29.4 (1zz)k(zν𝒵ν(z)) =zν-k𝒵ν-k(z),
(1zz)k(z-ν𝒵ν(z)) =z-ν-k𝒵ν+k(z).
10.29.5 𝒵ν(k)(z)=12k(𝒵ν-k(z)+(k1)𝒵ν-k+2(z)+(k2)𝒵ν-k+4(z)++𝒵ν+k(z)).