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10 Bessel FunctionsModified Bessel Functions

§10.29 Recurrence Relations and Derivatives

  1. §10.29(i) Recurrence Relations
  2. §10.29(ii) Derivatives

§10.29(i) Recurrence Relations

With 𝒵ν(z) defined as in §10.25(ii),

10.29.1 𝒵ν1(z)𝒵ν+1(z) =(2ν/z)𝒵ν(z),
𝒵ν1(z)+𝒵ν+1(z) =2𝒵ν(z).
10.29.2 𝒵ν(z) =𝒵ν1(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).

For results on modified quotients of the form z𝒵ν±1(z)/𝒵ν(z) see Onoe (1955) and Onoe (1956).

§10.29(ii) Derivatives

For k=0,1,2,,

10.29.4 (1zddz)k(zν𝒵ν(z)) =zνk𝒵νk(z),
(1zddz)k(zν𝒵ν(z)) =zνk𝒵ν+k(z).
10.29.5 𝒵ν(k)(z)=12k(𝒵νk(z)+(k1)𝒵νk+2(z)+(k2)𝒵νk+4(z)++𝒵ν+k(z)).