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10 Bessel FunctionsModified Bessel Functions

§10.44 Sums

Contents

§10.44(i) Multiplication Theorem

10.44.1 𝒵ν(λz)=λ±νk=0(λ2-1)k(12z)kk!𝒵ν±k(z),
|λ2-1|<1.

If 𝒵=I and the upper signs are taken, then the restriction on λ is unnecessary.

Examples

10.44.2 Iν(z) =k=0zkk!Jν+k(z),
Jν(z) =k=0(-1)kzkk!Iν+k(z).

§10.44(ii) Addition Theorems

Neumann’s Addition Theorem

10.44.3 𝒵ν(u±v)=k=-(±1)k𝒵ν+k(u)Ik(v),
|v|<|u|.

The restriction |v|<|u| is unnecessary when 𝒵=I and ν is an integer.

Graf’s and Gegenbauer’s Addition Theorems

For results analogous to (10.23.7) and (10.23.8) see Watson (1944, §§11.3 and 11.41).

§10.44(iii) Neumann-Type Expansions

10.44.4 (12z)ν=k=0(-1)k(ν+2k)Γ(ν+k)k!Iν+2k(z),
ν0,-1,-2,.
10.44.5 K0(z)=-(ln(12z)+γ)I0(z)+2k=1I2k(z)k,
10.44.6 Kn(z)=n!(12z)-n2k=0n-1(-1)k(12z)kIk(z)k!(n-k)+(-1)n-1(ln(12z)-ψ(n+1))In(z)+(-1)nk=1(n+2k)In+2k(z)k(n+k),

where γ is Euler’s constant and ψ=Γ/Γ5.2).

§10.44(iv) Compendia

For collections of sums and series involving modified Bessel functions see Erdélyi et al. (1953b, §7.15), Hansen (1975), and Prudnikov et al. (1986b, pp. 691–700).