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30 Spheroidal Wave FunctionsProperties

§30.5 Functions of the Second Kind

Other solutions of (30.2.1) with μ=m, λ=λnm(γ2), and z=x are

30.5.1 Qsnm(x,γ2),
n=m,m+1,m+2,.

They satisfy

30.5.2 Qsnm(-x,γ2)=(-1)n-m+1Qsnm(x,γ2),

and

30.5.3 Qsnm(x,0)=Qnm(x);

compare §14.3(i). Also,

30.5.4 𝒲{Psnm(x,γ2),Qsnm(x,γ2)}=(n+m)!(1-x2)(n-m)!Anm(γ2)An-m(γ2)(0),

with An±m(γ2) as in (30.11.4).

For further properties see Meixner and Schäfke (1954) and §30.8(ii).