where Aℓ+1ℓ=1, Aℓ+2ℓ=η/(ℓ+1), and
or in terms of the hypergeometric function (§§15.1, 15.2(i)),
Reported 2018-05-15 by Ian Thompson
where a=1+ℓ±iη and ψ(x)=Γ′(x)/Γ(x) (§5.2(i)).
The series (33.6.1), (33.6.2), and (33.6.5) converge for all finite values of ρ. Corresponding expansions for Hℓ±′(η,ρ) can be obtained by combining (33.6.5) with (33.4.3) or (33.4.4).