Derivation of Hankel function

Dear forum members,

I am encountering the problem related to finding derivations of Hankel functions presented below and would you please suggest me any idea for the problem if possible?

I would like to find derivations of H₀⁽²⁾(k₀.r) respect to k₀ and I found a recursive relation to find its derivatives can be found in the attachment.

And then I will calculate H₀⁽²⁾(k₁.r) by using Taylor expansion:

H₀⁽²⁾(k₁.r) = ( H₀⁽²⁾(k₀.r))^[0] + (k₁-k₀). ( H₀⁽²⁾(k₀.r))^[1]/1! + (k₁-k₀)². ( H₀⁽²⁾(k₀.r))^[2]/2!+ ....

However, the result only converge with small value of r (0.3 - 1.2) and it will diverge if r is large (100-1000)

k₀=21

k₁=27

Could you please propose me any idea for finding derivatives of Hankel function for large r in order to make the result converge?

Thank you very much

derivatives.doc