I want to verify the procedure of finding . . with p =-ve. This is usually found in Bessel's equation. It is well talked about if p=+ve. But books I have don't even talking about in general how to find the series representation when p=-ve. I worked this out and I want to verify with you guys whether I am correct for some simple numbers of p= -1/2, -3/2,-5/2.

I use as the bases to expand the equations below.

I start using this identity of p=1/2

For p=-1/2, I use the result from above and add on to it:

For p=-3/2

For p=-5/2

Please tell me whether I did this correct or not.