what is the difference between first- and second-type bessel functions?
The difference is in the characteristics of their behavior - eg. (and simplistically) the 1st kind are finite at the origin while the 2nd kind have a singularity there for example. So much for the differences - but you also want to know how come the numbering - well, considering these two classes, which one would you like to consider first?
It's a convention. Since the Bessel equation is 2nd order there have to be two independent solutions. I don't think there is anything especially profound about the order.
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