Re: Thoughts on Bessel functions


david petry wrote:

Anyway, your equation

x y'' + (n+1) y' = y

quickly leads to an interesting continued fraction:

y/y' = (n+1) + x/(y'/y'') = (n+1) + x/( (n+2) + x/( (n+3) + x/... ))

when y is an entire function.

Thanks for pointing that out; I knew there were interesting continued
fractions involving quotients of Bessel functions, but hadn't seen a
derivation. This proves I_n(2)/I_(n+1)(2)
has a cute continued fraction very nicely, and shows it can also be
seen as
B_n(1)/B_n'(1).

.