FTP, identify these special functions named for the German astronomer who discovered them and symbolized J and Y.
A derivative identity for them can be obtained in terms of Chebyshev polynomials, and their generating function is "exponential of x over 2 times quantity t minus one over t."  For small values of the argument, the ordinary one may be approximated as "x to the n over quantity 2 to the n times n factorial" and the nth modified one is related to the nth regular one by a factor of i to the n with multiplication of the argument by i.  Hankel functions are complex linear combination of them, and they arise as solutions of Laplace's equation in cylindrical coordinates, so they're also known as cylinder functions.  FTP, identify these special functions named for the German astronomer who discovered them and symbolized J and Y.
A derivative identity for them can be obtained in terms of Chebyshev polynomials, and their generating function is "exponential of x over 2 times quantity t minus one over t."  
Bessel functions