#2223 Parity of the Bessel functions bessel_j and bessel_i

[Dieter Kaiser]

The parity of the Bessel functions bessel_j and bessel_i is not fully implemented.

Maxima knows the rules:

bessel_j(-n, z) --> (-1)^n * bessel_j(n, z)
bessel_i(-n, z) --> (-1)^n * bessel_i(n, z)

for n an integer, but not the rules

bessel_j(n, -z) --> (-1)^n * bessel_j(n, z)
bessel_i(n, -z) --> (-1)^n * bessel_i(n, z)

for n an integer.

Maxima version: 5.24post
Maxima build date: 21:18 7/11/2011
Host type: i686-pc-linux-gnu
Lisp implementation type: SBCL
Lisp implementation version: 1.0.45

Simplification for a negative order:

(%i1) bessel_j(-1, z);
(%o1) -bessel_j(1,z)

(%i2) bessel_j(-2, z);
(%o2) bessel_j(2,z)

But no simplification for a negative argument:

(%i3) bessel_j(1, -z);
(%o3) bessel_j(1,-z)

(%i4) bessel_j(2, -z);
(%o4) bessel_j(2,-z)

Dieter Kaiser