The BESSELJ function ( section 6.16.13 in ODF1.3 draft) has the Constraints "N >= 0, INT(N)=N; Evaluators may evaluate expressions where N >= 0 returns a non-error value."

But the order N need not be non-negative, there exists reflection formulae. And the order N need not be integer, fractional order, especially half-integer is used too.

References:

http://functions.wolfram.com/webMathematica/FunctionPlotting.jsp?name=BesselJ&ptype=3&n=1&z=1&threeDtype2=0&part2=0&free1=n&free2=z&Min1=-5&Max1=5&Min2=-5&Max2=5

https://en.wikipedia.org/wiki/Bessel_function

https://www.boost.org/doc/libs/1_48_0/libs/math/doc/sf_and_dist/html/math_toolkit/special/bessel/bessel.html

https://stat.ethz.ch/R-manual/R-devel/library/base/html/Bessel.html