Bessel function
Transcendence

Siegel proved in 1929 that when ν is rational, all nonzero roots of J (x) and J ' (x) are transcendental, as are all the roots of K (x). It is also known that all roots of the higher derivatives J ν (n) (x) {\displaystyle J_{\nu }^{(n)}(x)} for n ≤ 18 are transcendental, except for the special values J 1 (3) (± 3) = 0 {\displaystyle J_{1}^{(3)}(\pm {\sqrt {3}})=0} and J 0 (4) (± 3) = 0 {\displaystyle J_{0}^{(4)}(\pm {\sqrt {3}})=0}.
combination:two_responses