Transcendentality of zeros of higher derivatives of functions involving Bessel functions (Q1895286)

It is a theorem of Siegel that for rational \(\nu\), the zeros of \(J_\nu (x)\), \(J_\nu' (x)\), \(x \neq 0\) are transcendental and that the ratio \(J_\nu' (x)/J_\nu (x)\) is transcendental if \(x\) is algebraic. Here \(J_\nu (x)\) is the usual Bessel function of the first kind and of order \(\nu\). The authors apply some surprisingly simple arguments to prove analogous results for zeros of higher derivatives of \(J_{\nu, \mu} (x) = x^{- \mu} J_\nu (x)\) with \(\mu\) algebraic and \(\nu\) rational. Indeed, suppose that \(x^n y'' + p_n (x) xy' + q_n (x)y = 0\), where \(p_n\), \(q_n\) are polynomials with algebraic coefficients. The authors point out that if for every algebraic \(x \neq 0\) one has \(y(x) \neq 0\) and \(y'(x)/y(x)\) transcendental, then a zero \(\xi \neq 0\) of \(y^{(n)} (x)\) is algebraic if and only if \(p_n (\xi) = q_n (\xi) = 0\). It follows that with \(\mu\) algebraic and \(\nu\) rational, then at most a finite number of the nonzero zeros of \(J^{(n)}_{\nu, \mu} (x)\) can be algebraic. Further ingenious calculation now provides detailed explicit results for a variety of specific cases \((\nu, \mu)\). As just one of several examples, if \(\nu\) is rational and \(\xi \neq 0\), then all zeros of \([x^{- 1/2} J_\nu (x)]^{(n)}\) are transcendental for \(n = 0,1, \dots, 18\) with the exception of the case \(n = 3\), \(\nu^2 = 25/4\), when there are zeros \(\pm \sqrt {12}\). There remain a substantial number of conjectures inviting numerical and theoretical investigation.