On the roots of the Bessel functions (Q2882829)

Using the method, proposed by \textit{Ch. Hermite} [C. R. 77, 74--79, 226--233, 285--293 (1873; JFM 05.0248.01)], the author studies the roots of the Bessel functions \(J_n(x)\) of the first kind and an integer order \(n\), i.e., of the functions that satisfy the differential equation \(x^2 J''_n(x) + x J'_n(x) - (x^2 - n^2) J_n(x) = 0\) and are given by the power series NEWLINENEWLINE\[NEWLINEJ_n(x) = \frac{x^n}{2^n}\sum_{m=0}^{\infty} NEWLINE\frac{(-1)^m}{m! (m+n)! 2^{2m}} x^{2m}.NEWLINE\]NEWLINENEWLINENEWLINE NEWLINEThe author proves that the values of the functions \(J'_n(x)/J_n(x)\) and \(J_n(x)/J'_n(x)\) are irrational numbers for every rational \(x\). As an immediate corollary, he concludes that every non-zero root of the functions \(J_n(x)\) and \(J'_n(x)\) is an irrational number.NEWLINENEWLINENEWLINEThese results are special cases of theorems on the roots of the Bessel functions NEWLINE\(J_\nu(x)\) of the first kind and a rational order \(\nu\) proved by \textit{C. L. Siegel} in NEWLINE[Abh. Akad. Berlin 1929, No. 1 (1929; JFM 56.0180.05)]. To be more precise, we recall Siegel's theorem:NEWLINENEWLINE (i) all zeros of \(J_\nu(x)\) and \(J'_\nu(x)\) are transcendental when \(\nu\) is rational, \(x\neq 0\), andNEWLINENEWLINENEWLINE(ii) \(J'_\nu(x)/J_\nu(x)\) is transcendental when \(\nu\) is rational and \(x\) algebraic. NEWLINENEWLINENEWLINE We should also mention that the transcendentality of the zeros of higher derivatives of functions \(x^\mu J_\nu(x)\) when \(\mu\) is algebraic and \(\nu\) rational was proved by \textit{L. Lorch} and \textit{M. E. Muldoon} [Int. J. Math. Math. Sci. 18, No. 3, 551--560 (1995; Zbl 0839.11029)].