Monotonicity of the zeros of the third derivative of Bessel functions (Q1898066)

The aim of the authors is to show that the zeros of the third derivative of the Bessel function \(J_\nu (\lambda)\) are increasing functions of the order \(\nu\). To this end they consider the boundary-value problem satisfied by \(y(x) = J_\nu (\lambda x)\), namely: \[ (xy')' = (\nu^2/x) y - \lambda^2 xy, \quad x \in (0,1) \qquad y(0) = 0, \quad y'''(1) = 0, \] which permits them to show that \(d\lambda/d \nu > 0\) when \(\nu > 0\), \(\lambda \geq \sqrt 3\) and \(\lambda \neq j'''_{11}\). Here \(j'''_{11}\) stands for the first zero of \(j'''_{11}(\lambda)\). Notice that all the results obtained in the paper concern the case where \(\nu\) is real (and positive).