On the zeros of derivatives of Bessel functions

Abstract

The positive zeros of J ν ‴ ( x ) and J ν ( n + 1 ) ( x ) are studied by using classical analysis and the properties of J ν ( x ) . It is proved that J ν ‴ ( x ) has a unique zero in specific intervals. Regarding J ν ( n + 1 ) ( x ) , it is proved that its positive zero j ν , m ( n + 1 ) is an increasing function with respect to ν, for ν > n . Moreover, the first two Rayleigh sums for j ν , m ( n ) are calculated. The obtained results extend and complement previously known results and also answer an open problem regarding the monotonicity of j ν , m ( n ) . As a consequence of these results, a lower bound for j ν , 1 ( n + 1 ) is deduced, as well as an inequality between j ν , 1 ( n + 1 ) and j ν , 1 ( n ) .