Collective interlacing and ranges of the positive zeros of Bessel functions (Q2661293)

The authors introduce the notion of collective interlacing of zeros of Bessel functions, thus generalizing the usual notion of interlacing in a natural way, and investigate whether the positive zeros of a family of Bessel functions $j_v,Y_v$ have such property, where the argument and order will be restricted to real numbers. It turns out that the order variation map, develops jump discontinuities across negative integers and behaves like a saw-tooth function on the interval $(-\infty,-1)$. Likewise, the other map develops jump discontinuities of orders $-n+\frac{1}{2}$, $n=1,2,\dots$ and behaves like a saw-tooth on $(-\infty,-\frac{1}{2})$. The aim is to specify those jump discontinuities in detail. As an application the paper deduces a set of collective interlacing properties for the positive zeros of families of Bessel functions. The paper studies Bessel function of the first kind, using a proved theorem and discussing different cases. All results are contained in several theorems, lemmas, remarks and figures, proofs are given in detail clear pattern. Likewise Bessel functions of the second kind were studied with results obtained using the same process as the first kind. The scientific arguments finish with a theorem stating that the positive zeros of two families of Bessel functions are collectively interlaced, referring to a graphical illustration for the case $n=1$. The paper ends with acknowledgments and a list of references.