  This paper investigates a generalized interlacing property between Bessel
functions, particularly $J_\nu$ and $J_\mu$, where the difference $|\nu-\mu|$
exceeds $2$. This interlacing phenomenon is marked by a compensatory
interaction with the zeros of Lommel polynomials, extending our understanding
beyond the traditional $|\nu-\mu| \le 2$ regime. The paper extends the
generalized interlacing property to cylinder functions and derivative of Bessel
functions, as an application. It is also discussed that Siegel's extension of
Bourget hypothesis to rational numbers of $\nu$ cannot be further improved to
arbitrary real numbers.
