Uniform asymptotic expansion of \(J_\nu(\nu a)\) via a difference equation (Q1597701)

The authors' purpose is to obtain an analytical representation of the Bessel functions \({J}_{\nu } (\nu a)\), when \(v \to \infty\), which holds uniformly for all \(a \geq 0\). The three term recurrence relation for the Bessel function \({J}_{\nu +1} (x) + {J}_{\nu -1} (x) = (2\nu /x){J}_{0} (x)\) is used to obtain the result.