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

Summary.

There are two ways of deriving the asymptotic expansion of \(J_\nu(\nu a)\), as \(\nu \to \infty\), which holds uniformly for \(a\geq 0\). One way starts with the Bessel equation and makes use of the turning point theory for second-order differential equations, and the other is based on a contour integral representation and applies the theory of two coalescing saddle points. In this paper, we shall derive the same result by using the three term recurrence relation \(J_{\nu+1}(x)+J_{\nu-1}(x)=(2\nu /x)J_\nu(x)\). Our approach will lead to a satisfactory development of a turning point theory for second-order linear difference equations.