WKB methods for difference equations I

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

“…Much earlier (in 1967), Dingle and Morgan [4,5] considered second order difference equations of the form…”

Asymptotic expansions for second-order linear difference equations with a turning point

“…(It is interesting to note that in most of the classical cases, we have α 1 = β 1 = 0 and hence β 1 = 0. Also, in [6] Dingle and Morgan have assumed that α 2s+1 = β 2s+1 = 0 for s = 0, 1, 2, · · · .) Thus we modify so that is a C ∞ -function in [0, ∞) and given by (4.28) only when t ≥ δ, 0 < δ < t + .…”

“…Despite the resemblance between equation (1.2) and (1.12), one can see, after a careful comparison, that they are not of the same type. There are two older papers by Dingle and Morgan [6,7] on WKB approximations for second-order linear difference equations, where they have briefly discussed the connection formulas for linking exponential and trigonometrical regions and the behavior near turning points. However, their argument is too sketchy and non-rigorous.…”

“…However, their argument is too sketchy and non-rigorous. For more comments on references [5] and [6,7], we refer to our previous paper [14].…”

Linear difference equations with transition points

“…It is interesting to note that in a paper on WKB methods for difference equations, Dingle and Morgan [8,9] also assumed this condition. In fact, they assumed the stronger condition that all coefficients α s and β s in (1.4) with odd indices vanish.…”