Asymptotic expansion of a Bessel function integral using hypergeometric functions

Generalized hypergeometric functions are used to extend, simplify, and complete the analysis of Stoyanov and Farrel (Math. Comput. 49 (1987) 275–279) and of Wong (Math. Comput. 50 (1998) 229–234), as well as putting their considerations within a wider framework: The integral∫0π/2sina θ cosb θ Jν(λ sin θ) Jμ(λ sin θ) dθ,where Jν is the Bessel function of order ν, is analyzed in terms of generalized hypergeometric functions and a complete asymptotic expansion is given for∫0π/2Jν(λ sin θ) Jμ(λ sin θ) dθ.