PCFs are used as basic approximating functions in the theory of contour integrals with a coalescing saddle point and an algebraic singularity, and in the theory of differential equations with two coalescing turning points; see §§2.4(vi) and 2.8(vi). For examples see §§13.20(iii), 13.20(iv), 14.15(v), and 14.26.
Sleeman (1968b) considers certain orthogonality properties of the PCFs and corresponding eigenvalues. In Brazel et al. (1992) exponential asymptotics are considered in connection with an eigenvalue problem involving PCFs.