§12.16 Mathematical Applications
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.
PCFs are also used in integral transforms with respect to the parameter, and
inversion formulas exist for kernels containing PCFs. See
Erdélyi (1941a),
Cherry (1948),
and Lowdon (1970).
Integral transforms
and sampling expansions are considered in Jerri (1982).
