Properties of the Generalized Airy Functions of W.H. Reid

In a paper titled Composite Solutions to the Orr-Sommerfeld Equation (Studies in Appl Math., 51, pp 361-368, 1972) W.H. Reid introduced a family of functions that he called generalized Airy functions. Reid and his co-workers demonstrated the usefulness of these functions in a number of papers and, especially, in the treatise he co-authored with P.G. Drazin (Hydrodynamic Stability, Cambridge University Press, first edition 1981, second edition 2004) an appendix to which contains the fullest exposition of the theory of these functions. Two families of Reid functions are A_k(z,p,q) and B_k(z,p,q), in which k∈{1,2,3}, z and p are complex numbers, and q is a natural number. Reid denotes the special cases A_k(z,p,0) and B_k(z,p,0) in the two-argument form A_k(z,p) and B_k(z,p). Reid's definitions include that of one more function, B₀(z,p). §9.13(ii) of the National Institute of Standards and Technology Digital Library of Mathematical Functions (DLMF) includes the two-argument forms of Reid's functions. Reid's definitions of his functions are in the form of integral representations of Laplace type. The appendix to Hydrodynamic Stability includes a great deal of additional information including asymptotic expansions, recursion relations, and values at the origin z=0. Although this exposition was satisfactory for Reid's purposes it does not include Maclaurin series expansions about z=0. The present work fills this lacuna and, in the process expresses A_k(z,p,0) and B_k(z,p,0) in terms of the G-function of C.S. Meijer.