I.—Induction Proofs of the Relations between certain Asymptotic Expansions and Corresponding Generalised Hypergeometric Series.

The subject of this paper was studied by Orr (Camb. Phil. Trans., vol. xvii, 1898, pp. 171–199; 1899, pp. 283–290), and later by Barnes (Proc. Lond. Math. Soc., ser. 2, vol. v, 1906, pp. 59–116), in whose paper a number of references to earlier work on the subject are given. Formulæ equivalent to (9), (18), and (20) below were given by, Barnes, who derived them by his well‐known method of integrating products and quotients of Gamma Functions. In this paper the formulæ are deduced by induction from simpler formulæ which are established in section 2. In order to simplify the notation, four functions, denoted by P, E, Q, and H, are introduced, the first two in section 3, the last two in section 5. The P and Q functions are merely generalised hypergeometric functions multiplied by convenient factors. The E function, which is equivalent to Barnes's contour integral, is defined as a multiple integral, and from it the asymptotic expansion, with a useful form for the remainder, is easily derived. The H function is a multiple of the E function.