Asymptotic method for Dougall's bilateral hypergeometric sums

By means of the modified Abel lemma on summation by parts, a recurrence relation for Dougall's bilateral -series is established with an extra natural number parameter m. Then the steepest descent method allows us to compute the limit for m→∞, which leads us surprisingly to a completely new proof of the celebrated bilateral -series identity due to Dougall (1907). The same approach applies also to the bilateral very well-poised -series identity [J. Dougall, On Vandermonde's theorem and some more general expansions, Proc. Edinburgh Math. Soc. 25 (1907) 114–132].