Asymptotic transformations of $q$-series

http://dx.doi.org/10.4153/CJM-1998-022-x
Canad. J. Math. 50(1998), 412-425

  Published:1998-04-01
 Printed: Apr 1998

For the $q$-series $\sum_{n=0}^\infty a^nq^{bn^2+cn}/(q)_n$ we construct a companion $q$-series such that the asymptotic expansions of their logarithms as $q\to 1^{\scriptscriptstyle -}$ differ only in the dominant few terms. The asymptotic expansion of their quotient then has a simple closed form; this gives rise to a new $q$-hypergeometric identity. We give an asymptotic expansion of a general class of $q$-series containing some of Ramanujan's mock theta functions and Selberg's identities.