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# Asymptotic transformations of $q$-series

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Published:1998-04-01
Printed: Apr 1998
• Richard J. McIntosh
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## Abstract

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.
 MSC Classifications: 11B65 - Binomial coefficients; factorials; $q$-identities [See also 05A10, 05A30] 33D10 - unknown classification 33D1034E05 - Asymptotic expansions 41A60 - Asymptotic approximations, asymptotic expansions (steepest descent, etc.) [See also 30E15]

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