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In this paper, the expansion of a basic confluent hypergeometric function in terms of an incomplete  q -gamma function is investigated and shown to be an asymptotic expansion for a basic confluent hypergeometric function and a  q -analogue of the Hadamard expansion. The general expansion associated with a basic hypergeometric function is introduced. In special cases, this provides alternative derivations of the  q -Hadamard expansion. Expansions and asymptotics for modified  q -Bessel functions are derived as special cases from the general expansion.

Expansions and asymptotics associated with basic hypergeometric functions and modified q -Bessel functions