Modified $q$-Bessel functions and $q$-Macdonald functions

The $q$-analogues of modified Bessel functions and Macdonald functions are defined in this paper. As in the case of $q$-Bessel functions introduced by Jackson, there are two kinds of these functions. Like their classical prototypes, they arise in harmonic analysis on quantum symmetric spaces. The definition is based on the power series expansion. Recurrence relations, difference equations, and $q$-Wronskians are obtained, as well as analogues of asymptotic expansions, which are convergent series if $q\ne 1$. Some relations for basic hypergeometric series, which result from this, are given.