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We define a q-analog of the modified Bessel and Bessel-Macdonald functions. As for the q-Bessel functions of Jackson there is a couple of functions of the both kind. They are arisen in the Harmonic analysis on quantum symmetric spaces similarly to their classical counterpart. Their definition is based on the power expansions. We derive the recurrence relations, difference equations, q-Wronskians, and an analog of asymptotic expansions which turns out is exact in some domain if q = 1. Some relations for the basic hypergeometric function which follow from this fact are discussed.1 is essentially noncommutative. The further progress in the Harmonic analysis demands to take into account the noncommutativity of the group algebra. It concerns, in particular, with the Poisson kernel for the Dirichlet problem in the quantum Lobachevsky space. In the classical case the Whittaker functions or, what is the same, BMF are the Fourier transform of the Poisson kernel. To reproduce this construction in the quantum case it is necessary to derive the most fundamental properties of q-MBF.As in the classical case we begin with definition of q-MBF as the power expansions. There are two q-MBF and they are related to q-Bessel functions of Jackson [1] as the classical ones. We derive the action of difference operators, recurrence relation, difference equation and q-Wronskian for them. The most of these results can be easily derived from [2,3], where q-Bessel functions were investigated. While the definition of q-MBF, based on the Jackson q-Bessel functions is strightforward, the definition of q-BMF is a rather subtle. We choose it in a such way that it becomes a holomorphic in the complex right half plane. We repeat the same program for q-BMF as for q-MBF. The most important part for the applications is the Laurent type expansion which is only asymptotics in the classical case and is represented by an convergent series in some domain in the quantum situation. Using this property we obtain in conclusion some relations for the basic hypergeometric function.

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