e-mail: vrogov@cemi.rssi.ru

**The q-Bessel functions and q-Macdonald-Hahn-Exton functions**

**Abstract**

The $q$-Bessel-Jackson functions were introduced on the begining of
the centure by Jackson. The $q$-Hahn-Exton functions were introdused resently
and their properties were considered by Koelink, Ismail and others. The
definitions of the modified $q$-Bessel-Jackson functions ($q$-MBF) and
$q$-Macdonald functions ($q$-MF) and their properties were given by Olshanetskii
and Rogov. The $q$-analogs of the modified Bessel functions and Macdonald
functions are interesting because they are present in the harmonic analysis
on the quantum homogeneous spaces as in the classical case. If suppose
that the

commutative relations in the universal enveloping algebra and the commutative
relations for the generators of the quantum homogeneous space are determined
by different parameters then the eigenvalue problem for the second Casimir
operator leads us to the difference equation depending on one parameter.
This equation is the second order one for three values of

this perameter. These values of parameter correspond the $q$-Bessel-Jackson
functions of kind 1 $J_\nu^{(1)}$ and kind 2

$J_\nu^{(2)}$, and the Hahn-Exton function $J_\nu^{(3)}$. This fact
allows to consider these functions with uniform point of view. In this
work we start from the definition of the $q$-MBF as a solution of the second
order difference equation. The $q$-MBFs are connected with $q$-Bessel functions
as in the classical case. We receive the actions of the difference operators,
the recurrence relations, and the $q$-Wronskians for them. The Laurent
reries for these functions are very important. They allow to define the
$q$-MF as a linear combination of the $q$-MBFs. Moreover the $q$MFs are
holomorphic functions in the right half-plane. At last we give the representations
of the $q$-MBFs and $q$-MFs as the

Jackson $q$-integral.