Vladimir Rogov, Moscow State University of Railway Communications (MIIT), Russia
e-mail: vrogov@cemi.rssi.ru

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


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.

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