In this paper we establish the spectral decomposition of the Berezin transforms on the space Mat(2, ℂ) of complex 2 × 2 matrices related to the two-sided action of U(2) × U(2). The eigenspaces are described explicitly by means of the matrix elements of a certain representation of GL(2, ℂ) and each eigenvalue is expressed as a finite sum involving the Meijer G-functions evaluated at 1 and the Hahn polynomials.
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