On changing between bases of the space of representations of group $SO(2,2)$

Three bases of the space of representations are considered. Matrix elements of the transformation operators are found for each pair of these bases: for the first pair, they are represented in terms of the Gauss hypergeometric function, for the second pair by the product of Whittaker functions, and for the third pair in terms of the Bessel functions, Bessel–Clifford functions, or Meijer G-function. Using various approaches (Cartan decomposition, intertwining operator, or subrepresentation), formulas for special functions based on the matrix elements are derived.