On Some Formulas for Single and Double Integral Transforms Related to the Group SO(2, 2)

We present a novel proof, using group theory, for a Meijer transform formula. This proof reveals the formula as a specific case of a broader generalized result. The generalization is achieved through a linear operator that intertwines two representations of the connected component of the identity of the group SO(2,2). Using this same approach, we derive a formula for the sum of three double integral transforms, where the kernels are represented by Bessel functions. It is particularly noteworthy that the group SO(2,2) is connected to symmetry in several significant ways, especially in mathematical physics and geometry.