Some formulas for Legendre functions related to the Poisson transform and Lorentz group representation

We consider the representation of Lorentz group in the spaces of homogeneous functions on cone and hyperboloid. Using the Poisson transform, which maps a function from the first space into the corresponding function from the second space, we derive some formulas containing Legendre functions. Using the restriction of the representation in the first space on a one-parameter subgroup of hyperbolic rotations, we obtain another formula for Legendre functions. All obtained formulas are new.