Computing the matrix elements of the linear operator, which transforms the spherical basis of SO(3, 1)-representation space into the hyperbolic basis, we present an integral formula involving the product of two Legendre functions of the first kind expressed in terms of 4 F 3 -hypergeometric function. Also, using the general Mehler-Fock transform, we obtain another integral formula for the Legendre function of the first kind. A relevant connection of one of the results presented here with a known integral formula is also pointed out.

The authors derive the matrix elements of the linear operators which appear under the representation of the group SO(2, 1) and correspond to some diagonal or block-diagonal matrices belonging to the above group. Then, by applying these matrix elements, that is, from a group theoretical point of view, the authors show how certain interesting integral and series representations of the Whittaker function of the second kind and some formulas for the (basic and modified) Bessel functions can be obtained. A special case of one of the results presented here is indicated to be also a special one of a known formula.

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