Certain relations between Bessel and Whittaker functions related to some diagonal and block-diagonal 3\times3-matrices

Abstract

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.

Integral and series representations of special functions related to the group SO(2, 2)

“…0 F 3 -functions are noted to be also related to Appell series F 3 and Kampé de Fériet functions [2]. Our preliminary calculations reveal that, with help of the above multi-index (hyper-bessel) functions, it is possible to obtain some new integral representations similar to those in this paper and [14].…”

“…It is noted that, for the group SO (2,1) in [14], the same situation arises. Yet, for SO (3,1), the dimension of H 1 and H 2 , which act transitively on the sphere and hyperboloid, respectively, is equal to 3 and γ 1 and γ 2 depend on 2 parameters.…”

“…

In this sequel to our earlier works [3,14,15], we aim to present certain integral and series representations for special functions by using some different group theoretical methods as follows: Restrictions of the representation matrix elements to some block-diagonal matrices; Poisson transform intertwining two realizations of the SO(2, 2)-representation; Invariant properties of the bilinear integral functionals which are used to obtain the matrix elements of bases transforms operators.

…”

“…The authors [3,14,15] considered the cases s = t = 2, s = 2 = t + 1 and s = 3 = t + 2, respectively, whose representation space was denoted by D and called the basic space. The similar space of (2 − s − t − σ )-homogeneous functions was denoted by D • and named the concerted representation space.…”

“…In this paper, we employ the approach used in [14] for the SO(2, 1), where we obtained the integral representations of the following type:…”