CHEBYSHEVSKII SBORNIK On one sum of Hankel–Clifford integral transforms of Whittaker functions

In [11], the authors considered the realization 𝑇 of 𝑆𝑂 (2 , 2) -representation in a space of homogeneous functions on 2 × 4 -matrices. In this sequel, we aim to compute matrix elements of the identical operator 𝑇 ( 𝑒 ) and representation operator 𝑇 ( 𝑔 ) for an appropriate 𝑔 with respect to the mixed basis related to two different bases in the 𝑆𝑂 (2 , 2) -carrier space and evaluate some improper integrals involving a product of Bessel-Clifford and Whittaker functions. The obtained result can be rewritten in terms of Hankel-Clifford integral transforms and their analogue. The first and the second Hankel-Clifford transforms introduced by Hayek and P´erez–Robayna, respectively, play an important role in the theory of fractional order differential operators (see, e.g., [6, 8]). The similar result have been derived recently by the authors for the regular Coulomb function in [12]. integral transform, Macdonald-Clifford integral transform, Whittaker functions, Bessel-Clifford functions. Bibliography: 15 titles.


Introduction and preliminaries
We recall the definitions and notations in [11]. The group (2,2), which preserves the quadratic form ℰ defined in R 4 whose matrix with respect to the canonical basis is a diagonal matrix 2,2 = diag(1, 1, −1, −1), which is called the split orthogonal group and consists of real 4 × 4 matrices satisfying the equality 2,2 t = 2,2 . Here and throughout, let C, R, R + , R − , Z and N be the sets of complex numbers, real numbers, positive real numbers, negative real numbers, integers and positive integers, respectively, and let N 0 := N ∪ {0}. Let be a real linear space consisting of real 2 × 4 matrices. We define the cone Λ in by the subset of matrices of rank 2 satisfying the equation 2,2 t = diag(0, 0). Let L be the complex linear space consisting of infinitely differentiable functions defined on Λ and satisfying the equality ( ) = | 11 | 1 | 22 | 2 ( ) for a fixed pair ( 1 , 2 ) ∈ C 2 and arbitrary non-degenerate matrix = In [14], with a view to investigating some special functions of matrix argument, this construction has been used.
Shilin and Choi [11] dealt with the spherical section 1 of Λ consisting of matrices In particular, they [11] showed that for any ∈ Λ there are a low triangular non-degenerate 2 × 2matrix and˜∈ 1 such that =˜. If L 1 is the linear space of restrictions of functions ∈ L on which is orthonormal with respect to the scalar product in L 1 . Writing Cartan decomposition = 1 2 3 for an arbitrary element of the group (2, 2), where 1 , 3 ∈ (2) × (2) and 2 ∈ exp they showed that in case | | ̸ = | | the matrix elements of the linear operator ( ) with respect to˜1 can be written as a product of four exponential functions, depending respectively on four parameters of the rotations 1 and 3 , and two Gaussian hypergeometric functions depending (respectively) on (︀ tanh ± 2 )︀ 2 . The parabolic section 2 of Λ has been defined as the subset consisting of matrices where 2 ∈ R and 2 ∈ [0, 2 ). If L 2 is the linear space of restrictions of functions ∈ L on 2 , then the canonical basis in L 2 can be defined as follows: They [11] established the one-to-one correspondence between the restrictions of ( ) to L 2 and integral operators whose kernels can be described in terms of some Bessel functions. )︂ inside the matrix ∈ Λ by Δ , and introduce the basis in L, consisting of functions

Two bases in
Obviously the restriction of 1 , 1 to 1 coincides with i 1˜1 ,− 1 : In this paper, we also use the basis .
It is easy to see that 2 , 2 is an extension of the function i 2˜2 , 2 to Λ. Let span( 1 , 1 ,^1 ,^1 ) be the subspace in L 1 . It is invariant with respect to the linear operator ( ) (for some fixed ) and its basis vectors 1 , 1 and^1 ,^1 which are not eigenfunctions of this operator. In this paper, we aim to establish dependence between the matrix elements of the operator ( ) with respect to the ordinary basis 2 and the mixed basis 2 | 1 and matrix elements of the operator id ≡ ( ) with respect to 2 | 1 . Choosing here the group element as follows: we will show that the above dependence can be rewritten as a representation of Whittaker function of the second kind in the form of integral involving Whittaker and Bessel-Clifford functions. The Bessel-Clifford functions are used, for example, for solution of wave equation [1] and are a particular case of more generalized so-called Bessel-Maitland functions (see [8]). The above-described approach, together with other methods, was used by Shilin and Choi [10] who considered another realization of the representation of the group (2, 2) and representation operators corresponding to some diagonal and block-diagonal matrices which belong to the split orthogonal group.

Functionals F 1 and F 2 and assorted spaces
Let us introduce the following bilinear functionals defined on the direct product L × L • of two representation spaces: where the functions on L • are ( • 1 , • 2 )-homogeneous.
It is clear that the equality F 1 = F 2 is equivalent to 1 + • 1 − 2 − • 2 + 4 = 0. 2 Further we assume that representation spaces L and L • are mutually assorted, i.e., the pair ( • 1 , • 2 ) for the representation space L • is connected with the pair ( 1 , 2 ) for L by the equality (3).

Matrix elements of the
In view of Lemma 1, we have We compute the matrix elements of the linear operator acting in L • and transforming the basis • 1 into • 2 , asserted by the following theorem.
where Γ is the gamma function, , is the Whittaker function of the second kind, and , is the Kronecker symbol.
Some similar results to those in Theorem 3 and formula (19) can be obtained from (17) in case 2 ∈ R − .
Using the following three integral transformations: