Some integrals involving Coulomb functions associated with the three-dimensional proper Lorentz group

1.   Introduction and preliminaries

The Coulomb wave functions $F_\sigma(\rho, \lambda)$ and $H^\pm_\sigma(\rho, \lambda)$ are functions belonging to the kernel of the Coulomb differential operator (see, e.g., [18,Chapter 33])

where $\lambda\in\mathbb{R}^+$, $\rho\in\mathbb{R}$ (Sommerfeld parameter), and $\sigma \in \mathbb{N}_0$ (angular momentum quantum number). Here and throughout, we denote $\mathbb{C}$, $\mathbb{R}$, $\mathbb{R}^+$, $\mathbb{Z}$, and $\mathbb{N}$ by the sets of complex numbers, real numbers, positive real numbers, integers, and positive integers, respectively, and let $\mathbb{N}_0: = \mathbb{N} \cup \{0\}$, ${\mathbb Z}_0^-: = \mathbb{Z}\setminus \mathbb{N}$. The functions $F_\sigma(\rho, \lambda)$ and $H^\pm_\sigma(\rho, \lambda)$ satisfy the following formulas (see ^[6]):

and

where ${\bf i} = \sqrt{-1}$, $M_{\mu, \nu}(z)$ and $W_{\mu, \nu}(z)$ are Whittaker functions of the first kind and the second kind (see, e.g., [1,Sections 4.3 and 4.4]), respectively, and the normalizing constant (Gamow factor) $N_\sigma(\rho)$ and Coulomb phase shift $n_\sigma(\rho)$ are given as follows:

and

where $\Gamma$ is the familiar Gamma function (see, e.g., [27,Section 1.1]). The functions $F_\sigma(\rho, \lambda)$ and $H^\pm_\sigma(\rho, \lambda)$ may also be continued analytically to $\lambda \in \mathbb{C} \setminus \{0\}$, and $\rho, \, \sigma \in \mathbb{C}$ (see, e.g., ^[5,8]; see also [18,Entry 33.13]).

Recall the generalized hypergeometric series $_pF_q$ $\left(p, \, q \in \mathbb{N}_0\right)$ defined by (see, e.g., [22,p. 73]):

where $(\lambda)_\nu$ denotes the Pochhammer symbol which is defined (for $\lambda, \, \nu \in \mathbb{C}$), in terms of the familiar Gamma function $\Gamma$, by

it being understood conventionally that $(0)_0: = 1$.

Since the two Whittaker functions are determined by the confluent hypergeometric function ${}_1F_1$ (see, e.g., [7,Entries 6.5(7),6.9(1) and 6.9(2)]) which satisfies the following Kummer's transformation formula (e.g., [7,Entry 6.3(7)])

the choice of ambiguous signs in (1.1) and (1.2) is immaterial, provided that either all upper signs are taken or all lower signs are taken (see, e.g., the comment after [18,Entry 33.2.5]).

Also recall (see, e.g., [18,Entry 33.2(ⅲ)])) that $H^+_\sigma(\rho, \lambda)$ and $H^-_\sigma(\rho, \lambda)$ are complex conjugates, and their real and imaginary parts are given by

Here $F_\sigma(\rho, \lambda)$ is the regular solution of the Coulomb wave equation $\mathfrak{d}[y] = 0$ while $G_\sigma(\rho, \lambda)$ and $H^\pm_\sigma(\rho, \lambda)$ are its irregular solutions.

Since the dimension of the space ${\rm Ker}\, \mathfrak{d}$ is $2$, the linearly independent functions $F_\sigma(\rho, \lambda)$ and $G_\sigma(\rho, \lambda)$ form a basis in ${\rm Ker}\, \mathfrak{d}$. Another basis consists of $H^+_\sigma(\rho, \lambda)$ and $H^-_\sigma(\rho, \lambda)$. Curtis ^[4] considered two other bases in ${\rm Ker}\, \mathfrak{d}$ consisting of functions which are still named Coulomb functions (see also ^[11,14]).

Recall another differential operator (see ^[12]):

The so-called Bessel-Clifford function of the first kind

is one of the functions belonging to Ker$\mathfrak{b}$. The function $\mathcal{C}_\sigma (\lambda)$ in (1) is related to the Bessel function $J_\sigma$ as follows:

Likewise the modified Bessel-Clifford function of the second kind $\mathcal{K}_\sigma (\lambda)$ can be expressed via the Macdonald function $K_\sigma$:

We also need the following integral transformations:

The first Hankel-Clifford integral transform (see [16,Eq. (2.7)])

The second Hankel-Clifford integral transform (see ^[13]; see also [16,Eq. (2.9)])

The Macdonald-Clifford transform

In this paper, we aim to present new formulas which involve Coulomb functions and are related to a representation of the three-dimensional proper Lorentz group in one appropriate carrier functional space (called here the representation space). The paper is organized as follows: In Section 2 we describe the representation space and two bases in it. In Section 3 we express the connection between above bases in terms of Coulomb functions (basically, in terms of $F_\sigma(\rho, \lambda)$). In Sections 4–8 we consider some restrictions of our representation to some subetaoups or their separate elements and, using the connections between bases, obtain some integral formulas. In particular, for ${\rm diag}(-1, 1, -1)$ and the (circle) rotations through $\pi$ and $\frac\pi2$ in the plane $Ox_2x_3$ we obtain the Bessel–Clifford and Mellin transforms or Coulomb functions, respectively, which are expressed in terms of Coulomb functions and Gaussian hypergeometric functions. Considering the product of the maximal nilpotent subetaoup and the subetaoup of hyperbolic rotations in the plane $Ox_1x_2$, we express one integral involving product of Coulomb functions in terms of Appell hypergeometric function.

The same problem for two different pairs of bases in the similar representation space for the four-dimensional analogue of our group has been considered in the papers ^[25] and ^[26], in which the related theorems have been formulated in terms of ${}_4F_3$ and Macdonald functions, respectively.

2.   Representation space, its bases, and functionals ${\sf F}_1$ and ${\sf F}_2$

We recall that the three-dimensional Lorentz group is the subetaoup of matrices $(g_{ij})$ in $GL(3, \mathbb{R})$ satisfying the equalities

where $E(\mu)$ denotes the integer part of $\mu \in \mathbb{R}$, and $g_{i1}g_{j1}-g_{i2}g_{j2}-g_{i3}g_{j3} = 0$ for different $i, j\in\{1, 2, 3\}$. In this paper, we consider the intersection $G$ of this group with $SL(3, \mathbb{R})$, calling $G$ the proper Lorentz group. Let $\sigma\in\mathbb{R}$ and $T$ be the representation of $G$ in the linear space $\mathfrak{D}$ consisting of $\sigma$-homogeneous and infinitely differentiable functions defined on the cone

acting according to rule $T(g)[f(x)] = f(g^{-1}x)$ (see ^[30]). The similar realization of the $G$-representation can be considered on hyperboloid $x_1^2-x_2^2-x_3^2 = r^2$ (see ^[32]).

We recall that the functions $x^\mu_\pm$ on $\mathbb{R}$, which generate the generalized functions $(x^\mu_\pm, f)$, are defined as follows (see ^[9]): For $\mu\in\mathbb{C}$, $x^\mu_\pm$ is equal to $|x|^\mu$ for $x\in\mathbb{R}^+$ and coincides with zero function otherwise. Here we deal with the bases (see ^[31])

and

In the following, we use two bilinear functionals defined on pairs of representation spaces in the same way as in ^[24]. In order to introduce them, we define the following subsets on $\Lambda$: parabola $\gamma_1:\, x_1+x_2 = 1$ and hyperbola $\gamma_2 = \gamma_{2, +}\cup\gamma_{2, -}$, where $\gamma_{2, \pm}:\, x_2 = \pm1$. Let $H_i$ be a subetaoup of $G$, which acts transitively on $\gamma_i$. We define ${\sf F}_1$ and ${\sf F}_2$ as

where ${\rm d}\gamma_i$ is a $H_i$-invariant measure on $\gamma_i$. Let $\gamma_1$ and $\gamma_{2, \pm}$ be parameterized as follows:

where $\alpha_1, \alpha_2\in\mathbb{R}$. Since the subetaoups $H_1$ and $H_2$ consist of matrices, respectively,

and

where $\theta_1, \theta_2\in\mathbb{R}$, and

we have ${\rm d}\gamma_i = {\rm d}\alpha_i$. Also these identities mean that $f_\lambda$ is an eigenfunction of the operator $T(h_1(\theta_1))$ and $f_{\rho, \pm}$ is an eigenfunction of $T(h_2(\theta_2))$ with eigenvalues $\exp(-\theta_1\lambda)$ and $\exp(-\theta_2\rho)$, respectively. It was shown in ^[24] that ${\sf F}_1$ and ${\sf F}_2$ coincide on pairs $(\mathfrak{D}, \mathfrak{D}^\bullet)$ such that degree of homogeneity of $\mathfrak{D}^\bullet$ is equal to $-\sigma-1$.

3.   Matrix elements of $B_1\rightleftarrows B_2$ and $B^\bullet_1\rightleftarrows B^\bullet_2$ transformations in terms of Coulomb functions

Express a function $f_\lambda\in B_1^\bullet$ as a linear combination of vectors belonging to $B_2^\bullet$:

Since

we have

where $\delta(\rho+\hat\rho)$ is the $\hat\rho$-delayed Dirac delta function. Therefore,

Likewise, if

then

Considering that $\sigma$ is the third argument (after $\rho$ and $\lambda$) of $c^\bullet_{\rho, \pm, \lambda}$, we derive from (3.4) that

Theorem 3.1. The $c^\bullet_{\lambda, \rho, +}$ in (3.3) is expressed in terms of the Coulomb wave function in (1.1) as follows:

Proof. Recall a known formula (see, e.g., [10,Entry 3.383-1] and [19,Entry 2.3.6-1])

where ${\rm B}(\alpha, \beta)$ is the familiar Beta function given by (see, e.g., [27,Section 1.1])

Then

where (3.7) is applied for the fifth equality. For the ${}_1F_1$ of the last expression, using the following known relation (see, e.g., [1,Eq. (4.3.2)], [17,p. 290])):

we find

which, in view of (3.8), is equivalent to

Since $\Gamma(z)$ is analytic in the half plane $\Re(z) > 0$ whose domain is symmetric with respect to the real axis $x = \Re(z) \in \mathbb{R}^+$ and $\Gamma(x) \in \mathbb{R}$ on the real axis $x = \Re(z) \in \mathbb{R}^+$, by reflection principle (see, e.g., [3,p. 57]), we have $\Gamma(\bar{z}) = \overline{\Gamma(z)}$ for each $z$ in the half plane $\Re(z) > 0$. Therefore (3.10) is rewritten as follows:

From (1.1) and (1.3),

Finally, using (3.12) in (3.11), we obtain the desired result (3.6).

Theorem 3.2. Let $-1 < \sigma < 0$ and $\lambda\ne0$. Then

Or, equivalently,

Proof. We have

Recalling an integral formula (see, e.g., [19,Entry 2.3.2.(3)])

and using (3.9) and (1.1), we obtain

where

(see also [10,Entry 9.220.(4)]). Using (1.2), we derive (3.13). Using (3.13) and (1.4), we obtain (3.14).

4.   Formulae related to the matrix ${\rm diag}(-1, 1, -1)$

For any $f^\bullet_\lambda$ and any $g\in G$, let us express $T^\bullet(g)[f^\bullet_\lambda]$ as the integral operator

where $t^\bullet_{\lambda\hat\lambda}$ are the matrix elements of the representation $T^\bullet$. Then

Choosing here $i = 1$, in view of

where $\delta(\hat\lambda+\tilde\lambda)$ is the $(-\tilde\lambda)$-delayed Dirac delta function, we have

Thus

From (3.1) and (4.1), for any $g\in G$, we have

where

On the other hand,

Since $f^\bullet_{\rho, +}$ is an eigenfunction of the linear operator $T^\bullet\big({\rm diag}(-1, 1, -1)\big)$ with eigenvalue $(-1)^{{\bf i}\rho}$, we have

We express the matrix elements $t_{\lambda\hat\lambda}\big({\rm diag}(-1, 1, -1)\big)$ in terms of Bessel-Clifford functions as in the following theorem.

Theorem 4.1. Let $-1 < \sigma < 0$. Then

and

Proof. Since

and

we find from (4.2) that

Using known integral formulas (see [19,Entry 2.5.24.4]), we obtain the following integral formulas: For $a\, b > 0$ and $|\Re(\alpha)| < 1$,

and

Using (4.7) and (4.8) to evaluate the integral (4.6), with the aid of the relations (1.5) and (1.6), we derive the expressions (4.4) and (4.5).

Using (4.3) and Theorems 3.1 and 3.2, we obtain the following identity

The identity (4.9) can be rewritten in the following theorem.

Theorem 4.2. Let $\lambda > 0$ and $-1 < \sigma < 0$. Then

Considering the case $\lambda < 0$ in (4.3), we obtain the following similar result.

Theorem 4.3. Let $\lambda < 0$ and $-1 < \sigma < 0$. Then

5.   Formulae related to the rotation through $\pi$ in the plane $Ox_2x_3$

Let $\mathbb{U}_n$ be the multiplicative group of all complex roots of the equation $z^n = 1$ $(n \in \mathbb{N})$. Then, for any $\varepsilon\in\mathbb{U}_2$, we have ${\rm diag}(1, \varepsilon, \varepsilon)\in G$ and

Therefore,

The trivial case $\varepsilon = 1$ gives $t_{\lambda\hat\lambda}({\rm id}) = \delta(\lambda-\hat\lambda)$. The case $\varepsilon = -1$ yields the following theorem.

Theorem 5.1. Let $\lambda > 0$ and $-1 < \sigma < 0$. Then

Proof. The proof of (5.1) would run parallel to that of Theorems 4.1 and 4.2. We omit the details.

Another result corresponding to (5.1) for the case $\lambda < 0$ is left to the interested reader.

6.   One formula related to the rotation through $\frac\pi 2$ in the plane $Ox_2x_3$

In this section we use the subetaoup $H^\circ$ consisting of (circle) rotations

Note that the subetaoup $H^\circ$ is isomorphic to the group $SO(2)$.

For any $g\in G$, let us express the function $T^\bullet(g)[f^\bullet_{\rho, \pm}]$ as linear combinations of functions belonging to the basis $B_2^\bullet$:

The equation (6.1) gives that, for any $k \in \mathbb{Z}$,

Choosing here $j = 1$ and considering (3.2), we have

Therefore,

Or, equivalently,

Theorem 6.1. Let $-1 < \sigma < 0$. Then

Proof. We find

Using a known integral formula (see [19,Entry 2.2.6.(1)])

we obtain

We also find

where

Thus, $t^\bullet_{\rho, +, \hat\rho, +}\left(h^\circ\Big(\frac\pi2\Big)\right)$ can be expressed in terms of Gamma functions (see, e.g., [19,Entry 2.3.4(1)]). Now, the statement of the theorem follows from (3.5), (6.2), and (6.4).

Remark 1. From Theorem 4.2 and throughout, $F_{-\sigma-1}(-\rho, -\lambda)$ can be changed by the linear combination of basis functions in ${\rm Ker}\, \mathfrak{d}$ (see ^[5])

where

7.   Formulae related to the hyperbolic rotations in the plane $Ox_1x_2$

In this section we deal with the subetaoup $H^*$ of matrices

in $G$.

Theorem 7.1. Let $-1 < \sigma < 0$ and $\tau > 0$. Then

where $Q^\mu_\nu (x)$ is the associated Legendre function of the second kind.

Proof. Computing the matrix elements of the subrepresentation $T^\bullet$ to $H^*$, we obtain

Since $\cosh\tau\cosh\alpha_2+\sinh\alpha_2+\sinh\tau > 0$ and $\sigma+1 > 0$, using a known integral formula (see [19,Entry 2.5.48.(6)])

we have

Also,

Here

where the last integral is evaluated by using (3.7). Therefore we have

Finally, considering (7.1) and (7.2), we complete the proof.

Remark 2. Cylindrical (ordinary) and spherical Bessel functions can be expressed in terms of Coulomb function, respectively, by the following formulas (see, e.g., [29,Eqs. (2.1a) and (2.1c)]

8.   Formulae related to the product of subgroups $H_1$ and $H^*$

In this section we show that the matrix elements of the subrepresentation $T^\bullet$ to the subetaoup $H_1H^*$ yield four integral representations of the Appell function $F_1$, which depend on the relations between parameters $\theta_1$ and $\tau$ and correspond to Figures 1–4.

Theorem 8.1. Let $-1 < \sigma < 0$ and $e^{-\tau} < |\theta_1\pm1|$. Then

Proof. We show that both functionals ${\sf F}_1$ and ${\sf F}_2$ are invariant with respect to the pair $(T^\bullet(h^*), T(h^*))$ of representation operators. Indeed, in view of homogeneity of $u$ and $v$,

Since $T^\bullet$ is a homomorphism (of $G$ to the group $GL(\mathfrak{D}^\bullet)$ of linear operators of $\mathfrak{D}^\bullet$ with trivial kernels) and $H_1$ is an one-parameter subetaoup (that is, $h_1^{-1}(\theta_1) = h_1(-\theta_1)$), we have

where $\Omega$ is the intersection of the segments $|\alpha_1|\le e^{-\tau}$ and $[\theta_1-1, \theta_1+1]$. Under the assumption $e^{-\tau} < |\theta_1\pm1|$ (Figure 1), this integral can be evaluated by using a known integral formula (see [10,Entry 3.211]): For $\min \{\Re(\mu), \Re(\nu)\} > 0$,

Also,

We thus have

Using (1.1), (3.7) and (3.9), we get

Using (3.5), we have

Considering (8.1) and (8.2), we complete the proof.

Employing the same proof as in Theorem 8.1 for the cases corresponding to Figures 2–4, we obtain the following results, without their proofs, which are given in Theorems 8.2, 8.3 and 8.4.

Theorem 8.2. Let r $-1 < \sigma < 0$ and $|\theta_1\pm1| < e^{-\tau}$. Then

Theorem 8.3. Let $-1 < \sigma < 0$, $\big|\theta_1+e^{-\tau}\big| < 1$, and $|\theta_1+1| < e^{-\tau}$. Then

Theorem 8.4. Let $-1 < \sigma < 0$, $\big|e^{-\tau}-\theta_1\big| < 1$, and $|\theta_1-1| < e^{-\tau}$. Then

Remark 3. In Theorems 8.1–8.4, the Appell function $F_1$ can be rewritten in terms of the hypergeometric function ${}_2F_1$ by using the following relation (see, e.g., [21,Entry 7.2.4.(63)])

9.   Concluding remarks

The group theoretic approach gives a natural (in some sense) technique to obtain formulas for integral transforms of special functions, since they occur in relations between kernels of integral operators of representations calculated in different ordinary and mixed bases and matrix elements of basis transformations. For instance, by evaluating the Mellin-Laplace transform of the Coulomb Sturmian radial function, Morales ^[15] used series decomposition and integration term-by-term to show that the integral

is equal up to multiplicative constant to

where $l, j\in\mathbb{N}_0$ and $\omega > -1$. For investigation of matrix elements of two-body Coulomb interaction in the lowest Landau level, Bentalha [2,Lemma 1] obtained the following integral formula for product of associated Laguerre polynomials $(l, m, n\in\mathbb{N}_0)$ expressed in terms of Appell $F_2$ function:

which may be also expressed in terms of ${}_3F_2$ (see, e.g., [20,Entry 2.19.14.(9)] and ^[28]). Since $L_n^{2l+1}(x)$ up to multiplicative constant coincides with ${}_1F_1(-n; 2l+2;x)$, we have in (9.1) and (9.2) Mellin-Laplace integrals for Whittaker functions of the first kind, which are related to regular Coulomb function $F_\sigma(\rho, \lambda)$. The results in Theorems 8.1–8.4 are concerned with the Mellin-Fourier transform for Coulomb functions and, in this regard, are closely related to the examples (9.1) and (9.2). It may be said that, in some sense, Theorems 8.1–8.4 are complexifications of these examples.

Theorems 4.1–8.4 are based on the fact (described in Theorems 3.1 and 3.2) that the matrix elements $c_{\rho, \pm, \lambda}$ and $c_{\lambda, \rho, \pm}$ (kernels of corresponding integral operators) of the transformations $B_1\longrightarrow B_2$ and $B_2\longrightarrow B_1$ can be expressed in terms of Coulomb wave function $F_\sigma(\rho, \lambda)$. In all these theorems (except Theorem 5.1), we have used only integrals of the following forms

Analogous integral transforms to those in this paper, containing kernels $c_{\rho, -, \lambda}$ and $c_{\lambda, \rho, -}$, can yield similar results. Also these kernels were considered for two dual cases $\sigma = -\frac12\pm\frac14$ in ^[23].

Acknowledgments

The authors are grateful to the anonymous referees for the constructive and valuable comments which improved this paper.

Conflict of interest

The authors declare that they have no conflict of interest.