Research article

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

  • Received: 01 May 2020 Accepted: 29 June 2020 Published: 02 July 2020
  • MSC : 33C10, 33C80, 33B15, 33C05

  • For two continual bases in the representation space, we obtain the matrix elements of the linear operator transforming the first basis into the second. These elements are expressed in terms of Coulomb wave functions. Computing the matrix elements of subrepresentations to some subgroups or their separate elements and using the connection between above bases, we evaluate some integrals involving Coulomb wave functions.

    Citation: I. A. Shilin, Junesang Choi, Jae Won Lee. Some integrals involving Coulomb functions associated with the three-dimensional proper Lorentz group[J]. AIMS Mathematics, 2020, 5(6): 5664-5682. doi: 10.3934/math.2020362

    Related Papers:

    [1] Shahid Mubeen, Rana Safdar Ali, Iqra Nayab, Gauhar Rahman, Thabet Abdeljawad, Kottakkaran Sooppy Nisar . Integral transforms of an extended generalized multi-index Bessel function. AIMS Mathematics, 2020, 5(6): 7531-7547. doi: 10.3934/math.2020482
    [2] A. Belafhal, N. Nossir, L. Dalil-Essakali, T. Usman . Integral transforms involving the product of Humbert and Bessel functions and its application. AIMS Mathematics, 2020, 5(2): 1260-1274. doi: 10.3934/math.2020086
    [3] Mohamed Abdalla . On Hankel transforms of generalized Bessel matrix polynomials. AIMS Mathematics, 2021, 6(6): 6122-6139. doi: 10.3934/math.2021359
    [4] İbrahim Aktaş . On some geometric properties and Hardy class of q-Bessel functions. AIMS Mathematics, 2020, 5(4): 3156-3168. doi: 10.3934/math.2020203
    [5] Ruma Qamar, Tabinda Nahid, Mumtaz Riyasat, Naresh Kumar, Anish Khan . Gould-Hopper matrix-Bessel and Gould-Hopper matrix-Tricomi functions and related integral representations. AIMS Mathematics, 2020, 5(5): 4613-4623. doi: 10.3934/math.2020296
    [6] D. L. Suthar, A. M. Khan, A. Alaria, S. D. Purohit, J. Singh . Extended Bessel-Maitland function and its properties pertaining to integral transforms and fractional calculus. AIMS Mathematics, 2020, 5(2): 1400-1410. doi: 10.3934/math.2020096
    [7] Rana Safdar Ali, Saba Batool, Shahid Mubeen, Asad Ali, Gauhar Rahman, Muhammad Samraiz, Kottakkaran Sooppy Nisar, Roshan Noor Mohamed . On generalized fractional integral operator associated with generalized Bessel-Maitland function. AIMS Mathematics, 2022, 7(2): 3027-3046. doi: 10.3934/math.2022167
    [8] Borhen Halouani, Fethi Bouzeffour . On the fractional Laplace-Bessel operator. AIMS Mathematics, 2024, 9(8): 21524-21537. doi: 10.3934/math.20241045
    [9] Gamal Hassan, Mohra Zayed . Expansions of generalized bases constructed via Hasse derivative operator in Clifford analysis. AIMS Mathematics, 2023, 8(11): 26115-26133. doi: 10.3934/math.20231331
    [10] Georgia Irina Oros, Gheorghe Oros, Daniela Andrada Bardac-Vlada . Certain geometric properties of the fractional integral of the Bessel function of the first kind. AIMS Mathematics, 2024, 9(3): 7095-7110. doi: 10.3934/math.2024346
  • For two continual bases in the representation space, we obtain the matrix elements of the linear operator transforming the first basis into the second. These elements are expressed in terms of Coulomb wave functions. Computing the matrix elements of subrepresentations to some subgroups or their separate elements and using the connection between above bases, we evaluate some integrals involving Coulomb wave functions.


    The Coulomb wave functions Fσ(ρ,λ) and H±σ(ρ,λ) are functions belonging to the kernel of the Coulomb differential operator (see, e.g., [18,Chapter 33])

    d:=d2dλ2+12ρλσ(σ+1)λ2,

    where λR+, ρR (Sommerfeld parameter), and σN0 (angular momentum quantum number). Here and throughout, we denote C, R, R+, Z, and N by the sets of complex numbers, real numbers, positive real numbers, integers, and positive integers, respectively, and let N0:=N{0}, Z0:=ZN. The functions Fσ(ρ,λ) and H±σ(ρ,λ) satisfy the following formulas (see [6]):

    Fσ(ρ,λ)=2σ1Nσ(ρ)(i)σ+1M±iρ,σ+12(±2iλ) (1.1)

    and

    H±σ(ρ,λ)=(i)σexp(πρ2±inσ(ρ))Wiρ,σ+12(2iλ), (1.2)

    where i=1, Mμ,ν(z) and Wμ,ν(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) Nσ(ρ) and Coulomb phase shift nσ(ρ) are given as follows:

    Nσ(ρ)=2σexp(πρ2)[Γ(2σ+2)]1|Γ(σ+1+iρ)| (1.3)

    and

    nσ(ρ)=argΓ(σ+1+iρ),

    where Γ is the familiar Gamma function (see, e.g., [27,Section 1.1]). The functions Fσ(ρ,λ) and H±σ(ρ,λ) may also be continued analytically to λC{0}, and ρ,σC (see, e.g., [5,8]; see also [18,Entry 33.13]).

    Recall the generalized hypergeometric series pFq (p,qN0) defined by (see, e.g., [22,p. 73]):

    pFq[α1,,αp;β1,,βq;z]=n=0(α1)n(αp)n(β1)n(βq)nznn!=pFq(α1,,αp;β1,,βq;z),

    where (λ)ν denotes the Pochhammer symbol which is defined (for λ,νC), in terms of the familiar Gamma function Γ, by

    (λ)ν:=Γ(λ+ν)Γ(λ)={1(ν=0;λC{0})λ(λ+1)(λ+n1)(ν=nN;λC),

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

    Since the two Whittaker functions are determined by the confluent hypergeometric function 1F1 (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)])

    1F1(a;b;z)=ez1F1(ba;b;z),

    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+σ(ρ,λ) and Hσ(ρ,λ) are complex conjugates, and their real and imaginary parts are given by

    H±σ(ρ,λ)=Gσ(ρ,λ)±iFσ(ρ,λ). (1.4)

    Here Fσ(ρ,λ) is the regular solution of the Coulomb wave equation d[y]=0 while Gσ(ρ,λ) and H±σ(ρ,λ) are its irregular solutions.

    Since the dimension of the space Kerd is 2, the linearly independent functions Fσ(ρ,λ) and Gσ(ρ,λ) form a basis in Kerd. Another basis consists of H+σ(ρ,λ) and Hσ(ρ,λ). Curtis [4] considered two other bases in Kerd consisting of functions which are still named Coulomb functions (see also [11,14]).

    Recall another differential operator (see [12]):

    b:=λd2dλ2+(σ+1)ddλ+1.

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

    Cσ(λ)=k=0(1)kλkk!Γ(σ+k+1)

    is one of the functions belonging to Kerb. The function Cσ(λ) in (1) is related to the Bessel function Jσ as follows:

    Cσ(λ)=λσ2Jσ(2λ). (1.5)

    Likewise the modified Bessel-Clifford function of the second kind Kσ(λ) can be expressed via the Macdonald function Kσ:

    Kσ(λ)=λσ2Kσ(2λ). (1.6)

    We also need the following integral transformations:

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

    H(1)σ[f](λ)=λσ0Cσ(λˆλ)f(ˆλ)dˆλ(λR+);

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

    H(2)σ[f](λ)=0ˆλσCσ(λˆλ)f(ˆλ)dˆλ(λR+);

    The Macdonald-Clifford transform

    Kσ[f](λ)=0ˆλσKσ(λˆλ)f(ˆλ)dˆλ(λR+).

    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 Fσ(ρ,λ)). 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 diag(1,1,1) and the (circle) rotations through π and π2 in the plane Ox2x3 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 Ox1x2, 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 4F3 and Macdonald functions, respectively.

    We recall that the three-dimensional Lorentz group is the subetaoup of matrices (gij) in GL(3,R) satisfying the equalities

    g2i1g2i2g2i3=(1)E(i2)(i{1,2,3}),

    where E(μ) denotes the integer part of μR, and gi1gj1gi2gj2gi3gj3=0 for different i,j{1,2,3}. In this paper, we consider the intersection G of this group with SL(3,R), calling G the proper Lorentz group. Let σR and T be the representation of G in the linear space D consisting of σ-homogeneous and infinitely differentiable functions defined on the cone

    Λ={(x1,x2,x3)R3|x21x22x23=0}

    acting according to rule T(g)[f(x)]=f(g1x) (see [30]). The similar realization of the G-representation can be considered on hyperboloid x21x22x23=r2 (see [32]).

    We recall that the functions xμ± on R, which generate the generalized functions (xμ±,f), are defined as follows (see [9]): For μC, xμ± is equal to |x|μ for xR+ and coincides with zero function otherwise. Here we deal with the bases (see [31])

    B1={fσλ(x)=(x1+x2)σexpλx3x1+x2λR}

    and

    B2={fσρ,±(x)=(x2)σiρ±(x1+x3)iρρR}.

    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 Λ: parabola γ1:x1+x2=1 and hyperbola γ2=γ2,+γ2,, where γ2,±:x2=±1. Let Hi be a subetaoup of G, which acts transitively on γi. We define F1 and F2 as

    Fi:(D,ˆD)C,(f,g)γif(x)g(x)dγi,

    where dγi is a Hi-invariant measure on γi. Let γ1 and γ2,± be parameterized as follows:

    γ1={x1=12(1+α21),x2=12(1α21),x3=α1,γ2,±={x1=coshα2,x2=±1,x3=sinhα2,

    where α1,α2R. Since the subetaoups H1 and H2 consist of matrices, respectively,

    h1(θ1)=12(2+θ21θ212θ1θ212θ212θ12θ12θ12)

    and

    h2(θ2)=(coshθ20sinhθ2010sinhθ20coshθ2),

    where θ1,θ2R, and

    T(h1(θ1))[fλ(α1)]=fλ(α1θ1),T(h2(θ2))[fρ,±(α2)]=fρ,±(α2θ2),

    we have dγi=dαi. Also these identities mean that fλ is an eigenfunction of the operator T(h1(θ1)) and fρ,± is an eigenfunction of T(h2(θ2)) with eigenvalues exp(θ1λ) and exp(θ2ρ), respectively. It was shown in [24] that F1 and F2 coincide on pairs (D,D) such that degree of homogeneity of D is equal to σ1.

    Express a function fλB1 as a linear combination of vectors belonging to B2:

    fλ(x)=R[cλ,ρ,+fρ,+(x)+cλ,ρ,fρ,(x)]dρ. (3.1)

    Since

    fρ,±|γ2,±=fρ,±|γ2,±=exp(iρα2)andfρ,±|γ2,=fρ,±|γ2,=0, (3.2)

    we have

    Fi(fλ,fˆρ,±)=cλ,ρ,±F2(fρ,±,fˆρ,±)dρ=cλ,ρ,±dρexp(i(ρ+ˆρ)α2)dα2=2πcλ,ρ,±δ(ρ+ˆρ)dρ=2πcλ,ˆρ,±,

    where δ(ρ+ˆρ) is the ˆρ-delayed Dirac delta function. Therefore,

    cλ,ρ,±=12πFi(fλ,fˆρ,±). (3.3)

    Likewise, if

    fρ,±(x)=cρ,±,λfλ(x)dλ,

    then

    cρ,±,λ=12πFi(fρ,±,fλ)=cλ,ρ,±. (3.4)

    Considering that σ is the third argument (after ρ and λ) of cρ,±,λ, we derive from (3.4) that

    cρ,±,λ(σ)=cλ,ρ,±(σ1). (3.5)

    Theorem 3.1. The cλ,ρ,+ in (3.3) is expressed in terms of the Coulomb wave function in (1.1) as follows:

    cλ,ρ,+=|Γ(σ+1+iρ)|πλσ+1exp(πρ2)Fσ(ρ,λ)(λ0,σ>1). (3.6)

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

    a0xν1(ax)μ1epxdx=B(μ,ν)aμ+ν11F1(ν;μ+ν;ap) (3.7)
    (aR0,min{(μ),(ν)}>0),

    where B(α,β) is the familiar Beta function given by (see, e.g., [27,Section 1.1])

    B(α,β)={10tα1(1t)β1dt(min{(α),(β)}>0)Γ(α)Γ(β)Γ(α+β)(α,βCZ0). (3.8)

    Then

    cλ,ρ,+=12πF1(fλ,fρ,+)=12π(1α212)σ+iρ+(1+α212+α1)iρexp(iλα1)dα1=2σ1π11(1α1)σ+iρ(1+α1)σiρexp(iλα1)dα1=2σ1πexp(iλ)20tσiρ(2t)σ+iρexp(iλt)dt=2σπexp(iλ)B(σ+1+iρ,σ+1iρ)1F1(σ+1iρ;2σ+2;2iλ),

    where (3.7) is applied for the fifth equality. For the 1F1 of the last expression, using the following known relation (see, e.g., [1,Eq. (4.3.2)], [17,p. 290])):

    Mμ,ν(z)=zν+12exp(z2)1F1(νμ+12;2ν+1;z), (3.9)

    we find

    cλ,ρ,+=(iλ)σ12πB(σ+1+iρ,σ+1iρ)Miρ,σ+12(2iλ),

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

    cλ,ρ,+=(iλ)σ12πΓ(σ+1+iρ)Γ(σ+1iρ)Γ(2σ+2)Miρ,σ+12(2iλ). (3.10)

    Since Γ(z) is analytic in the half plane (z)>0 whose domain is symmetric with respect to the real axis x=(z)R+ and Γ(x)R on the real axis x=(z)R+, by reflection principle (see, e.g., [3,p. 57]), we have Γ(ˉz)=¯Γ(z) for each z in the half plane (z)>0. Therefore (3.10) is rewritten as follows:

    cλ,ρ,+=(iλ)σ12π|Γ(σ+1+iρ)|2Γ(2σ+2)Miρ,σ+12(2iλ). (3.11)

    From (1.1) and (1.3),

    Miρ,σ+12(2iλ)=2(i)σ1exp(πρ2)Γ(2σ+2)|Γ(σ+1+iρ)|Fσ(ρ,λ). (3.12)

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

    Theorem 3.2. Let 1<σ<0 and λ0. Then

    cλ,ρ,=|Γ(σ+1+iρ)|2πλσ+1i2σ+1exp(πρ2)[Hσ(ρ,λ)H+σ(ρ,λ)]. (3.13)

    Or, equivalently,

    cλ,ρ,=(1)1σ|Γ(σ+1+iρ)|πλσ+1exp(πρ2)Fσ(ρ,λ). (3.14)

    Proof. We have

    cλ,ρ,=12πF1(fλ,fρ,)=12π(1α212)σ+iρ(1+α212+α1)iρexp(iλα1)dα1=2σ1π[exp(iλ)0tσiρ(t+2)σ+iρexp(iλt)dt+exp(iλ)0tσ+iρ(t+2)σiρexp(iλt)dt].

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

    0xμ1(x+y)ν1exp(sx)dx=yμ+ν1B(μ,1μν)1F1(μ;μ+ν;sy)+s1μνΓ(μ+ν1)1F1(1ν;2μν;sy) (3.15)
    ((μ)>0,|argy|<π,(s)>0;(s)=0,(μ+ν)<2)

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

    cλ,ρ,=12π(U++U),

    where

    U±:=B(σ+1±iρ,2σ1)Miρ,σ+12(2iλ)+Γ(2σ+1)Miρ,σ12(2iλ)(iλ)σ+1=Γ(σ+1±iρ)(iλ)σ+1[Γ(2σ1)Γ(iρσ)Miρ,σ+12(2iλ)+Γ(2σ+1)Γ(±iρ+σ+1)Miρ,σ12(2iλ)]=Γ(σ+1±iρ)(iλ)σ+1Wiρ,σ+12(2iλ)

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

    For any fλ and any gG, let us express T(g)[fλ] as the integral operator

    T(g)[fλ]=tλˆλ(g)fˆλdˆλ, (4.1)

    where tλˆλ are the matrix elements of the representation T. Then

    Fi(T(g)[fλ],f˜λ)=tλˆλ(g)Fj(fˆλ,f˜λ)dˆλ.

    Choosing here i=1, in view of

    F1(fˆλ,f˜λ)=exp(i[ˆλ+˜λ]α1)dα1=δ(ˆλ+˜λ),

    where δ(ˆλ+˜λ) is the (˜λ)-delayed Dirac delta function, we have

    Fi(T(g)[fλ],f˜λ)=tλˆλ(g)δ(ˆλ+˜λ)dˆλ=2πtλ,˜λ(g).

    Thus

    tλˆλ(g)=12πFi(T(g)[fλ],fˆλ). (4.2)

    From (3.1) and (4.1), for any gG, we have

    T(g)[fλ]=(τ+fρ,++τfρ,)dρ,

    where

    τ±=tλˆλ(g)cˆλ,ρ,±dˆλ.

    On the other hand,

    T(g)[fλ]=(cλ,ρ,+T(g)[fρ,+]+cλ,ρ,T(g)[fρ,])dρ.

    Since fρ,+ is an eigenfunction of the linear operator T(diag(1,1,1)) with eigenvalue (1)iρ, we have

    tλˆλ(diag(1,1,1))cˆλ,ρ,±dˆλ=(1)iρcλ,ρ,+. (4.3)

    We express the matrix elements tλˆλ(diag(1,1,1)) in terms of Bessel-Clifford functions as in the following theorem.

    Theorem 4.1. Let 1<σ<0. Then

    tλˆλ(diag(1,1,1))=2ˆλ2σ+1sin(σπ)(1)σπK2σ+1(λˆλ)(λˆλ<0) (4.4)

    and

    tλˆλ(diag(1,1,1))=(1)σ2cos(σπ)[ˆλ2σ+1C2σ+1(λˆλ)λ2σ1C2σ1(λˆλ)] (4.5)
    (λˆλ>0).

    Proof. Since

    fλ|γ1(diag(1,1,1))=(1)σ1α2σ21exp(iλα11)

    and

    fˆλ|γ1=exp(iˆλα1),

    we find from (4.2) that

    tλˆλ(diag(1,1,1))=12πF1(T(diag(1,1,1))[fλ],fλ)=1(1)σ+1π0α2σ21cos(λα11+ˆλα1)dα1. (4.6)

    Using known integral formulas (see [19,Entry 2.5.24.4]), we obtain the following integral formulas: For ab>0 and |(α)|<1,

    0xα1cos(ax+bx)dx=π2(ba)α21sin(απ2)[Jα(2ab)Jα(2ab)] (4.7)

    and

    0xα1cos(axbx)dx=2(ba)α2cos(απ2)Kα(2ab). (4.8)

    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

    (1)σ+iρλσ+1Fσ(ρ,λ)=2sin(σπ)π0ˆλσK2σ+1(λˆλ)Fσ(ρ,ˆλ)dˆλ+πcos(σπ)20ˆλσC2σ+1(λˆλ)Fσ(ρ,ˆλ)dˆλπcos(σπ)2λ2σ+10ˆλσ1C2σ1(λˆλ)Fσ(ρ,ˆλ)dˆλ (4.9)
    (λR+,1<σ<0).

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

    Theorem 4.2. Let λ>0 and 1<σ<0. Then

    (1)σ+iρλσ+1Fσ(ρ,λ)=πcos(σπ)2H(2)2σ+1[ˆλσ1Fσ(ρ,ˆλ)](λ)πcos(σπ)2H(1)2σ1[ˆλσ1Fσ(ρ,ˆλ)](λ)+2(1)σsin(σπ)πK2σ+1[ˆλσ1Fσ(ρ,ˆλ)](λ).

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

    Theorem 4.3. Let λ<0 and 1<σ<0. Then

    (1)iρλσ+1Fσ(ρ,λ)=πcos(σπ)2H(2)2σ+1[ˆλσ1Fσ(ρ,ˆλ)](λ)πcos(σπ)2H(1)2σ1[ˆλσ1Fσ(ρ,ˆλ)](λ)+2(1)σsin(σπ)πK2σ+1[ˆλσ1Fσ(ρ,ˆλ)](λ).

    Let Un be the multiplicative group of all complex roots of the equation zn=1 (nN). Then, for any εU2, we have diag(1,ε,ε)G and

    T(diag(1,ε,ε))[fρ,±]|γ2=(ε)σiρ±(coshα2+εsinhα2)iρ=fερ,±ε|γ2.

    Therefore,

    tλˆλ(diag(1,ε,ε))cˆλ,ρ,±dˆλ=cλ,ερ,±ε.

    The trivial case ε=1 gives tλˆλ(id)=δ(λˆλ). The case ε=1 yields the following theorem.

    Theorem 5.1. Let λ>0 and 1<σ<0. Then

    (1)σexp(πρ)λσ+1Fσ(ρ,λ)=2(1)σ+1sin(πσ)πK2σ+1[ˆλσ1Fσ(ρ,λ)](λ)+cos(σπ)2H(1)2σ1[ˆλσ1Fσ(ρ,λ)](λ)cos(σπ)2H(2)2σ+1[ˆλσ1Fσ(ρ,λ)](λ). (5.1)

    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 λ<0 is left to the interested reader.

    In this section we use the subetaoup H consisting of (circle) rotations

    h(ξ)=(1000cosξsinξ0sinξcosξ).

    Note that the subetaoup H is isomorphic to the group SO(2).

    For any gG, let us express the function T(g)[fρ,±] as linear combinations of functions belonging to the basis B2:

    T(g)[fρ,±](x)=[tρ,±,ˆρ,+(g)fˆρ,+(x)+tρ,±,ˆρ,(g)fˆρ,(x)]dˆρ. (6.1)

    The equation (6.1) gives that, for any kZ,

    Fi(T(g)[fρ,±],f˜ρ,±(1)k)=[tρ,±,ˆρ,+(g)Fj(fˆρ,+,f˜ρ,±(1)k)+tρ,±,ˆρ,+(g)Fj(fˆρ,,f˜ρ,±(1)k)]dˆρ.

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

    F2(fˆρ,±,f˜ρ,±(1)k)={2πδ(ˆρ+˜ρ)(evenk)0(oddk).

    Therefore,

    Fi(T(g)[fρ,±],f˜ρ,±(1)k)=2πtρ,±,˜ρ,±(1)k(g).

    Or, equivalently,

    tρ,±,ˆρ,±(1)k(g)=12πFi(T(g)[fρ,±],fˆρ,±(1)k).

    Theorem 6.1. Let 1<σ<0. Then

    λiˆρ1Fσ1(ρ,λ)dλ=22σ+1exp(πρ2)B(1+σ+iˆρ,σ+iρ)i1+σ+iˆρΓ(1+σiˆρ)|Γ(iρσ)|×2F1(σ+iρ,σiˆρ;1+i(ρ+ˆρ);1).

    Proof. We find

    tρ,+,ˆρ,+(h(π2))=12πF1(T(h(π2))[fρ,+],fˆρ,+)=2σπ(α1)σ1+iρ+α2iρ1(1α21)σiˆρ+(1+α1)2iˆρdα1=2σπ10ασ1+iρ1(1α1)σiˆρ(1+α1)σ+iˆρdα1. (6.2)

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

    ba(xa)μ1(bx)ν1(cx+d)κdx=(ba)μ+ν1(ac+d)κB(μ,ν)2F1(μ,κ;μ+ν;c(ab)ac+d)
    (min{(μ),(ν)}>0,|argbc+dac+d|<π), (6.3)

    we obtain

    tρ,+,ˆρ,+(h(π2))=2σπ1B(σ+1+iˆρ,σ+iρ)×2F1(σ+iρ,σiˆρ;1+i(ρ+ˆρ);1).

    We also find

    tρ,+,ˆρ,+(h(π2))=12πcρ,+,λFi(T(h(π2))[fλ],fˆρ,+)dα1,

    where

    Fi(T(h(π2))[fλ],fˆρ,+)=F1(T(h(π2))[fλ],fˆρ,+)=21σexp(iλ)11(1+α1)iˆρσ2(1α1)σiˆρexp(2iλα1+1)dα1=21σexp(iλ)+12(2u1)σiˆρexp(2iλu)du. (6.4)

    Thus, tρ,+,ˆρ,+(h(π2)) 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, Fσ1(ρ,λ) can be changed by the linear combination of basis functions in Kerd (see [5])

    Fσ1(ρ,λ)=cosθFσ(ρ,λ)+sinθGσ(ρ,λ),

    where

    θ:=(σ+12)π+cσ1(ρ)cσ(ρ).

    In this section we deal with the subetaoup H of matrices

    h(τ)=(coshτsinhτ0sinhτcoshτ0001)

    in G.

    Theorem 7.1. Let 1<σ<0 and τ>0. Then

    λ2Fσ1(0,λ)M1iˆρ,σ+32(2λeτ)dλ=22σ+4πexp(|ˆρ|π2τσ)Γ(σi|ˆρ|)Γ(σ)|Γ(σ)|B(σ+1+iˆρ,σ+1iˆρ)Qi|ˆρ|σ1(coshτ),

    where Qμν(x) is the associated Legendre function of the second kind.

    Proof. Computing the matrix elements of the subrepresentation T to H, we obtain

    tρ,+,ˆρ,+(h(τ))=12πF2(T(h(τ))[fρ,+],fˆρ,+)=12πγ2,+T(h(τ))[fρ,+]fˆρ,+)dγγ2,+=12π(coshτ+sinhτcoshα2)σ1iρ×(coshτcoshα2+sinhα2+sinhτ)iρexp(iˆρα2)dα2. (7.1)

    Since coshτcoshα2+sinhα2+sinhτ>0 and σ+1>0, using a known integral formula (see [19,Entry 2.5.48.(6)])

    0cosbxdx(a+coshcx)ν=exp(bπc)Γ(νibc)c(a21)ν2Γ(ν)Qbicν1(aa21)
    (bR+,(cν)>0,a[1,1]),

    we have

    t0,+,ˆρ,+(h(τ))=exp(|ˆρ|π)Γ(σi|ˆρ|)πΓ(σ)Qi|ˆρ|σ1(coshτ).

    Also,

    tρ,+,ˆρ,+(h(τ))=12πcρ,+,λFi(T(h(τ))[fλ],fˆρ,+)dλ.

    Here

    Fi(T(h(τ))[fλ],fˆρ,+)=F2(T(h(τ))[fλ],fˆρ,+)=exp(τσ)(coshα2+1)σ1exp(i[λsinhα2eτ(coshα2+1)ˆρα2])=exp(τσ+λeτ)2σ+110tσ+iˆρ(1t)σiˆρexp(2λeτt)dt,

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

    tρ,+,ˆρ,+(h(τ))=exp(2τσ)B(σ+1+iˆρ,σ+1iˆρ)(1)σ22σ+4π×λσ2cρ,+,λexp(3λeτ)M1iˆρ,σ+32(2λeτ)dλ. (7.2)

    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)]

    Jν(λ)=(2πλ)12Fν12(0,λ)andjν(λ)=1λFν(0,λ).

    In this section we show that the matrix elements of the subrepresentation T to the subetaoup H1H yield four integral representations of the Appell function F1, which depend on the relations between parameters θ1 and τ and correspond to Figures 14.

    Figure 1.  Theorem 8.1.
    Figure 2.  Theorem 8.2.
    Figure 3.  Theorem 8.3.
    Figure 4.  Theorem 8.4.

    Theorem 8.1. Let 1<σ<0 and eτ<|θ1±1|. Then

    λ2σexp(iθ1λ)Fσ1(ˆρ,λ)Fσ1(ρ,eτλ)dλ=πexp(2σττπ(ρ+ˆρ)2)22σ+2|Γ(σ+iρ)Γ(σ+iˆρ)|(1θ1eτ)σiˆρ×(1+θ1+eτ)σ+iˆρB(σ+iρ,σiρ)×F1(σ+iρ,σ+iˆρ,σiˆρ;2σ;2eτeτ+θ11,2eτeτ+θ1+1).

    Proof. We show that both functionals F1 and F2 are invariant with respect to the pair (T(h),T(h)) of representation operators. Indeed, in view of homogeneity of u and v,

    F2(T(h(τ))[u],T(h(τ))[v])=F1(T(h(τ))[u],T(h(τ))[v])=u(eτ+eτα212,eτeτα212,α1)×v(eτ+eτα212,eτeτα212,α1)dα1=u(1+e2τα212,1e2τα212,eτα1)×v(1+e2τα212,1e2τα212,eτα1)eτdα1=F1(u,v)=F2(u,v).

    Since T is a homomorphism (of G to the group GL(D) of linear operators of D with trivial kernels) and H1 is an one-parameter subetaoup (that is, h11(θ1)=h1(θ1)), we have

    tρ,+,ˆρ,+(h1(θ1)h(τ))=12πF1(T(h1(θ1)h(τ))[fρ,+],fˆρ,+)=1π(1(α1θ1)2)σ+iˆρ+(e2τα21)σ+1+iρ+(eτ+α1)2iρ(1+α1θ1)2iˆρdα1=1πΩ(1+α1θ1)σiˆρ(1α1+θ1)σ+iˆρ(eτ+α1)σ+1iρ(eτα1)σ+1+iρdα1, (8.1)

    where Ω is the intersection of the segments |α1|eτ and [θ11,θ1+1]. Under the assumption eτ<|θ1±1| (Figure 1), this integral can be evaluated by using a known integral formula (see [10,Entry 3.211]): For min{(μ),(ν)}>0,

    10xμ1(1x)ν1(1ux)κ(1vx)ωdx=B(μ,ν)F1(μ,κ,ω;μ+ν;u,v).

    Also,

    tρ,+,ˆρ,+(h1(θ1)h(τ))=12πFi(T(h1(θ1)h(τ))[fρ,+],fˆρ,+)=12πFi(T(h(τ))[fρ,+],T(h11(θ1))[fˆρ,+]=12πcˆρ,+,λF1(T(h(τ))[fρ,+],T(h11(θ1)[fλ])dλ.

    We thus have

    tρ,+,ˆρ,+(h1(θ1)h(τ))=exp(iθ1λτστ)2σπcˆρ,+,λdλ×(e2τα21)σ1iρ+(eτ+α1)2iρexp(iλα1)dα1=exp(iθ1λτστ)2σ1πcˆρ,+,λexp(iλeτ)dλ×2eτ0tσ1+iρ(2eτt)σ1iρexp(iλt)dt.

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

    tρ,+,ˆρ,+(h1(θ1)h(τ))=2exp(iθ1λ+πρ2)π|Γ(iρσ)|1cˆρ,+,λFσ1(ρ,λeτ)dλ.

    Using (3.5), we have

    tρ,+,ˆρ,+(h1(θ1)h(τ))=2π2exp(π(ρ+ˆρ)2)|Γ(σ+iρ)Γ(σ+iˆρ)|×λ2σexp(iθ1λ)Fσ1(ˆρ,λ)Fσ1(ρ,eτλ)dλ. (8.2)

    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 24, 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<σ<0 and |θ1±1|<eτ. Then

    λ2σexp(iθ1λ)Fσ1(ˆρ,λ)Fσ1(ρ,eτλ)dλ=4σπ(eτ+θ11)σ1+iρB(σ+1iˆρ,σ+1+iˆρ)(eτ+1θ1)σ+1+iρexp(π(ρ+ˆρ)2)|Γ(σ+iρ)Γ(σ+iˆρ)|×F1(σ+1iˆρ,σ+1iρ,σ+1+iρ;2σ+2;21θ1eτ,21θ1+eτ).

    Theorem 8.3. Let 1<σ<0, |θ1+eτ|<1, and |θ1+1|<eτ. Then

    λ2σexp(iθ1λ)Fσ1(ˆρ,λ)Fσ1(ρ,eτλ)dλ=π(1+θ1+eτ)i(ρ+ˆρ)(1θ1eτ)σiˆρB(σ+iρ,σ+1+iˆρ)2σ+2+iρexp(π(ρ+ˆρ)2+τ(1+σ+iρ))|Γ(σ+iρ)Γ(σ+iˆρ)|×F1(σ+iρ,σ+iˆρ,σ+1+iρ;1+i(ρ+ˆρ);eτ+θ1+1eτ+θ11,eτ(θ1+1)+12).

    Theorem 8.4. Let 1<σ<0, |eτθ1|<1, and |θ11|<eτ. Then

    λ2σexp(iθ1λ)Fσ1(ˆρ,λ)Fσ1(ρ,eτλ)dλ=21+iˆρπ(eτ+θ11)σ1iρB(σ+1iˆρ,σiρ)(1+eτθ1)i(ρ+ˆρ)exp(π(ρ+ˆρ)2)|Γ(σ+iρ)Γ(σ+iˆρ)|×F1(σ+1iˆρ,σ+1iρ,σiˆρ;1i(ρ+ˆρ);1+eτθ11eτθ1,1+eτθ12).

    Remark 3. In Theorems 8.1–8.4, the Appell function F1 can be rewritten in terms of the hypergeometric function 2F1 by using the following relation (see, e.g., [21,Entry 7.2.4.(63)])

    F1(a,b,ˆb;b+ˆb;w,z)=(1z)a2F1(a,b;b+ˆb;wz1z).

    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

    0x2l+j+1exp((1+ω)x)1F1(1+ln;2l+2;2x)dx (9.1)

    is equal up to multiplicative constant to

    (1+ω)22ljΓ(2+2l+j)2F1[1+ln,2+2l+j2+2l21+ω],

    where l,jN0 and ω>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,nN0) expressed in terms of Appell F2 function:

    0xl12exp(2x)Lln(x)Llm(x)dx=(n+l)!(m+l)!Γ(l+12)2l+12n!m!(l!)2F2(l+12;nm;l+1,l+1;12,12), (9.2)

    which may be also expressed in terms of 3F2 (see, e.g., [20,Entry 2.19.14.(9)] and [28]). Since L2l+1n(x) up to multiplicative constant coincides with 1F1(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 Fσ(ρ,λ). 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 cρ,±,λ and cλ,ρ,± (kernels of corresponding integral operators) of the transformations B1B2 and B2B1 can be expressed in terms of Coulomb wave function Fσ(ρ,λ). In all these theorems (except Theorem 5.1), we have used only integrals of the following forms

    cρ,+,λdλandcλ,ρ,+dρ.

    Analogous integral transforms to those in this paper, containing kernels cρ,,λ and cλ,ρ,, can yield similar results. Also these kernels were considered for two dual cases σ=12±14 in [23].

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

    The authors declare that they have no conflict of interest.



    [1] G. E. Andrews, R. Askey and R. Roy, Special Functions, Cambridge University Press, Cambridge, 1999.
    [2] Z. Bentalha, Representations of the Coulomb matrix elements by means of Appell hypergeometric function F2, Math. Phys. Anal. Geom., 21 (2018), 10.
    [3] J. W. Brown and R. V. Churchill, Complex Variables and Applications, Sixth Edi., McGraw-Hill International Editions, 1996.
    [4] A. R. Curtis, Coulomb Wave Functions, Cambridge University Press, Cambridge, 1964.
    [5] A. Dzieciol, S. Yngve and P. O. Froman, Coulomb wave functions with complex values of the variable and the parameters, J. Math. Phys., 40 (1999), 6145-6166. doi: 10.1063/1.533083
    [6] A. Erdélyi, M. Kennedy and J. L. McGregory, Asymptotic Forms of Coulomb Wave Functions, Part I, California Institute of Technology, Pasadena, 1955.
    [7] A. Erdélyi, W. Magnus, F. Oberhettinger, et al. Higher Transcendental Functions, McGraw-Hill Book Company, New York, Toronto and London, 1953.
    [8] D. Gaspard, Connection formulas between Coulomb wave functions, J. Math. Phys., 59 (2018), 112104.
    [9] I. M. Gel'fand and G. E. Shilov, Generalized Functions: AMS Chelsea Publishing, 2016.
    [10] I. S. Gradshteyn and I. M. Ryzhik, Table of Integrals, Series, and Products, Academic Press, 2007.
    [11] D. Hartree, The Calculation of Atomic Structures, Ney York, Wiley, 1957.
    [12] N. Hayek, Estudio de la ecuación diferencial xy'' + (ν+ 1) y' + y = 0 y de sus aplicaciones, Collect. Math., 18 (1966-1967), 57-174.
    [13] N. Hayek, Sobre la transformación de Hankel, in Actas de la VIII Reunión Anual de Matemáticos Epañoles, (1967), 47-60.
    [14] H. Jeffreys and B. Jeffreys, Methods of Mathematical Phisics, Cambridge University Press, Cambridge, 1956.
    [15] D. A. Morales, On the evaluation of integrals with Coulomb Sturmian radial functions, J. Math. Chem., 54 (2016), 682-689. doi: 10.1007/s10910-015-0588-1
    [16] J. M. R. Méndez Pérez and M. M. Socas Robayna, A pair of generalized Hankel-Clifford transformations and their applications, J. Math. Anal. Appl., 154 (1991), 543-557. doi: 10.1016/0022-247X(91)90057-7
    [17] A. F. Nikiforov and V. B. Uvarov, Special Functions of Mathematical Physics, Birkhauser, Basel, Boston, 1988.
    [18] F. W. J. Olver, D. W. Lozier, R. F. Boisvert, et al. NIST Handbook of Mathematical Functions hardback and CD-ROM, Cambridge University Press, 2010.
    [19] A. P. Prudnikov, Yu. A. Brychkov and O. I. Marichev, Integrals and Series, Vol. 1: Elementary Functions, OPA (Overseas Publishers Association), Amsterdam B. V. Published under the license of Gordon and Breach Science Publishers, 1986.
    [20] A. P. Prudnikov, Yu. A. Brychkov and O. I. Marichev, Integrals and Series, Vol. 2: Special Functions, OPA (Overseas Publishers Association), Amsterdam B. V. Published under the license of Gordon and Breach Science Publishers, 1986.
    [21] A. P. Prudnikov, Yu. A. Brychkov and O. I. Marichev, Integrals and Series, Vol. 3: More Special Functions, OPA (Overseas Publishers Association), Amsterdam B. V. Published under the license of Gordon and Breach Science Publishers, 1986.
    [22] E. D. Rainville, Special Functions, Macmillan Company, New York, 1960; Reprinted by Chelsea Publishing Company, Bronx, New York, 1971.
    [23] I. A. Shilin, On some integral transforms of Coulomb functions related to three-dimensional proper Lorentz group, arXiv: 1904.03729.
    [24] I. A. Shilin and J. Choi, Certain relations between Bessel and Whittaker functions related to some diagonal and block-diagonal 3×3-matrices, J. Nonlinear Sci. Appl., 10 (2017), 560-574. doi: 10.22436/jnsa.010.02.20
    [25] I. A. Shilin and J. Choi, Certain connections between the spherical and hyperbolic bases on the cone and formulas for related special functions, Integr. Transf. Spec. F., 25 (2014), 374-383. doi: 10.1080/10652469.2013.860454
    [26] I. A. Shilin and J. Choi, Some connections between the spherical and parabolic bases on the cone expressed in terms of the Macdonald function, Abs. Appl. Anal., 2014 (2014), 741079.
    [27] H. M. Srivastava and J. Choi, Zeta and q-Zeta Functions and Associated Series and Integrals, Elsevier Science Publishers, Amsterdam, London and New York, 2012.
    [28] H. M. Srivastava, H. A. Mavromatis and R. S. Alassar, Remarks on some associated Laguerre integral results, Appl. Math. Lett., 16 (2003), 1131-1136. doi: 10.1016/S0893-9659(03)90106-6
    [29] I. J. Thompson and A. R. Barnett, Coulomb and Bessel functions of complex arguments and order, J. Comput. Phys., 64 (1986), 490-509. doi: 10.1016/0021-9991(86)90046-X
    [30] N. J. Vilenkin and A. U. Klimyk, Representations of Lie Groups and Special Functions, Vol. 2, Dordrecht, Kluwer Academic Publishers, 1993.
    [31] N. J. Vilenkin and M. A. Sleinikova, Integral relations for the Whittakers functions and the representations of the three-dimensional Lorentz group, Mathematics of the USSR-Sbornik, 10 (1970), 173-180. doi: 10.1070/SM1970v010n02ABEH001593
    [32] R. F. Wehrhahn, Y. F. Smirnov and A. M. Shirokov, Symmetry scattering on the hyperboloid S O(2, 1)/S O(2) in different coordinate systems, J. Math. Phys., 33 (1992), 2384-2389. doi: 10.1063/1.529979
  • This article has been cited by:

    1. I. A. Shilin, J. Choi, Method of Continual Addition Theorems and Integral Relations between the Coulomb Functions and the Appell Function F1, 2022, 62, 0965-5425, 1486, 10.1134/S0965542522090068
    2. A. Belafhal, S. Chib, F. Khannous, T. Usman, Evaluation of integral transforms using special functions with applications to biological tissues, 2021, 40, 2238-3603, 10.1007/s40314-021-01542-2
    3. I. A. Shilin, J. Choi, Maximal subalgebras in $${{\mathfrak {s}}}{{\mathfrak {o}}}(2,1)$$, addition theorems and Bessel–Clifford functions, 2023, 31, 0971-3611, 719, 10.1007/s41478-022-00435-9
    4. J. Choi, I. A. Shilin, On Changing Between Bases of the Space of Representations of Group SO(2,2), 2021, 61, 0965-5425, 1219, 10.1134/S0965542521080066
  • Reader Comments
  • © 2020 the Author(s), licensee AIMS Press. This is an open access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0)
通讯作者: 陈斌, bchen63@163.com
  • 1. 

    沈阳化工大学材料科学与工程学院 沈阳 110142

  1. 本站搜索
  2. 百度学术搜索
  3. 万方数据库搜索
  4. CNKI搜索

Metrics

Article views(4166) PDF downloads(254) Cited by(4)

Figures and Tables

Figures(4)

Other Articles By Authors

/

DownLoad:  Full-Size Img  PowerPoint
Return
Return

Catalog