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ЧЕБЫШЕВСКИЙ СБОРНИК
Том 20. Выпуск 3.
УДК 517.444, 517.588 DOI 10.22405/2226-8383-2019-20-3-349-360
Об одной сумме интегральных преобразований Ганкеля^Клиффорда функций Уиттекера
Джунесанг Чой, А. И. Нижников, И. А. Шилин
Джунесанг Чой — доктор наук, заслуженный профессор Донггукского университета (г. Кён-джу, Республика Корея). e-mail: [email protected]
Нижников Александр Иванович — кандидат физико-математических наук, профессор, заведующий кафедрой прикладной математики, информатики и ИТ, Московский педагогический государственный университет (г. Москва). e-mail: ainizhnikov@mail. ru
Шилин Илья Анатольевич — кандидат физико-математических наук, доцент, доцент кафедры высшей математики НИУ МЭИ и кафедры алгебры, Московский педагогический государственный университет (г. Москва). e-mail: [email protected]
Аннотация
В статье fll] авторами рассматривалась реализация Т представления группы SO(2, 2) в одном пространстве однородных функций, заданных на 2 х 4-матрицах. Настоящее продолжение этой статьи посвящено вычислению матричных элементов тождественного оператора Т(е) и операторов представления Т(д) для подходящих элементов д группы относительно смешанного базиса, соответствующего двум различным базисам пространства представления, и вычислению некоторых несобственных интегралов, содержащих произведение функций Бесселя-Клиффорда и Уиттекера. Полученные результаты могут быть переписаны на языке интегральных преобразований Ганкеля-Клиффорда и их аналога. Первое и второе преобразования Ганкеля-Клиффорда, введенные сооответственно Хайе-ком и Перезом-Робайной, играют важную роль в теории дифференциальных операторов дробного порядка (см., например, [6, 8]). Близкий результат получен авторами недавно [12] для регулярной кулоновской функции.
Ключевые слова: группа SO(2,2), матричные элементы представления, интегральные преобразования Ганкеля-Клиффорда, интегральное преобразование Макдональда-Клиффорда, функции Уиттекера, функции Бесселя-Клиффорда.
Библиография: 15 названий. Для цитирования:
Джунесанг Чой, А. И. Нижников, И. А. Шилин. Об одной сумме интегральных преобразований Ганкеля-Клиффорда функций Уиттекера // Чебышевский сборник, 2019, т. 20, вып. 3, с. 349-360.
CHEBYSHEVSKII SBORNIK Vol. 20. No. 3.
UDC 517.444, 517.588 DOI 10.22405/2226-8383-2019-20-3-349-360
On one sum of Hankel-Clifford integral transforms of Whittaker
functions
J. Choi, A. I. Nizhnikov, I. A. Shilin
Junesang Choi — Doctor of Sciences, Emeritus Professor of Dangguo University (Gveongju,
Republic of Korea).
e-mail: [email protected]
Nizhnikov Alexander Ivanovich — candidate of physical and mathematical Sciences, Professor, Head of the Department of Applied Mathematics, Computer Science and IT, Moscow State Pedagogical University (Moscow). e-mail: ainizhnikovQmail. ru
Shilin Ilya Anatolevich — candidate of physical and mathematical Sciences, Associate Professor, Department of Higher Mathematics, NRU MEI and the Department of Algebra, Moscow State Pedagogical University (Moscow). e-mail: [email protected],
Abstract
In [11], the authors considered the realization T of SO(2,2)-representation in a space of homogeneous functions on 2 x 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-Cliffcrd integral transforms and their analogue. The first and the second Hankel-Clifford transforms introduced by Hayek and Perez-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].
Keywords: group SO(2,2), matrix elements of representation, Hankel-Clifford integral transform, Macdonald-Clifford integral transform, Whittaker functions, Bessel-Clifford functions.
Bibliography: 15 titles. For citation:
J. Choi, A. I. Nizhnikov, I. A. Shilin, 2019, "On one sum of Hankel-Clifford integral transforms of Whittaker functions" , Chebyshevskii sbornik, vol. 20, no. 3, pp. 349-360.
1. Introduction and preliminaries
We recall the definitions and notations in [11]. The group SO(2, 2), which preserves the quadratic form £ defined in R4 whose matrix with respect to the canonical basis is a diagonal matrix e2,2 = diag(1,1, —1, — 1) which is called the split orthogonal group and consists of real 4 x 4 matrices g satisfying the equality ge2,2 gt = e2,2. 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 No := N U {0}. Let L be a real linear space
consisting of real 2 x 4 matrices. We define the cone A in L by the subset of matrices of rank 2 satisfying the equation x e2,2xt = diag(0,0). Let L be the complex linear space consisting of infinitely differentiable functions defined on A and satisfying the equality f(bx) = |bn|<J1 1b22|°"2 f(x)
for a fixed pair ( a\,a2) £ C2 and arbitrary non-degenerate matrix
= 1 0
e consider
21 22
the SO(2, 2)-representation T in L defined by formula T(<jf)[/(x)] = f(xg). 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 wi of A consisting of matrices
o \ ( cos« -sin« cos Pi -sin^i \ x(«,Pi) = . ■ a a I («1, A £ [0,2^)). 1
y sin« cos« sin A cos Pi J
In particular, they [11] showed that for any x £ A there are a low triangular non-degenerate 2 x 2-matrix b and x £ w such that x = bx. If Li is the linear space of restrictions of functions f £ L on wi, we can realize the representations T as the same representation in Li. They also showed that the function f = exp(ipi«) exp(ig1P1) defined on w\ does not belong to f = exp(ipi«) exp(ig1P1) if and only if the sum pi + q\ is not divisible by 2, and defined the canonical basis
Bi = {fpi,qi(«i,Pi)=exp(ipi«) exp(ig1P1) | pi, ql £ Z,pi + qi = ta(od2)},
which is orthonormal with respect to the scalar product
1 C 2ir c 2ir
f*9 = 4^2/ I /(«i,Pi )g(<xi,Pi )d«i dPi
/0 J 0
in L i. Writing Cartan decomposition g = g ig2g3 for an arbitrary element g of the group SO(2, 2), where g\, g3 £ SO(2) x SO(2) and
n a exp( 0 diag(^ v)\
92 £ eXHdiag(/x, u) 0 ),
they showed that in case \u| = the matrix elements of the linear operator T(g) with respect to B\ can be written as a product of four exponential functions, depending respectively on four parameters of the rotations g\ and g3, and two Gaussian hvpergeometric functions depending (respectively) on (tanh ^ )2.
The parabolic section w2 of A has been defined as the subset consisting of matrices
_ , \ ( 1 «2 cos P2 - «2 sin P2 sin P2 + «2 cos P2 \ x(«2,P2) = n 2 - a a ,
y 0 1 — sin p2 cos p2 J
where a2 £ R and p2 £ [0,2n).li L2 is the linear space of restrictions of functions f £ L on w2,
L2
B2 = {fP2,q2 («2, P2) = exp(ip2«2) exp(i(?2P2) | P2 £ R, <?2 £ [0,2^)}.
T( ) L2
integral operators whose kernels can be described in terms of some Bessel functions.
2. Two bases in L and our purpose
(x\ x\ \
m n inside the matrix x £ A bv Am n and introduce x2m x2n) " ,
the basis
B = {fpi,qi (x) | pi,qi £ Z,p 1 +qi= ft(d2)}
in £, consisting of functions
/pi.il (a) = (^11 + ^2)-2- |A1;2|CT2-91 (xn - tel2)P1-91 (Ai,3 + iAM)9i.
Obviously the restriction of /Pl,9l to Wi coincides with i91 fPl—qi:
fpi.qi Li = i?1 fpi,-qi. (2)
In this paper, we also use the basis
B2 = [fP2 ,q2 (x) | P2 e R, Q2 e Z},
where
W(*) = 1|CT1-CT2 |Ai,2|CT2-92 (Ai,s + iAM)® exp ^^.
X\ 1
It is easy to see that fP2,q2 is an extension of the function i92/P2,Ç2 to A.
Let span(/Pi)gi, /pi,çi) be the subspace in L1. It is invariant with respect to the linear operator T(g) (for some fixed g) and its basis vectors fPi,qi and /pi,çi which are not eigenfunctions of this operator. In this paper, we aim to establish dependence between the matrix elements of the operator T(g) with respect to the ordinary basis B2 and the mixed basis B2\B1 and matrix elements of the operator id = T(e) with respect to B2\B1. Choosing here the group element as follows:
h* =diag((j , diag(1,1)) ,
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 fl] 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 SO(2,2) and representation operators corresponding to some diagonal and block-diagonal matrices which belong to the split orthogonal group.
3. Transitive subgroups and invariant measures
It is obvious that w1 is an orbit of the sub group H1 ~ SO (2) x SO(2), consisting of the matrices
( cos <p1 — sin (f1
h1(iß1,1p1) =
0 \
Sin (fi1
0 0
cos (fi1
0 cos ip1 — sin ip1 0 sin ip1 cos ip1 /
Let us consider the matrices
h(Q) =
(2 q 0 Q\
— Q 2 Q 0
0 Q 2 Q
V Q 0 — Q 2 J
and the points x(a2, (32) and x(a2,$2) belong to the subset w2 of A. Since the matrix elements hij(q) of the matrix h( g) satisfy the equalities
hn( q) + hi2( q) — hi3( q) — ha( q) = 4sign(2.5 — i ),
0
0
0
we get ^h( q) g SO(2,2). It is easy to see that
x(a,2,fÎ2)hi(0, -fc)= x(a>2,0) = ^ 1 1 ^ , 1x(a2, 0)h(aa2 — a2) = x(aa2, 0)
and
x(aa2, 0)h\(0,fa2) = x(a2,f2).
Thus the matrix
h2(cx2 — a2,f2 — f 2) = 2 hi(0, —f2 )h(a2 — «2)hi(0, f2)
transforms the point x(a2,f2) into the point x(a2,f2). It means that the subgroup
H2 = ^2(^2) | ^2 g R, Ф2 g [0; 2^)}
acts transitively on w2. Also we find that dw2 = da1 df2 is an #2-invariant measure on w2. It is found that /P2,Ç2 is an eigenfunction of the linear operator T(h2(<p2,^2)), more exactly,
T (h2((fi2^2))[ fP2,q2 ] = exP(i) fP2 m .
Similarly dw1 = da1 df 1 ш an fl1-invariant measure от the spherical section ш1 and fPl,qi is an eigenfunction of the operator T(h1((p1,^1)), namely
T (h1(<p1,^1))[fPi,gi ] = exp(ip 1^1)exp(ig 1Ф1) fPuqi.
4. Functionals Fi and F2 and assorted spaces
Let us introduce the following bilinear functionals defined on the direct product L x L* of two representation spaces:
Fi : (u, v^)—^JJu(ai, fi ) v\ai,fi)dui (г = 1, 2),
Ui
where the functions on L* аде (a^, a£)-homogeneous.
Lemma 1. The functional F1 coincides with F2 if and only if
a1 — a2 = 02 — 01 — 4. (3)
Доказательство. It was shown in [11] that for any point x g Л there are a low triangular non-degenerate 2 x 2-matrix bx and the point x(a2,f2)x g w2 such that x = bxx(a2, f2)x, and b11 = x11, b21 = x21, a2 = x^- In particular, to ад arbitrary point (1) belonging to W]^, we have
'cos a1 0
/cos a1 0 \ ysin a1 sec a1j ,
bx(ai,l31) x sin «i sec
and, therefore, the correspondence x(«i,(i) ^ x(«2, P2)x(ai,i3i) is one-to-one. Since the operands u £ ^d v* £ L* of the functional Fi are (ai,a2)- and (a^, ^-homogeneous, respectively, we have
U(«1,A) vVi,Pi) = (cos «1 r+^-^-^U^,^) V(«2,(32).
Considering that Wi-coordinates depend on ^-coordinates according to the formulae
a = arccos | ±—. 1 I , A = arctan
V VT+oiJ
a2 cos /2 + sin /2 л/ТГа|Г ^ ^™"a2 sin /2 - cos/21
d(a\,a2)
d(a2, a2)
= (l + a2)
21
we get
cr /7 / a \ a\A A a /7 u(a2,@2) v'(a2,^2)da2 d^2
f1(u,v ) = JJ u(ai,Pi) v (ai,^) d«i dA = JJ (1 + -^+4 •
Wl W2
It is clear that the equality Fi = F2 is equivalent to a1 + a\ — a2 — &2 +4 = 0. □
Further we assume that representation spaces £ and £• are mutually assorted, i.e., the pair
(a1 ,a2) f°r the representation space £• is connected with the pair (a1,a2) for £ bv the equalitv i2
5. Matrix elements of the B^ —> basis transformation
Let us express the function fP2 q2 as a linear combination of the functions belonging to the basis
B-2:
fp2,12 CPl,Ql,P2,Q2 fpi,qi (x). (4)
pi, qiEZ
In view of Lemma 1, we have
F2(f P2,q2, fpl,gi
)= Y1 CPi ,9i,P2,92 F1( fPi,gi, fpiAi ).
Pi,qi<EZ
Since
/П fK
/ exp(i[pi + j6i]ai) exp(i[g 1 +qi]fti)dai d/i,
-Ж J —Ж
' — Ж J --K
we obtain
cpi,qi,p2,q2 = f2(fP2m, f—pi,—qi).
We compute the matrix elements of the linear operator acting in £• and transforming the basis B^ into asserted bv the following theorem.
Теорема 1. Let p\, q\, q2 e Z, p2 e R \ {0}, and Re(ai - a2) > -3. Then
3 _i . . a1 -<72 +1
cpi,qi ,P2,q2 = 2 2 ж "Я1,—Я2 \'Р2\ 2
г I -02 + (qi - Pi )sign P2 ^
—1
WM -Pi) signp2 a2-ai-3 (2|P21) , 2 ' 2
where Г is the gamma function, is the Whittaker function of the second kind, and 5s,t is the Kronecker symbol.
Доказательство. Using iterated integrals for
/OO fK _ _ ,
(1 + a2)CT2-CT1-P1 91 —2 (l - ia2)pi—
-OJ—Ж
x exp (i['2a2 + (qi + 92)/2]) da2 d/2,
x
we find that cP1 qi P2 q2 can be expressed as an exponential Fourier transform:
/<x
(1 + i«)
-oo
• \a2-IJ1+P1-n 2 !• \ 1
x (1 - i a2) 2 exp(ip2a2) da2.
(6)
-2-'-2
(7)
The integral in (6) can be evaluated by the following known formulae (see, e.g., [4, Entry 3.2.(12)])
/<x
( « + ix)-211 (3 - ix)-2" exp(—ixy) dx = 2tt(« + p)-v-1 [r(2z/)]-i
-<x
x exp (i3-^) f+1-i Wv-it 2-„-„([« + 3]y) + v) > 1, min{Re(«), Re(3)} > 0; y £ R+^J
/<x
( « + ix)-21 (3 - ix)-2v exp(-ixy) dx = 2^т(a + p)-1-v [r(2^)]-i
-<x
x ) (-yy+1-i Wi-Vt2-v-i(-[« + 3]y)
+ v) > 1, min{Re(«), Re(3)} > 0; y £ R-^ .
and
(8)
□
6. Matrix elements of the operator T^(h*) with respect to
For any g £ SO(2, 2), let t*2 q2 -2 —(g) be a matrix element of the linear operator T•(g) with respect to the basis that is,
fOO
T^(9)[fP2,q2]=^/ t'P2,q2,i>2A2 (9) f^M dp2. (9)
,2ezJ 0
In view of Lemma 1, we get
f <x
Fi(T(9)[ fP2 m ],fp2 m ) = ^ / tP2,q2,P2,q2 (9) F2( fp2,q2 , fp2,q2 )dp2
0
q2& 0
f <x
= 4r2 tP>2,q2,P2,-q2 (9) 6(p2 + p2) ^ 0
where S(p2 + p2) is the (-p2)-delaved Dirac delta function. We therefore have
tp2,q2,P2,q_2(9) = 4^2 Fi(T (^)[fp2,q2], f-i>2,-q2).
In Theorem 2, we show that the matrix elements tp r, p - (h*) can be described in terms of ' P2, q2 ,P2, q2y '
either Bessel-Clifford functions of the first kind
(z) = z-2 Jv (2^) (10)
or modified Bessel-Clifford functions of the second kind
Kv(z) = z-K2KV (2^) (11)
depending on sign(p2p2) in both cases (see [2]). Here Ju and Kv are Bessel functions of the first kind and modified Bessel functions of the second kind, respectively, (see, e.g., [13, Chapter 9]).
and
Теорема 2. Let, p2,p2 e R \ {0} q2, <?2 e [0,2ж), and 2 < Re( a2 - a1) < 4. Then
^«A,7(h*) = -2\P2Г ——3 sin to-r1^ Kb —.i—3(-P2P2) (12)
2
('2 p2 e R—)
(h*) = ^ <W cos ( ^2 - ^
p2,^,p2 A2V1 > 2 2 (13)
x [|p2|CTi—+3C.i—+3(P2P2) - |P2Г ———.i—3(P2P2)]
(P2 P2 e R+).
Доказательство. Since the right shift of the subset w2 by the matrix h* permutates the first and the second columns of any point belonging to w2, we have
T (h*)[ fp2,q2 (x)] = |Ж12Г—'72 |Ai,2|CT2 —^ (A2,3 + iA2,4)92 exp ( - .
V x12 J
In view of Lemma 1, the T^ (h*)-image of the restrict ion of /p2,92 to w2 is given as follows: T*(h*)[fp2,q2](a2,/2) = (-1)^ M"2 —CTi—4 exp(ig2/2) exp (-.
We obtain
^2,®,p2,7(h ) = 4^2 F2(t (h )[$ pirn], f—p2,—92)
i2^—<72 _4 /г p21\
= 4jt2 J exp(i[q2 - <?2]/2) d/2 J Ы^2 exp i -i P2a2 + — ) da2.
Considering here the principle value of the last integral and using the following known formulae (see, e.g., [9, Entires 2.5.24.4 and 2.5.24.7])
a
J xa—1 cos ^ ax + dx = ^ sin ^J—a(2^a&) - Ja(2Vab)
and
a
J xa—1 cos ^ax - — ^ dx = 2 ^ cos ^a^) Ka(2Vab)
(a, b e R+; |Re( a)| < l) , with the aid of (10) and (11), we complete the proof. □
7. Matrix elements of the operator T*(h*) with respect to
th.6 IXlliXGcl "bclSlS fp2 q2 | fp 1 Qi
From (2) and (9), we find
T (9)[fp2,q2 ] = I ^^ I tp2,q2,p2,72 (5') Cpi,9i,p2,7 dp2 ) fpi ,qi.
pi,gi€Z GZ J — O
oo
The expression in the brackets in (14) can be characterised as a matrix element of the operator T*(h*) with respect to the so-called mixed basis fP2,q21 fPl,qi (see, e.g., [3, p. 204]).
On the other hand, these matrix elements may be obtained in the following way. It is not hard to check that the linear subspace span(fPl,qi, f2qi-Pl,qi) in £i is invariant with respect to the linear operator T*(h*), namely, in view of (2),
T-(h*)[ fPl,qik] = T(h*)[fpim] = iPl+* exp(ibiai - QiPi) = iPl fPl,qiL. (15) TEOPEMA 3. Let pi, qi e Z such that pi + qi = 0r(od2), p2 e Rand 1 < Re(0) < 2. Then
ю
те r i
2 cos(07r) pt1 pt3e С3-2в(p2p2) _3 (2p2)
- 1cos(07) p2e_4 '[¿Г9 C2e_3(p>2p2) W41-P1 3 (2p2)
2 г (qi-pi - 9) 1
+ (-!)* 7 г (pi2- qi sin(0 7) p2_4 p\_e ^2e_3(p2i>2) W^ fi_ 3 (2p2)J dp2
Pl+n
= (-1) 2 W41 -P1 й 3 (2p2).
2 'u 2
Доказательство. From (14) and (15) we have
T )ifp2,q2] = cPi,Qi,P2,q2 T (9)[fpi,qi] = ^^ ^ cpi,qi,p2,q2 fpi,qi. (16)
Pl,qi<EZ Pi,qi&
Since the matrix element cPimP2A2 is equal to zero in case q1 = — q2 and the matrix element tp2 q2 p2 (ft*) is equal to zero in case q2 = q2, considering (14) and (16), we have
/те
tp2,_qi,p2 ,_qi ) CPi,qi,P2,_qi dp2 = ^ cPi ,qi,P2,_qi. (1^)
-те
Using, for (16), the results from Theorems 1 and 2, and letting в = a2_ai, we complete the proof. □
8. Concluding Remarks
Setting pi = qi = 0 in (14) and considering the following relation between Macdonald functions Kv and Whittaker functions W0,u (see, e.g., Entry [15, 7.8.8]):
K (2) = (7) 2 W0tV(z), (18)
K
[4]) of the linear combination of the Bessel-Clifford functions:
f^ r 1 ft-3 1 3-ft
- cos(07r) Pft 3 p45-3ft Cs-2ft(p2p2) - 2 cos(07) pf-4* pi C2ft-3(p2p2)
Jo ft 2 (19)
2 ( —l)ft ft i
sin(07r) plft-4/5 p2 K,2ft-3 (p2p2) Kft- 3 (p2)dp2 = K- 3 fo)
7 J 2 2
(p2 e R+, 1 < Re(0) < 2) .
Some similar results to those in Theorem 3 and formula (19) can be obtained from (17) in case p2 e R-.
Using the following three integral transformations:
(i) The first Hankel-Clifford integral transform (see, e.g., [7, Eq. (2.7)])
f <x
H^1)[/](A) = AM Ca(A\)f(X)d\ (A e R+) ; Jo
(ii) The second Hankel-Clifford integral transform (see, e.g., [5]; see also [7, Eq. (2.9)])
r <x
42)[/](A)= / Ca(A\)f(X)d\ (A e R+); o
(iii) The Macdonald-Clifford transform (see [12])
r <x
K<t[/](A)=/ MAA)/(A)dA (A e R+) ,
o
we can rewrite the identity in Theorem 3 in the following form:
cos(0 ж) 2
H(i)
pt™ Wn-P1 3 (2iP2) Ы - H$_3 3 (2P2) Ы
0 г (gi _pi _
4-30
W 51_ __
2 ^2
p2T3yWPi-qi 3 (2^2) Ы
2
+ 2 (-1)e Г -g) K
+ ж Г (^ - 0) 20_3
= (-1) ^ p4_3'Wpi-i 3 (2p2).
2 2
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6. Kirvakova, V. k, Hernanden Suarez, V. Bessel-Clifford third order differential operator and corresponding Laplace type integral transform // Dissertationes Mathematicae. 1995. Vol. 340. P. 143-161.
7. Méndez Pérez J.M.R., Socas Robavna M.M. A pair of generalized Hankel-Clifford transformations and their applications //J. Math. Anal. Appl. 1991. Vol. 154, №2, P. 543-557.
8. Paneva-Konovska J. Bessel type functions: Relations with integrals and derivatives of arbitrary orders // AIP Conference Proceedings. 2018. Vol. 2048. 050015. doi: 10.1063/1.5082114.
9. Prudnikov A. P., Brvchkov Yu.A., Marichev O.I. Integrals and Series, Vol. 1: Elementary Functions. OPA (Overseas Publishers Association), Amsterdam, 1986.
10. Shilin I. A., Choi J. Integral and series representations of special functions related to the group SO(2,2) // Ramanujan J. 2017. Vol. 44. №1. P. 133-153.
11. Shilin I. A., Choi J. On matrix elements of the SO(2, 2)-representation in a space of functions on 2 x 4-matrices // Integral Transforms Spec. Funct. 2018. Vol. 29. №10. P. 761-770.
12. Shilin I. A., Choi J. Some integrals involving Coulomb functions related to three-dimensional proper Lorentz group. (Submitted.)
13. Temme N. M. Special Functions: An Introduction to the Classical Functions of Mathematical Physics. John Wiley and Sons, New York, 1996.
14. Vilenkin N.Ya., Pavlvuk A. P. Representations of some semisimple Lie groups and special functions of the matrix argument // In: Group Theoretical Methods in Physics, vol. 1. Harwood Academic Publishers, Chur, London, Paris, New York, 1985.
15. Wang Z.X., Guo D.R. Special functions. World Scientific, Singapore, New Jersey, London, Hong Kong, 1989.
REFERENCES
1. Abramowitz, M. 1950, "Coulomb wave functions expressed in terms of Bessel-Clifford and Bessel functions", Stud. Appl. Math., vol. 29, no. 1-4, pp. 303-308.
2. Clifford, WT. K. 1882, "On Bessel's functions", In: Mathematical Papers. Oxford University Press, London, pp. 346-349.
3. Gilmore, R 2015, "Group theory", In: Mathematical Tools for Physicists. Wiley-VCH, Weinheim, pp. 159-210.
4. Erdélvi, A., Magnus, WT., Oberhettinger F. k Tricomi, F.G 1954. Tables of Intefral Transforms, Vol. I. McGraw-Hill Book Company, New York, Toronto and London.
5. Hayek, N. 1967, "Sobre la transformación de Hankel", Actas de la VIII Reunión Anual de Matemáticos Epañoles, pp. 47-60.
6. Kirvakova, V. k Hernanden Suarez, V. 1995, "Bessel-Clifford third order differential operator and corresponding Laplace type integral transform", Dissertationes Mathematicae, vol. 340, pp. 143-161.
7. Méndez Pérez, J.M.R. k Socas Robavna, M.M. 1991, "A pair of generalized Hankel-Clifford transformations and their applications", J. Math. Anal. Appl, vol. 154, no. 2, pp. 543-557.
8. Paneva-Konovska, J. 2018, "Bessel type functions: Relations with integrals and derivatives of arbitrary orders", AIP Conference Proceedings, vol. 2048, 050015, doi: 10.1063/1.5082114.
9. Prudnikov, A. P., Brvchkov, Yu.A. k Marichev, O.I 1986. Integrals and Series, Vol. 1: Elementary Functions. OPA (Overseas Publishers Association), Amsterdam.
10. Shilin, I. A. k Choi, J. 2017, "Integral and series representations of special functions related to S O(2, 2)
S O(2, 2)
2 x 4
12. Shilin, I.A. k Choi, J. 2019, "Some integrals involving Coulomb functions related to three-dimensional proper Lorentz group", submitted.
13. Temme, N. M 1996. Special Functions: An Introduction to the Classical Functions of Mathematical Physics. John Wiley & Sons, New York.
14. Vilenkin, N.Ya. &Pavlvuk, A. P. 1985, "Representations of some semisimple Lie groups and special functions of the matrix argument", In: Group Theoretical Methods in Physics, vol. 1. Harwood Academic Publishers, Chur, London, Paris k New York.
15. Wang, Z.X. k Guo, D.R. 1989, Special functions. World Scientific, Singapore, New Jersey, London k Hong Kong.
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