Ramanujan J DOI 10.1007/s11139-016-9799-8
Integral and series representations of special functions related to the group S O(2, 2) I. A. Shilin1,2 · Junesang Choi3
Received: 23 October 2015 / Accepted: 28 March 2016 © Springer Science+Business Media New York 2016
Abstract 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 S O(2, 2)-representation; Invariant properties of the bilinear integral functionals which are used to obtain the matrix elements of bases transforms operators. Keywords Whittaker function · Bessel functions · Generalized hypergeometric functions p Fq · Special orthogonal group S O(s, t) · Poisson transformation · Matrix elements of representation Mathematics Subject Classification 33C10 · 33C80 · 33B15 · 33C05
1 Introduction and preliminaries It is known that the irreducible representation of the special orthogonal group S O(s, t) can be realized as the left or right quasiregular representation in the space of infinitely
B
Junesang Choi
[email protected] I. A. Shilin
[email protected];
[email protected]
1
Department of Algebra, Moscow State Pedagogical University, Malaya Pirogovskaya 1, Moscow, Russia 119991
2
Department of Higher Mathematics, National Research University “Moscow Power Engineering Institute”, Krasnokazarmennaya 14, Moscow, Russia 111250
3
Department of Mathematics, Dongguk University, Gyeongju 38066, Republic of Korea
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I. A. Shilin and J. Choi
t t 2 = 0 and satisfy differentiable functions defined on the cone : i=1 xi2 − i=1 xs+i σ the condition of σ -homogeneity f (αx) = |α| f (x) for the fixed σ and an arbitrary α [10]. 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. The representation T in D maps a group element g into the non-degenerate linear operator T (g), which acts by the formula f (x) −→ f (g −1 x). Moreover, by means of homogeneity, it is possible to realize the above representation as the restriction of T to the linear subspace V of D, which consists of functions defined on a conic section intersecting all or almost all generatrices. Let γ be one of these conic sections and the subgroup H of S O(2, 2) acts transitively on γ . Throughout this paper, let C, R, and Z be the sets of complex numbers, real numbers and integers, respectively. In [3], the authors considered the following sections: γ1 = {(sin α1 , cos α1 , sin β1 , cos β1 ) | α1 , β1 ∈ [−π, π )} ; γ2 = {(cosh α2 sin β2 , cosh α2 cos β2 , sinh α2 , ±1) | α2 ∈ R, β2 ∈ [−π, π )} ; 1−α32 +β32 1+α32 −β32 γ3 = α3 , , β3 , α3 , β3 ∈ R . 2 2 The dimension of the corresponding group H is equal to the number of parameters of γi . Since H can be expressed as a direct product of two one-parameter subgroups, it implies that the eigenfunction of the linear operators T (g) (g ∈ H ) is a product of two exponential functions. These eigenfunctions form a basis in V , which can be easily continued to D via the σ -homogeneity and continuity of functions. As in [3], in this paper, we deal with the bases B1 , B2 , and B3 , which consist, respectively, of the functions σ − p1 −q1
f p1 ,q1 (x)
:= (x12 + x22 ) 2 (x2 + ix1 ) p1 (x4 + ix3 )q1 , iq2 p2 +1 σ −iq2 2 2 ∗ x1 + x2 + x3 (x2 + ix1 ) p2 (x12 + x22 )− 2 , f p2 ,q2 ,± (x) := (x4 )± f p∗∗3 ,q3 (x)
+q3 x3 ) := |x2 + x4 |σ exp i( p3xx21+x , 4
where p1 , q1 , p2 ∈ Z and q2 , p3 , q3 ∈ R and the homogeneous generalized functions (x)λ± are defined by (x)λ±
0, if sign x = ∓1 or x = 0, = λ |x| , if sign x = ±1.
. The similar bases in D• consist of the analogous functions f p•1 ,q1 , f p∗•2 ,q2 ,± and f p∗∗• 3 ,q3 We recall that the Hi -invariant measures on γi were given as follows: (dx)γ1 = dα1 dβ1 ; (dx)γ2 = cosh α2 dα2 dβ2 ; (dx)γ3 = dα3 dβ3 .
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Integral and series representations related to the group S O(2, 2)...
It is noted that, for the group S O(2, 1) in [14], the same situation arises. Yet, for S O(3, 1), the dimension of H1 and H2 , which act transitively on the sphere and hyperboloid, respectively, is equal to 3 and γ1 and γ2 depend on 2 parameters. So the corresponding basis function is a product of two exponential functions and one of the following well-known special functions of one variable, which is determined by corresponding conic section, Gegenbauer polynomial and Legendre function, respectively (see [15]). At the same time, the group H3 , whose homogeneous space is paraboloid, can be expressed in the form H3 = exp[R(e12 − e21 − e23 − e32 ) + R(e14 + e41 + e43 − e34 )], where ekl ≡ (ai j ) is the 4 × 4- matrix such that ai j =
1, if k = i and l = j, 0, if k = i or l = j.
So it is possible to choose the corresponding basis which consists of functions +q3 x3 ) |x1 + x4 |σ exp i( p3xx12+x . However, in [15], the authors chose the basis whose basis 4 function is a product of the exponential function and Bessel function. In this paper, we employ the approach used in [14] for the S O(2, 1), where we obtained the integral representations of the following type: cos(π σ ) 2
+∞
0
1 ˆ J−2σ −1 2 λλˆ − J2σ +1 2 λλˆ dλˆ λˆ − 2 W−k,σ + 1 (2λ) 2
+∞
) sin(π σ ) − π (−k−σ (k−σ )
=
(−1)k
λ
− 21
1 ˆ K 2σ +1 2 λλˆ dλˆ λˆ − 2 Wk,σ + 1 (2λ) 2
0
W−k,σ + 1 (2λ) (λ > 0) 2
and derive the generalization π −1
sin(π σ ) (−2σ − 1)
+{ (−σ + k)}−1 =
(−1)k
+∞
{ (−σ − k)}−1
+∞
0
λˆ σ W−k,σ + 1 (2λˆ ) dλˆ
λˆ σ Wk,σ + 1 (2λˆ ) dλˆ
0 22σ +1 (−2σ
2
− 1)−1 {B(k
− σ, −k − σ )}−1
2
−1 < (σ ) < − 21
of the following known formula (see, e.g., [7, Entry 6.516.16]): +∞
x μ K ν (ax) dx
0
= 2μ−1 a −μ−1
1+μ+ν 2
1+μ−ν 2
( (μ + 1 ± ν); (a) > 0).
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I. A. Shilin and J. Choi
We give a more detailed account of the approach here. In [3], we considered the following functionals: ˜ −→ C Fj : D × D defined by u(x) v(x) (dx)γ j ( j ∈ {1, 2, 3})
F j ((u, v)) = γj
˜ = D• . For any i, arbitrary functions u, v ∈ Bi and proved that F1 = F2 = F3 if D are mutually orthogonal with respect to Fi . Using these functionals, more exactly, computing (2π )−2 Fi (u, v) (u ∈ Bi• , v ∈ B j ), we evaluated the matrix elements of the linear operators, which transform Bi• into B j . Using these matrix elements, they derived some formulas for special functions. Let Symm M be the group of permutations of a nonempty finite set M. For instance, let us fix the permutations τ ∈ Symm {1, 2}, τ˜ ∈ Symm {3, 4} and consider the intersection γτ,τ˜ of by the hyperplane xτ (2) + xτ (4) = 0. It is easy to see that γ3 = γid,id . Considering the basis Bτ,τ˜ in D, which consists of functions f p∗∗3 ,q3 (τ, τ˜ , x) := (xτ (2) + xτ˜ (4) )σ exp
i( p3 xτ (1) + q3 xτ˜ (3) ) xτ (2) + xτ˜ (4)
( p3 , q3 ∈ R) ,
we see that the matrix elements of the linear operator B3 −→ Bτ,τ˜ coincide with the corresponding matrix elements of the linear operator T (g ), where g is the matrix whose first and second rows coincide with the τ (1)-th and τ (1)-th rows, respectively, and third and fourth rows coincide with τ˜ (1)-th and τ˜ (1)-th rows, respectively. Thereto, the operator T (g ) transforms the function u ∈ B1 into the function v ∈ B1 up to a multiplicative constant. With this, in Sects. 2 and 3, by using the same argument for the group S O(2, 1)) in [3], we obtain the integral representations of Whittaker functions Wμ,ν or their products and we also present a formula for the linear combination of 3 F2 functions. Let the point y = (y1 , y2 , y3 , y4 ) belongs to the hyperboloid H : y12 + y22 − y32 − 2 y4 = 1. Then the function p(x) := |x1 y1 + x2 y2 − x3 y3 − x4 y4 |−σ −2 (x = (x1 , x2 , x3 , x4 ) ∈ ) belongs to D• . Let T1 and T2 be representations of a group G in the linear spaces L 1 and L 2 , respectively. We recall that the linear operator S : L 1 −→ L 2 intertwines the representations T1 and T2 if, for each g ∈ G, the compositions ST1 (g) and T2 (g)S coincide. The linear operator u(x) p(x) (dx)γ ,
P : u −→ γ
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Integral and series representations related to the group S O(2, 2)...
which maps the function u ∈ D into the function p(u) on H, intertwines the representation T and the realization of the representation in the space of functions on the hyperboloid H (see [17]). It is noted that this operator P is called the Poisson transform, certain asymptotic properties of P were investigated in [1], and the problem concerning the generalization of the Funk–Hecke theorem for P−1 , called the Gelfand–Graev operator [9], was considered in [16]. It is also noted that P(u) = Fi (u, p), that is, P does not depend on a section γ . In Sect. 5, by using the Poisson transform, we obtain certain formulas for double series converging to either Macdonald function or difference of two Bessel functions. In Sect. 6, some simple consequences of the equality F2 = F3 are considered.
2 Matrices gε1 ,ε2 and gχ1 ,χ2 and relations induced by them Here we show that the restrictions of the representation T to the block-diagonal matrices ⎛
gε1 ,ε2
1 ⎜0 := ⎜ ⎝0 0
0 1 0 0
⎞ 0 0⎟ ⎟ (ε1 , ε2 ∈ {1, −1}) ε1 ⎠ 0
0 0 0 ε2
give double integral representations of products of two Whittaker functions Wκ
1 p1 ,
1+σ +κ2 q1 2
(2μ) and Wλ
1 q1 ,
1+σ +λ2 p1 2
(2μ) (κi , λi ∈ {1, −1}) .
Let ε := ε1 ε2 . For arbitrary p3 and q3 , we express the function f p∗∗3 ,q3 through the basis B1 : f p∗∗3 ,q3 (x) =
c p3 ,q3 , p1 ,q1 f p1 ,q1 (x)
(1)
p1 ∈Z q1 ∈Z
and express the function T (gε1 ,ε2 )[ f p∗∗3 ,q3 ] through the basis B3 : T (gε1 ,ε2 )[ f p∗∗3 ,q3 ](x) =
t p3 ,q3 , pˆ3 ,qˆ3 (gε1 ,ε2 ) f p∗∗ ˆ 3 ,qˆ3 (x) d pˆ 3 dqˆ3 ,
(2)
R2
where t p3 ,q3 , pˆ3 ,qˆ3 (g) is the matrix element of the linear operator T (g) with respect to the basis B3 . By combining these expressions and noting that T (gε1 ,ε2 )[ f p1 ,q1 ] = (ε1 i)q1 f p1 ,−εq1 , we have ⎛ T (gε1 ,ε2 )[ f p∗∗3 ,q3 ] =
⎜ ⎝ p1 ∈Z a1 ∈Z
⎞ ⎟ t p3 ,q3 , pˆ3 ,qˆ3 (gε1 ,ε2 ) c pˆ3 ,qˆ3 , p1 ,q1 d pˆ 3 dqˆ3 ⎠ f p1 ,q1
R2
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I. A. Shilin and J. Choi
and T (gε1 ,ε2 )[ f p∗∗3 ,q3 ] = =
p 1 ∈Z a 1 ∈Z p1 ∈Z a1 ∈Z
c p3 ,q3 , p1 ,q1 T (gε1 ,ε2 )[ f p1 ,q1 ] c p3 ,q3 , p1 ,−εq1 (ε1 i)−εq1 f p1 ,q1 ,
which leads to t p3 ,q3 , pˆ3 ,qˆ3 (gε1 ,ε2 ) c pˆ3 ,qˆ3 , p1 ,q1 d pˆ 3 dqˆ3 = (ε1 i)−εq1 c p3 ,q3 , p1 ,−εq1 .
(3)
R2
The matrix elements c•p3 ,q3 , p1 ,q1 of the linear operator of the space D• that transforms the basis B1• into B3• have been computed in [3] under the condition that
(σ − p1 ) < 0, (σ + q1 ) < 0, | p3 | = |q3 |. Here we represent them for the case of the space D in the form c p3 ,q3 , p1 ,q1 = 2
p1 −q1 2
c( ˜ p3 + q3 , q3 − p3 , p1 , q1 ),
where c( ˜ p3 + q 3 , q 3 − p3 , p1 , q 1 ) σ +q1 +1
σ − p1 +1
:= | p3 + q3 | 2 | p3 − q3 | 2 −1 ( p3 +q3 ) p1 +q1 sign ( p3 −q3 )−σ × − σ +q1 + p1 sign − 1 − 1 2 2 ×W p sign ( p +q ),− σ +q1 +1 (2| p3 + q3 |) Wq sign (q − p ), p1 −σ −1 (2| p3 − q3 |). 1
3
3
1
2
3
3
2
From (2), for any i ∈ {1, 2, 3}, we have
•∗∗ )= Fi (T (gε1 ,ε2 )[ f p∗∗3 ,q3 ], f τ,ρ
•∗∗ t p3 ,q3 , pˆ3 ,qˆ3 (gε1 ,ε2 ) Fi (T (g1 )[ f p∗∗ ˆ 3 ,qˆ3 ], f τ,ρ ) d pˆ 3 dqˆ3 .
R2
In particular, •∗∗ ) (gε1 ,ε2 )[ f p∗∗3 ,q3 ], f τ,ρ Fi (T
•∗∗ ) d pˆ dqˆ = R2 t p3 ,q3 , pˆ3 ,qˆ3 (gε1 ,ε2 ) F3 (T (gε1 ,ε2 )[ f p∗∗ ], f τ,ρ 3 3 ˆ 3 ,qˆ3
= R2 t p3 ,q3 , pˆ3 ,qˆ3 (gε1 ,ε2 ) δ( pˆ 3 + τ ) δ(qˆ3 + ρ) d pˆ 3 dqˆ3 = 4π 2 t p3 ,q3 ,−τ,−ρ (gε1 ,ε2 ),
where δ(x − a) denotes the a-delayed Dirac delta function with respect to x. It means that t p3 ,q3 , pˆ3 ,qˆ3 (gε1 ,ε2 ) =
123
1 Fi (T (gε1 ,ε2 )[ f p∗∗3 ,q3 ], f −•∗∗ pˆ 3 ,−qˆ3 ). 4π 2
(4)
Integral and series representations related to the group S O(2, 2)...
Let us introduce the following functions depending on positive arguments: F1 (a, b) :=
exp(−iε2 q3 ) 2σ +3
cos2 σ2π
|q32 − p32 | ab
σ +1 2
√ √ × Jσ +1 2|εq3 − p3 |a − J−σ −1 2|εq3 − p3 |a √ √ × Jσ +1 2|εq3 + p3 |b − J−σ −1 2|εq3 + p3 |b ; σ +1 2 √ |q32 − p32 | exp(−iε2 q3 ) sin(π σ ) F2 (a, b) := K σ +1 2|εq3 − p3 |a ab 2σ +2 π √ √ × Jσ +1 2|εq3 + p3 |b − J−σ −1 2|εq3 + p3 |b ; σ +1 |q32 − p32 | 2 2 πσ 2 q3 ) F3 (a, b) := exp(−iε sin 2 ab 2σ −1 π 2 √ √ ×K σ +1 2|εq3 − p3 |a K σ +1 2|εq3 + p3 |b ; σ +1 π σ |q32 − p32 | 2 exp(−iε2 q3 ) 2 F4 (a, b) := 2σ −1 π 2 sin 2 ab √ √ ×K σ +1 2|εq3 + p3 |a K σ +1 2|εq3 − p3 |b .
Lemma 1 The matrix elements t p3 ,q3 , pˆ3 ,qˆ3 (gε1 ,ε2 ) can be represented as follows: 1. For εq3 > 0, | p3 | < q3 , | pˆ 3 | < qˆ3 , we have t p3 ,q3 , pˆ3 ,qˆ3 (gε1 ,ε2 ) = exp(iε1 qˆ3 ) F1 (qˆ3 + pˆ 3 , qˆ3 − pˆ 3 ). 2. For pˆ 3 < 0, |q3 | < − p3 , | pˆ 3 | < qˆ3 , we have t p3 ,q3 , pˆ3 ,qˆ3 (gε1 ,ε2 ) = exp(iε1 qˆ3 ) F3 (qˆ3 − pˆ 3 , qˆ3 + pˆ 3 ). 3. For |q3 | < p3 , | pˆ 3 | < qˆ3 , we have t p3 ,q3 , pˆ3 ,qˆ3 (gε1 ,ε2 ) = exp(iε1 qˆ3 ) F2 (qˆ3 + pˆ 3 , qˆ3 − pˆ 3 ). 4. For εq3 < 0, | p3 | < |q3 |, | pˆ 3 | < |qˆ3 |, we have t p3 ,q3 , pˆ3 ,qˆ3 (gε1 ,ε2 ) = exp(iε1 qˆ3 ) F4 (qˆ3 + pˆ 3 , qˆ3 − pˆ 3 ). 5. For εq3 > 0, | p3 | < |q3 |, |qˆ3 | < pˆ 3 , we have t p3 ,q3 , pˆ3 ,qˆ3 (gε1 ,ε2 ) = exp(iε1 qˆ3 ) F3 (|qˆ3 − pˆ 3 |, qˆ3 + pˆ 3 ). 6. For p3 < 0, |q3 | < − p3 , |qˆ3 | < pˆ 3 , we have t p3 ,q3 , pˆ3 ,qˆ3 (gε1 ,ε2 ) = exp(iε1 qˆ3 ) F1 (qˆ3 + pˆ 3 , |qˆ3 − pˆ 3 |).
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7. For εq3 < 0, | p3 | < |q3 |, |qˆ3 | < pˆ 3 , we have t p3 ,q3 , pˆ3 ,qˆ3 (gε1 ,ε2 ) = exp(iε1 qˆ3 ) F2 (qˆ3 + pˆ 3 , |qˆ3 − pˆ 3 |). 8. For |q3 | < p3 , qˆ3 < 0, |qˆ3 | < pˆ 3 , we have t p3 ,q3 , pˆ3 ,qˆ3 (gε1 ,ε2 ) = exp(iε1 qˆ3 ) F4 (qˆ3 + pˆ 3 , |qˆ3 − pˆ 3 |). 9. For εq3 > 0, | p3 | < |q3 |, pˆ 3 < 0, |qˆ3 | < − pˆ 3 , we have t p3 ,q3 , pˆ3 ,qˆ3 (gε1 ,ε2 ) = exp(iε1 qˆ3 ) F2 (|qˆ3 + pˆ 3 |, qˆ3 − pˆ 3 ). 10. For p3 < 0, , |q3 | < − p3 , pˆ 3 < 0, |qˆ3 | < − pˆ 3 , we have t p3 ,q3 , pˆ3 ,qˆ3 (gε1 ,ε2 ) = exp(iε1 qˆ3 ) F4 (|qˆ3 + pˆ 3 |, qˆ3 − pˆ 3 ). 11. For |q3 | < p3 , pˆ 3 < 0, |qˆ3 | < − pˆ 3 , we have t p3 ,q3 , pˆ3 ,qˆ3 (gε1 ,ε2 ) = exp(iε1 qˆ3 ) F4 (|qˆ3 + pˆ 3 |, qˆ3 − pˆ 3 ). 12. For εq3 < 0, | p3 | < |q3 |, pˆ 3 < 0, |qˆ3 | < − pˆ 3 , we have t p3 ,q3 , pˆ3 ,qˆ3 (gε1 ,ε2 ) = exp(iε1 qˆ3 ) F3 (|qˆ3 − pˆ 3 |, |qˆ3 + pˆ 3 |). 13. For εq3 > 0, | p3 | < |q3 |, qˆ3 < 0, | pˆ 3 | < −qˆ3 , we have t p3 ,q3 , pˆ3 ,qˆ3 (gε1 ,ε2 ) = exp(iε1 qˆ3 ) F4 (|qˆ3 + pˆ 3 |, |qˆ3 − pˆ 3 |). 14. For p3 < 0, |q3 | < − p3 , qˆ3 < 0, | pˆ 3 | < −qˆ3 , we have t p3 ,q3 , pˆ3 ,qˆ3 (gε1 ,ε2 ) = exp(iε1 qˆ3 ) F1 (|qˆ3 + pˆ 3 |, |qˆ3 − pˆ 3 |). 15. For εq3 < 0, | p3 | < |q3 |, qˆ3 < 0, | pˆ 3 | < −qˆ3 , we have t p3 ,q3 , pˆ3 ,qˆ3 (gε1 ,ε2 ) = exp(iε1 qˆ3 ) F1 (|qˆ3 + pˆ 3 |, |qˆ3 − pˆ 3 |). 16. For |q3 | < p3 , qˆ3 < 0, | pˆ 3 | < −qˆ3 , we have t p3 ,q3 , pˆ3 ,qˆ3 (gε1 ,ε2 ) = exp(iε1 qˆ3 ) F3 (|qˆ3 − pˆ 3 |, |qˆ3 + pˆ 3 |). Proof We find from (4) that t p3 ,q3 , pˆ3 ,qˆ3 (gε1 ,ε2 ) = 4π1 2 F3 (T (gε1 ,ε2 )[ f p∗∗3 ,q3 ], f −•∗∗ ) pˆ 3 ,−qˆ3
2 p3 α3 +ε2 q3 (1+α32 −β32 ) 2 −σ −2 −2 2 σ =2 π |(β3 + ε1 ) − α3 | exp i (β +ε )2 −α R2 × exp(−i pˆ 3 α3 )
123
3
exp(−iqˆ3 β3 ) dα3 dβ3 .
1
3
Integral and series representations related to the group S O(2, 2)...
Introducing the new variables z i := β3 + ε1 + (−1)i α3 (i ∈ {0, 1}), we obtain t p3 ,q3 , pˆ3 ,qˆ3 (gε1 ,ε2 ) =
exp(i[ε1 qˆ3 −ε2 q3 ]) 2σ +1 π 2
×
+∞
1
|z i |σ cos
i=0 0
qˆ3 +(−1)i pˆ 3 zi 2
+
εq3 −(−1)i p3 zi
dz i .
Finally using the following known formulas (see, e.g., respectively, [12, Entry 2.5.24.4] and [12, Entry 2.5.24.7]): +∞
x α−1 cos ax + bx dx (5) 0 √ √ α = π2 ab 2 sin απ (2 ab) − J (2 ab) (min{a, b} > 0; | (α)| < 1) J −α α 2
and +∞
x α−1 cos ax − bx dx 0 √ α K α (2 ab) (min{a, b} > 0; | (α)| < 1), = 2 ab 2 cos απ 2
(6)
we are led to the desired results. Now let us rewrite the equality (3) for the case |q3 | < p3 : +∞
+∞
0
+ + + + +
F2 c˜ pˆ3 ,qˆ3 , p1 ,q1 pˆ 3 +∞
pˆ3 0 − pˆ 3 +∞
− pˆ3 0 −∞
0 +∞
−∞ − pˆ 3
0 − pˆ3 −∞ pˆ 3
0 pˆ3
exp(i ε1 qˆ3 ) dqˆ3 d pˆ 3
F4 c˜ pˆ3 ,qˆ3 , p1 ,q1 exp(i ε1 qˆ3 ) dqˆ3 d pˆ 3 F2 c˜ pˆ3 ,qˆ3 , p1 ,q1 exp(i ε1 qˆ3 ) dqˆ3 d pˆ 3 F2 c˜ pˆ3 ,qˆ3 , p1 ,q1 exp(i ε1 qˆ3 ) dqˆ3 d pˆ 3 F4 c˜ pˆ3 ,qˆ3 , p1 ,q1 exp(i ε1 qˆ3 ) dqˆ3 d pˆ 3
F2 c˜ pˆ3 ,qˆ3 , p1 ,q1 exp(i ε1 qˆ3 ) dqˆ3 d pˆ 3 −∞ −∞ = (ε1 i)−εq1 c˜ p3 ,q3 , p1 ,−εq1
and introduce new variables μ := |qˆ3 + pˆ 3 | and ν := |qˆ3 − pˆ 3 |. Applying the same procedure for the case p < 0, |q| < − p and for the case | p| < |q|, we obtain the following theorem.
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I. A. Shilin and J. Choi
Theorem 1 The following integral representations hold true: For |q3 | < p3 , Wp
1,
σ +εq1 +1 2
(2( p3 + q3 )) Wεq
= (ε1 i)εq1 | p3 + q3 |
σ +εq1 2 +1
1,
σ + p1 +1 2
(2( p3 − q3 ))
| p3 − q 3 |
σ + p1 2 +1
σ +εq1 − p1 2
σ + p1 −εq1 2
i(μ+ν) exp i(μ+ν) F F3 (μ, ν) × (μ, ν) + exp − 2 2 2 μ,ν>0
+ cos
μ−ν 2
F4 (μ, ν) c˜μ,−ν, p1 ,−εq1 dμ dν.
For p3 < 0, |q3 | < − p3 , W− p
1,
σ +εq1 +1 2
(−2( p3 + q3 )) W−εq
= (ε1 i)εq1 | p3 + q3 |
σ +εq1 2 +1
1,
σ + p1 +1 2
| p3 − q 3 |
(2(q3 − p3 ))
σ + p1 2 +1
σ +εq1 + p1 2
σ + p1 +εq1 2
i(μ+ν) exp i(μ+ν) F F1 (μ, ν) × (μ, ν) + exp − 2 2 2 μ,ν>0
+
1 2
exp
i(μ−ν) 2
F1 (μ, ν) +
1 2
exp
i(ν−μ) 2
F4 (μ, ν) c˜μ,−ν, p1 ,−εq1 dμ dν.
For | p3 | < |q3 |, Wp
1 sign ( p3 +q3 ),
σ +εq1 +1 2
= (ε1 i)εq1 | p3 + q3 |
(2|q3 + p3 |) Wεq
σ +εq1 2 +1
| p3 − q 3 |
1 sign ( p3 −q3 ),
σ + p1 2 +1
σ + p1 +1 2
(2|q3 − p3 |)
σ +εq1 + p1 2
σ + p1 +εq1 2
exp − i(μ+ν) F1 (μ, ν) + exp i(μ−ν) F2 (μ, ν) × 2 2 μ,ν>0
+ exp
i(ν−μ) 2
F3 (μ, ν) + exp
i(μ+ν) 2
F4 (μ, ν)+ c˜μ,−ν, p1 ,−εq1 dμ dν.
It is noted in passing that other similar formulas as in Theorem 1 can be derived by using the linear operators T (gχ1 ,χ2 ), where ⎛
gχ1 ,χ2
123
0 ⎜χ2 := ⎜ ⎝0 0
χ1 0 0 0
0 0 1 0
⎞ 0 0⎟ ⎟ (χ1 , χ2 ∈ {1, −1}). 0⎠ 1
Integral and series representations related to the group S O(2, 2)...
3 Relations induced by the compositions gχ1 ,χ2 gε1 ,ε2 Using a similar argument as in Sect. 2, we have t p3 ,q3 , pˆ3 ,qˆ3 (gχ1 ,χ2 gε1 ,ε2 ) =
1 F ( f ∗∗ , f •∗∗ ) 4π 2 3 p3 ,q3 − pˆ 3 ,−qˆ3
=
1 4π 2
R2
|χ2 α3 + ε2 β3 |σ exp(−i pˆ 3 α3 ) exp(−iqˆ3 β3 )
ε q +χ p +(ε q3 −χ1 p3 )(α32 −β32 ) dα3 dβ3 . × exp i 1 3 1 32(χ2 α1 3 +ε 2 β3 ) Setting new variables s := ε1 q3 + χ1 p3 and t := ε1 q3 − χ1 p3 , we obtain t p3 ,q3 , pˆ3 ,qˆ3 (gχ1 ,χ2 gε1 ,ε2 ) 1 = 2π δ (qˆ3 − pˆ3 )−(ε21 q3 −χ1 p3 ) ×
+∞
s σ cos
0
pˆ 3 +qˆ3 2 s
−
ε1 q3 +χ1 p3 2s
ds.
As in Sect. 2, the following lemma can be proved. Lemma 2 The matrix elements t p3 ,q3 , pˆ3 ,qˆ3 (gχ1 ,χ2 gε1 ,ε2 ) can be written in the form: ( pˆ 3 + qˆ3 )(χ1 p3 + ε1 q3 ) > 0, t p3 ,q3 , pˆ3 ,qˆ3 (gχ1 ,χ2 gε1 ,ε2 ) =−
sin( π2σ ) 2π 2
χ1 p3 +ε1 q3 pˆ 3 +qˆ3
σ +1 2
×K σ +1 2 ( pˆ 3 + qˆ3 )(χ1 p3 + ε1 q3 ) δ (qˆ3 − pˆ3 )−(ε21 q3 −χ1 p3 ) .
( pˆ 3 + qˆ3 )(χ1 p3 + ε1 q3 ) < 0, t p3 ,q3 , pˆ3 ,qˆ3 (gχ1 ,χ2 gε1 ,ε2 ) σ +1 χ1 p3 +ε1 q3 2 δ (qˆ3 − pˆ3 )−(ε21 q3 −χ1 p3 ) pˆ3 +qˆ3 × J−σ −1 2 −( pˆ 3 + qˆ3 )(χ1 p3 + ε1 q3 )
=
cos( π2σ ) 2π 2
−Jσ +1 2 −( pˆ 3 + qˆ3 )(χ1 p3 + ε1 q3 ) . 3
t0,0, pˆ3 ,qˆ3 (gχ1 ,χ2 gε1 ,ε2 ) = 2σ π 2 |εqˆ3 |−σ −1 δ( pˆ 3 − χ εqˆ3 ).
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I. A. Shilin and J. Choi
Let χ := χ1 χ2 . Similarly as in Sect. 2, by using the equality
t p3 ,q3 , pˆ3 ,qˆ3 (gχ1 ,χ2 gε1 ,ε2 ) c pˆ3 ,qˆ3 , p1 ,q1 d pˆ 3 dqˆ3 pˆ 3 +qˆ3 <0
+
pˆ 3 +qˆ3 >0
t p3 ,q3 , pˆ3 ,qˆ3 (gχ1 ,χ2 gε1 ,ε2 ) c pˆ3 ,qˆ3 , p1 ,q1 d pˆ 3 dqˆ3
= (χ1 i)−χ p1 (ε1 i)−εq1 c p3 ,q3 ,−χ p1 ,−εq1
and choosing the case p3 + q3 > 0, we obtain the following theorem. Theorem 2 The following integral representation holds true: cos
"−1 π σ ! p1 +εq1 −σ −1 2 2
×
+∞
λ−
εq1 +1 2
√ √ J−σ −1 2 (χ1 p3 + ε1 q3 )λ − Jσ +1 2 (χ1 p3 + ε1 q3 )λ
0
×W p ×
1,
+∞
εq1 −σ −1 2
λ−
(2λ) dλ − sin
π σ ! εq1 −σ − p1 "−1 −1 2 2
√ K σ +1 2 (χ1 p3 + ε1 q3 )λ W p
εq1 +1 2
0
σ −εq1
σ +1
1,
εq1 −σ −1 2
χp
εq
(2λ) dλ
= 4π 2 | p3 + q3 | 2 |χ1 p3 + ε1 q3 |− 2 χ1 1 ε1 1 i−χ p1 −εq1 −1 × εq1 −σ − p1 2sign( p3 +q3 ) − 1 W p sign( p +q ), εq1 −σ −1 (2| p3 + q3 |). 1
3
3
2
4 The relation induced by the matrix diag(1, 1, 1, −1) Lemma 3 For −1 < (σ ) < 0, we have t0,0, pˆ3 ,qˆ3 (diag(1, 1, 1, −1)) =
! "2 24σ +1 σ +1 2 ! "2 . π 2 |qˆ32 − pˆ 32 |σ +1 − σ2
Proof Indeed, we find t0,0, pˆ3 ,qˆ3 (diag(1, 1, 1, −1)) = =
1 ∗∗ ], f •∗∗ F (T (diag(1, 1, 1, −1))[ f 0,0 ) − pˆ 3 ,−qˆ3 4π 2 3 +∞ +∞
σ
σ s cos |qˆ3 +2 pˆ3 | s ds t cos |qˆ3 −2 pˆ3 | t 2 0 0
dt.
Then using the following known integral formula (see, e.g., [12, Entry 2.5.4.14]): +∞
0
=
123
x α−1 cos2n+1 (bx) dx √
n π (2n+1)! ( α2 ) (2n−2k+1)−α k! (2n−k+1)! (b 1−α α b 2 k=0
2α−2n−1
(7) > 0, 0 < (α) < 1),
Integral and series representations related to the group S O(2, 2)...
we are led to the desired result. Introducing the variables s := α3 + β3 and t := α3 − β3 , we have c0,0, p1 ,−q1 = =
1 F ( f ∗∗ , f • ) 4π 2 3 0,0 − p1 ,q1 2σ −1 π2
×
+∞
−∞ +∞
−∞
(1 + is)
(1 + it)−
p1 +q1 −σ 2
p1 +q1 +σ 2
−1
−1
(1 − is)
(1 − it)
q1 − p1 −σ 2
q1 − p1 −σ 2
−1
−1
ds
dt.
Using here the following known formula (see, e.g., [12, Entry 2.2.6.31]): +∞
−∞
dx (a+ix)μ (b−ix)ν
=
2π (μ+ν−1) (a+b)μ+ν−1 (μ) (ν)
(min{a, b} > 0, μ + ν > 1),
we obtain the following lemma. Lemma 4 For p1 < 0 and q1 > 0, we have c0,0, p1 ,−q1 =
2q1 − p1 −σ −1 (1 + σ + p1 ) (1 + σ − q1 ) ! "2 . 1 + σ − p21 −q1 1 + σ + p21 +q1 1 + σ +q21 − p1
Similar arguments as in Sects. 2 and 3 will give the following formula: t0,0, pˆ3 ,qˆ3 (diag(1, 1, 1, −1)) c pˆ3 ,qˆ3 , p1 ,q1 = (−1)q1 c0,0, p1 ,−q1 . R2
Dividing the domain of this integral into 6 parts and using a known formula (see, e.g., [13, Entry 2.19.2.1]): +∞
x α−1 Wρ,τ (cx) dx
0
=
1 1 2 +α+τ 2 +α−τ cα (1+α−ρ)
×2 F1
1 2
+ α + τ,
1 2
+ α − τ ; 1 + α − ρ;
1 2
(c) > 0, | (τ )| < (α) + 21 , we derive the following summation formula asserted in Theorem 3.
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I. A. Shilin and J. Choi
Theorem 3 For p1 < 0 and q1 > 0, we have (1 + δ1,kl ) (−1)k+1 p1 +
1 1 k=0 l=0
⎡
×2 F1 ⎣
=
1 2
3−σ +q1 2
⎤
− σ, 23 − p1 ;
⎡
(−1)l+1 q1 + 1 2
3−σ − p1 2
⎤
− σ, 23 + q1 ;
1⎦ 1⎦ 2 F1 ⎣ 3−σ − p1 2 3−σ +q1 2 k+1 ; p1 + ; (−1) 1+ 2 2 2 3q1 −3 p1 σ − p1 −q1 σ + p1 +q1 σ +q1 − p1 2 21−5σ + π 2 1+ 1+ (− σ2 ) 1+ 2 2 2 2 . (−1)q1 (1+σ + p1 ) (1+σ −q1 ) 23 − p1 23 +q1 σ +1 21 −σ 2
(−1)l+1 q
5 Relations induced by the Poisson transforms Let u ∈ H such that y2 + y4 = 0 and C :=
y32 −y12 y2 +y4 .
Then we find
P( f p∗∗3 ,q3 ) = F3 ( f p∗∗3 ,q3 , p)
p3 y1 +q3 y3 4 −σ −2 = y2 +y exp i 2 y2 +y4
2 2 −σ × |α3 − β3 + C| −2 exp(i p3 α3 ) exp(iq3 β3 ) dα3 dβ3 . R2
Here setting the new variables s := α3 + β3 and t := α3 − β3 , we have p3 y1 + q3 y3 P( f p∗∗3 ,q3 ) = 2σ +1 |y2 + y4 |−σ −2 exp i I, y2 + y4 where
I :=
|st + C|
−σ −2
R2
=2
i( p3 + q3 )s exp 2
i( p3 − q3 )t exp 2
ds dt
( ' +∞ ( p3 − q3 )C ( p3 + q3 )s − ds |s|−σ −2 exp i 2 2s
−∞ +∞
×
t
−σ −2
| p3 − q3 |t cos 2
dt.
0
It is noted that the inner integral can be computed via the following formula (see, e.g., [12, Entry 2.5.3.10]): +∞
0
x α−1 cos(bx) dx = b−α cos
απ 2
(α) (b > 0, 0 < (α) < 1)
and the exterior integral can be computed via either (5) or (6).
123
Integral and series representations related to the group S O(2, 2)...
It is seen that y12 + y22 = 0 for every y ∈ H. Suppose that y32 + y42 = 0 and introduce the following notations: sin ϕ :=
y1 y12 + y22
cos ϕ :=
, cos ϕ := y4 y32 + y42
y2
, sin ψ :=
y12 + y22 ) y32 + y42 N := . y12 + y22
,
y3 y32 + y42
,
Then we have P( f p1 ,q1 ) = F1 ( f p1 ,q1 , p) σ
= 4(y12 + y22 )− 2 −1 exp(i p1 ϕ) exp(iq1 ψ)
π
π × cos(|q1 |β1 ) dβ1 | cos α1 − N cos β1 |−σ −2 cos(| p1 |α1 ) dα1 . 0
0
Here, since 0 < N < 1, we can represent the inner integral as follows: arccos(N cos β1 )
[cos α1 − N cos β1 ]−σ −2 cos(| p1 |α1 ) dα1
0 π −arccos(N cos β1 )
+(−1)
[cos α1 − cos(π − arccos(N cos β1 )]−σ −2
p1 0
× cos(| p1 |α1 )dα1 , which can be evaluated by applying the following known formula (see, e.g., [12, Entry 2.5.16.1]):
a 0
(cos x − cos a)ν−1 cos(bx) dx 1 1 −ν = π2 {sin a}ν− 2 (ν) P 2 1 (cos a) (b > 0, (ν) > 0, 0 < a < π ).
(8)
b− 2
Using the formula for multiple-angle cosine and introducing the new variable t := N cos β1 , we have P( f p1 ,q1 ) σ
3
= 2 2 (y12 + y22 )− 2 −1 exp(i p1 ϕ) exp(iq1 ψ) (−σ − 1)
×
|q1 | 2
k=0
(−1)k N 2k−|q1 |
|q1 | N k
−N
σ
5
σ + 32
t
| p1 |− 21
N
t |q1 |−2k (1 − t 2 )2k− 2 − 4 P
dt,
123
I. A. Shilin and J. Choi
where [x] is the integer function of the real argument x. Recalling the following known formulas (see, e.g., [7, Entries 8.736.2 and 8.705], respectively) Pνμ (−x) = cos[(ν + μ)π ] Pνμ (x) −
2 sin[(ν + μ)π ] Q μ ν (x) (x ∈ (0, 1)) π
and π 2 sin(μπ )
Qμ ν (x) =
(ν + μ + 1) −μ Pν (x) cos(μπ ) Pνμ (x) − (x ∈ (−1, 1)), (ν − μ + 1)
we obtain P( f p1 ,q1 ) σ
3
= 2 2 (y12 + y22 )− 2 −1 exp(i p1 ϕ) exp(iq1 ψ) (−σ − 1)
|q1 | 2
×
(−1)k
k=0
N −|q1 | |qk1 |
N
'
(−1)q1 sin([σ +| p1 |]π ) (| p1 |+σ +2) sin(σ π ) (| p1 |−σ −1) σ + 32
t
0
| p1 |− 21
N
0
| p1 |− 2
σ
5
× t |q1 |−2k (1 − t 2 )− 2 − 4 (N 2 − t 2 )2k P
dt
+(1 − (−1)q1 {cos([σ + | p1 |]π ) + cos([σ + | p1 |]π ) cot(σ π )}) (
N |q |−2k −σ − 32 t − σ2 − 54 2 2 2 2k 1 × t (1 − t ) (N − t ) P 1 N dt . σ
5
Now, in order to compute P( f p1 ,q1 ), we can use the Maclaurin series for (1 − t 2 )− 2 − 4 and the known formula (see, e.g., [13, Entry 2.17.2.1]):
a
μ x a
x α−1 (a 2 − x 2 )β−1 Pν
0
=
2μ−1 a 2β+α−2 (α2) β− μ2 (1−μ) β+ α−μ 2
(
1+ν−μ , 2
dx
)
− ν+μ 2 ,
μ 2;
×3 F2 β− 1 − μ, β + (min{a, (α), (2β − μ)} > 0).
(9)
α−μ 2 ;1
From (1) we have P( f p∗∗3 ,q3 ) =
c p3 ,q3 , p1 ,q1 P( f p1 ,q1 ).
(10)
p1 ∈Z q1 ∈Z
From the arguments given here, we obtain certain interesting identities asserted in Theorem 4.
123
Integral and series representations related to the group S O(2, 2)...
Theorem 4 The following series representations hold true: |y2 + y4 |σ +2 csc (y12 +
σπ
2 σ σ +1 y22 ) 2 +1 | p32 − q32 | 2
×
|q1 |
p3 y1 + q3 y3 exp(−i[ p1 ϕ + q1 ψ]) exp −i y2 + y4
2 ∞
(−1)k c p3 ,q3 , p1 ,q1
p1 ∈Z q1 ∈Z l=0 k=0
σ 5 + 2 4
[l!]−1 N 4k+2l+3
l
|q1 | k
' |q1 | + 1 (−1)q1 2σ sin([σ + | p1 |]π ) (| p1 | + σ + 2) +l −k 2 sin(σ π ) (| p1 | − σ − 1) ⎡ | p |−σ −1 ⎤ +1 1 , − | p1 |+σ , 2k + 41 − σ2 ; 2k + 41 − σ2 2 2 3 F2 ⎣ × 1⎦ −σ − 21 k + l + 43 + |q1 2|−σ −σ − 21 , k + l + 43 + |q1 2|−σ ; ×
+(1 − (−1)q1 {cos([σ + | p1 |]π ) + cos([σ + | p1 |]π ) cot(σ π )}) ⎡ | p |+σ ⎤ 1 ( + 1, σ −|2 p1 | + 1, 2k + σ2 + 74 ; 2−σ −3 2k + σ2 + 74 2 3 F2 ⎣ 1⎦ × σ + 25 k + l + 49 + |q1 2|+σ σ + 25 , k + l + 94 + |q1 2|+σ ; ⎧ σπ ⎪ 2 2 2 2 ⎪ ⎪ ⎨2 sin 2 K σ +1 2 | p3 − q3 | , ifsign( p3 − q3 ) = sign C, =
⎪ ⎪ π ⎪ ⎩
( ' cos( σ2π ) J−σ −1 2 | p32 −q32 | −Jσ +1 2 | p32 −q32 | 2
, if sign( p32 − q32 ) = −sign C.
6 Relations arising from the invariance of F i with respect to i Expressing the function f p∗∗3 ,q3 through the basis B2 , we have f p∗∗3 ,q3 (x)
=
p2 ∈Z R
d p3 ,q3 , p2 ,q2 ,± f p∗2 ,q2 ,± (x) dq2
(11)
and, therefore, Fi ( f p∗∗3 ,q3 , f p∗• ) ˆ 2 ,qˆ2 ,±
d p3 ,q3 , p2 ,q2 ,± F2 ( f p∗2 ,q2 ,± , f p∗• = ˆ ,qˆ 2
p2 ∈Z R
=
p2 ∈Z R
∞
×
−∞
d p3 ,q3 , p2 ,q2 ,±
π −π
2 ,±
) dq2
exp(i( p2 + pˆ 2 ]β2 ) dβ2
exp(i(q2 + qˆ2 ]α2 ) dα2 dq2 = 4π 2 d p3 ,q3 ,− pˆ2 ,−qˆ2 ,± .
We thus have d p3 ,q3 , p2 ,q2 ,± =
1 Fi ( f p∗∗3 ,q3 , f −∗•p2 ,−q2 ,± ). 4π 2
123
I. A. Shilin and J. Choi
Lemma 5 For −1 < (σ ) < 23 , we obtain d0,0, p2 ,0,−
⎤ ⎡ σ +| p2 |+1 σ −| p2 |+1 , , σ + 1; (1−3 sin(π σ )) { (σ +1)}2 43 − σ2 2 2 1⎦ = 3 F2 ⎣ 3 2σ +2 π 2 σ + 23 σ2 + 74 σ + 23 , σ2 + 47 ; ⎡ ⎤ | p2 |−σ −1 , − σ +|2 p2 | , 21 ; 2σ cos(π σ ) (σ +1) (| p2 |−σ ) 43 − σ2 2 3 F2 ⎣ − 2 1⎦ . π (| p2 |+σ +1) 21 −σ 45 − σ2 1 5 σ 2 − σ, 4 − 2 ;
Proof Consider the equality d0,0, p2 ,0,− = =
1 F ( f ∗∗ , f ∗• ) 4π 2 2 0,0 − p2 ,0 +∞
π 1 | cosh α2 cos β2 4π 2 −∞ −π
1
− 1|σ exp(−i p2 β2 ) {cosh α2 } 2 dβ2 dα2 .
Now the inner integral arccos sech α2
(cos β2 − sech α2 )σ cos(| p2 |β2 ) dβ2
0
+(−1) p2
π −arccos
sech α2
(cos β2 − (−sech α2 ))σ cos(| p2 |β2 ) dβ2
0
can be computed via the formula (8). Introducing the new variable t := sech α2 , we have (σ + 1) d0,0, p2 ,0,− = √ 2π 3
1
σ
1
1
t 2 −σ (1 − t 2 ) 2 − 4
' P
−σ − 12
| p2 |−21
(t)+(−1) p2 P
( (−t) dt. 1
−σ − 12
| p2 |− 2
0
Recalling the following known formula Pνμ (−x) = cos[(ν + μ)π ] Pνμ (x) − and Qμ ν (x)
2 sin[(ν + μ)π ] Q μ ν (x) (x ∈ (0, 1)) π
(ν + μ + 1) −μ μ P (x) cos(μπ ) Pν (x) − (x ∈ (−1, 1)), (ν − μ + 1) ν
π = 2 sin(μπ )
we obtain d0,0, p2 ,0,−
.
= (σ + 1) −2
123
1 2
(1−3 sin(π σ ))
1
π
0
1 22
3 2
cos(π σ ) (| p2 |−σ ) π
5 2
(| p2 |+σ +1)
1 0
1
σ
1
1
t 2 −σ (1 − t 2 ) 2 − 4 P
−σ − 12
| p2 |− 21
σ
1
t 2 −σ (1 − t 2 ) 2 − 4 P
σ + 12
| p2 |− 21
(t) dt /
(t) dt .
Integral and series representations related to the group S O(2, 2)...
Finally using (9), we are led to the desired result. Similarly, the following formula can be derived. Lemma 6 For −1 < (σ ) < 23 , we have d0,0, p2 ,0,+
. | p2 |−σ σ +| p2 | 2σ −1 (1 − 3 sin(π σ )) 43 − σ2 2 , − 2 , = F 3 2 1 5 π 21 − σ 45 − σ2 2 − σ, 4 − cos(π σ ) (σ ) (| p2 | − σ ) 43 − σ2 − 3 5 2σ +1 π 2 (| p2 | + σ + 1) 23 + σ 47 + σ2 ⎡ ×F2 ⎣
| p2 |+σ +1 1+σ −| p2 | , , 2 2 3 2
+ σ,
σ + 1; 7 4
+
σ 2
;
1 2 σ 2
; ;
/ 1
⎤ 1⎦ .
∗∗ , f ∗• Using the identity d0,0, p2 ,0,− = 4π1 2 F3 ( f 0,0 − p2 ,0 ), we get the following integral formula formula asserted in Theorem 5.
Theorem 5 The following integral formula holds true: For −1 < (σ ) < p2 ∈ Z,
β32 −α32 <1
3 2
and
p2 −1
(1+2α32 +2β32 +(α32 −β32 )2 ) 2 dα3 dβ3 (1+α32 −β32 )σ +1 ([1+iα3 ]2 +β32 ) p2
⎤ ⎡ σ +| p2 |+1 σ −| p2 |+1 , , σ + 1; (1−3 sin(π σ )) { (σ +1)}2 43 − σ2 2 2 = 1⎦ 3 F2 ⎣ 3 3 σ 7 2σ +2 π 2 σ + 32 σ2 + 74 σ + 2, 2 + 4 ; ⎡ ⎤ | p |−σ −1 σ +| p2 | 1 2 , − , ; 2σ cos(π σ ) (σ +1) (| p2 |−σ ) 43 − σ2 2 2 2 3 F2 ⎣ − 1⎦ . π 2 (| p2 |+σ +1) 21 −σ 54 − σ2 1 5 σ − σ, − ; 2 4 2
It is noted that another similar formula as in Theorem 5 can be obtained from the ∗∗ , f ∗• ∗∗ ∗• equality F2 ( f 0,0 − p2 ,0 ) = F3 ( f 0,0 , f − p2 ,0 ).
7 Concluding remarks Here and in [14] for groups S O(2, 2) and S O(2, 1), respectively, we expressed the matrix elements (with respect to a parabolic basis) of the restrictions of the representation T to block-diagonal matrices in terms of Bessel functions (in wider sense). More exactly, these matrix elements contain the basic Bessel functions of the first kind and the modified Bessel functions of the second kind (Macdonald function). Our recent preliminary calculations show that for S O(3, 1) they disappear and the generalized
123
I. A. Shilin and J. Choi
hypergeometric functions of the type 0 F3 take their place. However, 0 F3 leads to the natural multi-index analogues of Jα (x) and Iα (x): For example, Iν1 ,...,νn (z) :=
ν1 +...+νn
n
−1
(νi + 1) i=1 ' n+1 ( z , ×0 Fn ν1 + 1, . . . , νn + 1; n+1 z n+1
where ν1 , . . . , νn ∈ / {−1, −2, . . .} (see, e.g., [5]). Besides the fact that these functions are interesting in themselves [11], they are closely related to fractional calculus (hyper-Bessel differential equations, Poisson–Dimovsky integral operator) [4,8] and the generalized Mittag-Leffler functions [6]. 0 F3 -functions are noted to be also related to Appell series F3 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]. Acknowledgments The authors would like to express their deep thanks for the reviewer’s helpful comment to improve this paper.
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