Some Connections between the Spherical and Parabolic Bases on the Cone Expressed in terms of the Macdonald Function

Hindawi Publishing Corporation Abstract and Applied Analysis Volume 2014, Article ID 741079, 8 pages http://dx.doi.org/10.1155/2014/741079 Research Article Some Connections between the Spherical and Parabolic Bases on the Cone Expressed in terms of the Macdonald Function I. A. Shilin1,2 and Junesang Choi3 1 Department of Mathematics, Sholokhov Moscow State University for the Humanities, Verkhnyaya Radishevskaya 16-18, Moscow 109240, Russia 2 Department of Mathematical Modeling, Moscow Aviation Institute, Volokolamskoe Shosse 4, Moscow 125993, Russia 3 Department of Mathematics, Dongguk University, Gyeongju 780-714, Republic of Korea Correspondence should be addressed to Junesang Choi; Received 7 November 2013; Accepted 26 December 2013; Published 11 February 2014 Academic Editor: Kwang Ho Shon Copyright © 2014 I. A. Shilin and J. Choi. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. Computing the matrix elements of the linear operator, which transforms the spherical basis of 𝑆𝑂(3, 1)-representation space into the hyperbolic basis, very recently, Shilin and Choi (2013) presented an integral formula involving the product of two Legendre functions of the first kind expressed in terms of 4 𝐹3 -hypergeometric function and, using the general Mehler-Fock transform, another integral formula for the Legendre function of the first kind. In the sequel, we investigate the pairwise connections between the spherical, hyperbolic, and parabolic bases. Using the above connections, we give an interesting series involving the Gauss hypergeometric functions expressed in terms of the Macdonald function. 1. Introduction and Preliminaries For completeness and an easier reference, we begin by just recalling some parts of [1, Section 1]. Let Λ be the cone in the Euclidean space R4 defined by Λ := {(𝑥1 , 𝑥2 , 𝑥3 , 𝑥4 ) ∈ R4 | 𝑥12 − 𝑥22 − 𝑥32 − 𝑥42 = 0} , (1) where R denotes the set of real numbers. Let 𝑆𝑂(3, 1) be a multiplicative group consisting of all 4 × 4 matrices 𝑔 in R which satisfies the following two properties: det 𝑔 = 1, 𝑔𝑇 𝐼1,3 𝑔 = 𝐼1,3 , (2) where 𝑔𝑇 denotes (as usual) the transpose of the matrix 𝑔 and 𝐼1,3 is a 4 × 4 matrix given by 1 0 0 0 0 −1 0 0 𝐼1,3 := ( ). 0 0 −1 0 0 0 0 −1 (3) Remark 1. 𝑆𝑂(3, 1) is called the special pseudoorthogonal group. 𝑆𝑂(3, 1) is a group of linear operators preserving the quadratic form 𝑥12 − 𝑥22 − 𝑥32 − 𝑥42 , that is, the special classical Lorentz group. Similarly, for the group 𝑆𝑂(2, 1), see [2–4]. For a 𝜎 ∈ C where C is the set of complex numbers, D𝜎 denotes the linear space consisting of infinitely differentiable 𝜎-homogeneous functions on Λ. The representation of 𝑆𝑂(3, 1) in D𝜎 is a homomorphism 𝑔 󳨃→ 𝑇𝜎 (𝑔), where the operator 𝑇𝜎 (𝑔) acts as 𝑓(𝑥) 󳨃→ 𝑓(𝑔−1 𝑥) in the space D𝜎 . Let 𝛾1 be the intersection of Λ and the hyperplane 𝑥1 = 1, 𝛾2 the intersection of Λ and the pair of hyperplanes 𝑥4 = ±1, and 𝛾3 the intersection of Λ and the hyperplane 𝑥1 + 𝑥4 = 0. In other words, 𝛾1 is a sphere with radius √2, 𝛾2 is a twosheet hyperboloid, and 𝛾3 is a paraboloid. It is seen that, for 𝑖 ∈ {1, 2, 3}, 𝛾𝑖 are two-parameter manifolds on Λ explicitly given by 𝛾1 = {(1, sin 𝛼1 sin 𝛽1 , sin 𝛼1 cos 𝛽1 , cos 𝛼1 ) | 𝛼1 ∈ [0, 𝜋] , 𝛽1 ∈ [0, 2𝜋)} , 2 Abstract and Applied Analysis 𝛾2 = {(cosh 𝛼2 , sinh 𝛼2 sin 𝛽2 , sinh 𝛼2 cos 𝛽2 , ±1) | 𝛼2 > 0, 𝛽2 ∈ [0, 2𝜋)} , 𝛾3 = {( 1 + 𝛼32 1 − 𝛼32 , 𝛼3 sin 𝛽3 , 𝛼3 cos 𝛽3 , ) | 𝛼3 > 0, 2 2 𝛽3 ∈ [0, 2𝜋] } . (4) Let 𝐻𝑖 denote subgroups acting transitively on 𝛾𝑖 (𝑖 = 1, 2, 3). It is noted, in particular, that 𝐻1 ≃ 𝑆𝑂(3) and 𝐻2 ≃ 𝑆𝑂(2, 1), where 𝑆𝑂(𝑛) is the group of rotations of 𝑛-dimensional Euclidean space R𝑛 (for more details of this group, see [5, Chapter IX]). In order to describe the group 𝐻3 , in detail, on the linear space of all diagonal matrices 𝑎 := diag(𝑎1 , 𝑎2 ), we introduce a scalar product as 𝑎 ⋅ 𝑏 := tr(𝑎𝑏𝑇 ). Then 𝛾3 is a homogeneous space of the subgroup 𝐻3 consisting of all matrices 1 1 + |𝑎|2 𝑎 2 ( 𝑎𝑇 diag (1, 1) ( 1 − |𝑎|2 2 −𝑎 ), (5) 1 1 − |𝑎|2 2 ) Here id and 𝜕(𝑥2 , 𝑥3 )/𝜕(𝛼𝑗 , 𝛽𝑗 ), respectively, denote the identical permutation and the corresponding Jacobian determinant. Let 𝑢 ∈ D𝜎 and V ∈ D−𝜎−2 . Then F2 (𝑢, V) 2 = ∫ [sech 𝛼2 ] 𝑢 (1, tanh 𝛼2 sin 𝛽2 , tanh 𝛼2 cos 𝛽2 , ± sech 𝛼2 ) 𝛾2 ⋅ V (1, tanh 𝛼2 sin 𝛽2 , tanh 𝛼2 cos 𝛽2 , ± sech 𝛼2 ) d𝑥𝜏(2) d𝑥𝜏(3) 󵄨󵄨 󵄨 , 󵄨󵄨𝑥𝜏(4) 󵄨󵄨󵄨 Setting cos 𝜑 := ± sech 𝛼2 , we obtain d𝛼2 = ± cosh 𝛼2 d𝜑 and sinh 𝛼2 = (1/2) sin 2𝜑. We therefore find that F2 (𝑢, V) (6) where 𝜏 is an arbitrary permutation of the set {2, 3, 4}. The 𝐻2 -invariant measure on 𝛾2 is = ∫ 𝑢 (1, sin 𝜑 sin 𝛽2 , sin 𝜑 cos 𝛽2 , cos 𝜑) 𝛾1 ⋅ V (1, sin 𝜑 sin 𝛽2 , sin 𝜑 cos 𝛽2 , cos 𝜑) sin 𝜑d𝜑d𝛽2 , (12) which implies F2 = F1 . In the same way, F3 (𝑢, V) −2 (d𝑥)𝛾2 = d𝑥𝜏(1) d𝑥𝜏(2) 󵄨󵄨 󵄨󵄨 󵄨󵄨𝑥𝜏(3) 󵄨󵄨 (10) (11) where |𝑎|2 denotes the square of the length of the matrix 𝑎. The 𝐻1 -invariant measure on 𝛾1 can be written as (d𝑥)𝛾1 = 󵄨󵄨 𝜕 (𝑥 , 𝑥 ) 󵄨󵄨 󵄨 2 3 󵄨󵄨 d𝛼1 d𝛽1 󵄨󵄨 󵄨 󵄨 = sin 𝛼1 d𝛼1 d𝛽1 , (d𝑥)𝛾1 = 󵄨󵄨󵄨󵄨 󵄨󵄨 𝜕 (𝛼1 , 𝛽1 ) 󵄨󵄨󵄨 󵄨󵄨󵄨𝑥4 󵄨󵄨󵄨 󵄨󵄨 𝜕 (𝑥 , 𝑥 ) 󵄨󵄨 󵄨 2 3 󵄨󵄨 d𝛼2 d𝛽2 󵄨󵄨 󵄨 󵄨 = sinh 𝛼2 d𝛼2 d𝛽2 . (d𝑥)𝛾2 = 󵄨󵄨󵄨󵄨 󵄨󵄨 𝜕 (𝛼2 , 𝛽2 ) 󵄨󵄨󵄨 󵄨󵄨󵄨𝑥1 󵄨󵄨󵄨 × sinh 𝛼2 d𝛼2 d𝛽2 . 1 2 |𝑎| 2 𝑎𝑇 Proof. Choose 𝜏 = id in (6) and 𝜏 = (1 2 3) in (7) and write the measures (d𝑥)𝛾𝑗 (𝑗 ∈ {1, 2}) in (𝛼𝑗 , 𝛽𝑗 )-coordinate system, respectively: (𝜏 ∈ S3 ) , (7) = 4 ∫ (1 + 𝛼32 ) 𝑢 (1, 𝛾3 ⋅ V (1, where S3 is the symmetric group. The 𝐻3 -invariant measure on 𝛾3 is (d𝑥)𝛾3 = d𝑥2 d𝑥3 . (8) We define the bilinear functionals F𝑗 : D𝜎 × D𝜎̂ → C (𝑗 = 1, 2, 3) given by 2𝛼3 sin 𝛽3 2𝛼3 cos 𝛽3 1 − 𝛼32 , , ) 1 + 𝛼32 1 + 𝛼32 1 + 𝛼32 2𝛼3 sin 𝛽3 2𝛼3 cos 𝛽3 1 − 𝛼32 , , ) 𝛼3 d𝛼3 d𝛽3 . 1 + 𝛼32 1 + 𝛼32 1 + 𝛼32 (13) Setting here cos 𝜑 := (1 − 𝛼32 )/(1 + 𝛼32 ), we obtain sin 𝜑 = 2𝛼32 /(1 + 𝛼32 ) and d𝛼3 = ((1 + 𝛼32 )d𝜑)/2. It means that F3 (𝑢, V) = ∫ 𝑢 (1, sin 𝜑 sin 𝛽3 , sin 𝜑 cos 𝛽3 , cos 𝜑) (𝑢, V) 󳨃󳨀→ ∫ 𝑢 (𝑥) V (𝑥) (d𝑥)𝛾𝑗 . 𝛾𝑗 (9) Then we observe the invariant property for the functionals F𝑗 asserted by the following lemma. Lemma 2. If 𝜎̂ = −𝜎 − 2, then, for any 𝑖, 𝑗 ∈ {1, 2, 3}, F𝑖 = F𝑗 . 𝛾1 ⋅ V (1, sin 𝜑 sin 𝛽3 , sin 𝜑 cos 𝛽3 , cos 𝜑) sin 𝜑d𝜑d𝛽3 . (14) We thus see that F1 = F3 . Since the relation = defined on the set {F1 , F2 , F3 } is transitive, we have F2 = F3 . The proof is complete. Abstract and Applied Analysis 3 In [5, Chapter IX], Vilenkin constructed the canonical basis Ξ𝑙𝐾 on a sphere. Here, continuing this canonical basis from the sphere 𝛾1 to the cone Λ via 𝜎-homogeneity, we obtain the basis consisting of 𝑓𝑝1 ,𝑞1 : 𝑓𝑝1 ,𝑞1 (𝑥) 𝜎−|𝑞1 | = 𝑥1 |𝑞 |+(1/2) 𝐶𝑝 1−|𝑞 | 1 1 𝑥 |𝑞 | ( 4 ) (𝑥3 + i𝑥2 sign 𝑞1 ) 1 𝑥1 (15) 󵄨 󵄨 (i = √−1; 𝑝1 , 𝑞1 ∈ Z; 𝑝1 ≥ 0, 󵄨󵄨󵄨𝑞1 󵄨󵄨󵄨 ≤ 𝑝1 ) , 𝑓𝑝∗2 ,𝑞2 ,± (𝑥) −|𝑞2 |/2 −|𝑞2 | 𝑃−(1/2)+i𝑝 ( 2 𝜎 |𝑞2 | × (𝑥3 + i𝑥2 sign 𝑞2 ) 𝑥1 ) (𝑥4 )± 2. Description of the Connection between the Spherical and Hyperbolic Bases in terms of 4 𝐹3 Functio (...truncated)