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Some Connections between the Spherical and Parabolic Bases on the Cone Expressed in terms of the Macdonald Function
Computing the matrix elements of the linear operator, which transforms the spherical basis of -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 -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 be the cone in the Euclidean space defined by where denotes the set of real numbers. Let be a multiplicative group consisting of all 4 × 4 matrices in which satisfies the following two properties: where denotes (as usual) the transpose of the matrix and is a 4 × 4 matrix given by
Remark 1. is called the special pseudoorthogonal group. is a group of linear operators preserving the quadratic form , that is, the special classical Lorentz group. Similarly, for the group , see [2–4].
For a where is the set of complex numbers, denotes the linear space consisting of infinitely differentiable -homogeneous functions on . The representation of in is a homomorphism , where the operator acts as in the space .
Let be the intersection of and the hyperplane , the intersection of and the pair of hyperplanes , and the intersection of and the hyperplane . In other words, is a sphere with radius , is a two-sheet hyperboloid, and is a paraboloid. It is seen that, for , are two-parameter manifolds on explicitly given by
Let denote subgroups acting transitively on . It is noted, in particular, that and , where is the group of rotations of -dimensional Euclidean space (for more details of this group, see [5, Chapter ]). In order to describe the group , in detail, on the linear space of all diagonal matrices , we introduce a scalar product as . Then is a homogeneous space of the subgroup consisting of all matrices where denotes the square of the length of the matrix .
The -invariant measure on can be written as where is an arbitrary permutation of the set . The -invariant measure on is where is the symmetric group. The -invariant measure on is
We define the bilinear functionals given by
Then we observe the invariant property for the functionals asserted by the following lemma.
Lemma 2. If , then, for any , .
Proof. Choose in (6) and in (7) and write the measures in -coordinate system, respectively:
Here and , respectively, denote the identical permutation and the corresponding Jacobian determinant.
Let and . Then Setting , we obtain and . We therefore find that which implies .
In the same way, Setting here , we obtain and . It means that We thus see that .
Since the relation = defined on the set is transitive, we have . The proof is complete.
In [5, Chapter ], Vilenkin constructed the canonical basis on a sphere. Here, continuing this canonical basis from the sphere to the cone via -homogeneity, we obtain the basis consisting of : where denotes the set of integers and are the Gegenbauer polynomials (see, e.g., [6, Chapter 17]; see also [5, page 458, Equation ]). We will call it the spherical basis. Similarly, the hyperbolic basis consists of functions of the following forms: where are the Legendre functions of the first kind (see, e.g., [7, pages 194–228]; see also [8, 9]) and About the nonhomogeneous hyperbolic part of the hyperbolic basis, see, for instance, [5, Chapter ]. Finally, the parabolic basis consists of the functions where , , and are the Bessel functions. The main idea of this basis is described in [5, Chapters and ]; the multiplier is its -homogeneous part.
Computing the matrix elements of the linear operator, which transforms the spherical basis of -representation space into the hyperbolic basis, very recently, Shilin and Choi  presented an integral formula involving the product of two Legendre functions of the first kind expressed in terms of -hypergeometric function and, using the general Mehler-Fock transform (see, e.g., [10, 11]), another integral formula for the Legendre function of the first kind. In the sequel, using the matrix elements of the linear operators, which transform the spherical basis into the parabolic basis and the hyperbolic basis into the parabolic basis, respectively, here, in this paper, we also give certain connections between the spherical and parabolic bases on the cone and an interesting series involving the Gauss hypergeometric functions, both expressed in terms of the Macdonald function.
2. Description of the Connection between the Spherical and Hyperbolic Bases in terms of Function
Using and the orthogonality relation (see, e.g., [6, page 281, Equations and ]; see also [5, page 462, Equations and ]) where is the Kronecker symbol, we find from the decomposition that where is the familiar Gamma function (see, e.g., [12, Section 1.1]). Shilin and Choi  then observed that the numbers give a relationship between the spherical and hyperbolic bases asserted by the following theorem.
Theorem 3. Each of the following formulas holds true: and, for ,provided that one of the numerator parameters in is zero or a negative integer, and are the generalized hypergeometric functions (see, e.g., [12, Section 1.5]).
Here we provide a more general result than the one in Theorem 3 for all matrix elements instead of . We omit the corresponding proof of Theorem 3 to reformulate Theorem 3 for a general case asserted by Theorem .
Theorem . LetThen each of the following formulas holds true:and, for , provided that one of the numerator parameters in is zero or a negative integer, and are the generalized hypergeometric functions (see, e.g., [12, Section ]).
3. Description of the Connection between the Spherical and Parabolic Bases in terms of the Macdonald Function
Using the same method used in Section 2, from the decomposition we derive
Theorem 4. The following formula holds true: for , where is the Macdonald function (the modified Bessel function of the second kind) (see, e.g., [13, page 675]).
Proof. We have Using the explicit representation (see, e.g., [14, page 175, Equation ]) being the greatest integer less than or equal to , we obtain In case of , we can use the following known formula (see [15, page 179, Entry -28]): which holds for , , and .
Corollary 5. The following formula holds true: for ,
Further we will deal only with the matrix elements in case of , but let us pay attention to general case. Let . As by using the binomial theorem for the factor in (33) and formula (34), apparently, we can represent numbers as a sum whose terms contain a product of the Macdonald function and the generalized hypergeometric function. Without loss of generality, we demonstrate it, for example, under condition . In this case, where As (regarding the last formula see, e.g., [15, Entry -2]), we have and, therefore, It means that
It should be remarked in passing that the same reasoning can be applied to any values of .
4. Composition of Basis Transformations and a Representation of the Macdonald Function
Let be the matrix element of the linear operator acting in and mapping the hyperbolic basis into the parabolic basis; that is, The linear operator can be represented as the composition of the linear operators and ; namely, From this equality, we have, in particular,
Theorem 6. The following formula holds true: for ,
Proof. Note that the restrictions of the functions and to are, respectively, Setting , we can rewrite (45) as Changing the order of integration and using formula (see, e.g., [15, Entry -9]) we find, in the case of , , and , that Using here the decomposition we obtain To evaluate the right-sided integral, considering new variable , we rewrite it as and use a known formula (see, e.g., [16, page 315, Entry 3.197-8]) This completes the proof.
Conflict of Interests
The authors declare that there is no conflict of interests regarding the publication of this paper.
The authors would like to express their deep thanks for the reviewers’ helpful and critical comments to improve this paper. The first-named author is supported by the Ministry of Education and Science of the Russian Federation (1.8551.2013). The second-named author was supported by Dongguk University (Gyeongju) Research Fund.
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Copyright © 2014 I. A. Shilin and Junesang 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.