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Abstract and Applied Analysis
Volume 2014 (2014), Article ID 741079, 8 pages
Research Article

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

1Department of Mathematics, Sholokhov Moscow State University for the Humanities, Verkhnyaya Radishevskaya 16-18, Moscow 109240, Russia
2Department of Mathematical Modeling, Moscow Aviation Institute, Volokolamskoe Shosse 4, Moscow 125993, Russia
3Department of Mathematics, Dongguk University, Gyeongju 780-714, Republic of Korea

Received 7 November 2013; Accepted 26 December 2013; Published 11 February 2014

Academic Editor: Kwang Ho Shon

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.


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.