## On a Neumann-type series for modified Bessel functions of the first kind

HTML articles powered by AMS MathViewer

- by L. Deleaval and N. Demni PDF
- Proc. Amer. Math. Soc.
**146**(2018), 2149-2161 Request permission

## Abstract:

In this paper, we are interested in a Neumann-type series for modified Bessel functions of the first kind which arises in the study of Dunkl operators associated with dihedral groups and as an instance of the Laguerre semigroup constructed by Ben Said-Kobayashi-Orsted. We first revisit the particular case corresponding to the group of square-preserving symmetries for which we give two new and different proofs other than the existing ones. The first proof uses the expansion of powers in a Neumann series of Bessel functions, while the second one is based on a quadratic transformation for the Gauss hypergeometric function and opens the way to derive further expressions when the orders of the underlying dihedral groups are powers of two. More generally, we give another proof of De Bie et al.’s formula expressing this series as a $\Phi _2$-Horn confluent hypergeometric function. In the course of the proof, we shed light on the occurrence of multiple angles in their formula through elementary symmetric functions and get a new representation of Gegenbauer polynomials.## References

- George E. Andrews, Richard Askey, and Ranjan Roy,
*Special functions*, Encyclopedia of Mathematics and its Applications, vol. 71, Cambridge University Press, Cambridge, 1999. MR**1688958**, DOI 10.1017/CBO9781107325937 - Salem Ben Saïd, Toshiyuki Kobayashi, and Bent Ørsted,
*Laguerre semigroup and Dunkl operators*, Compos. Math.**148**(2012), no. 4, 1265–1336. MR**2956043**, DOI 10.1112/S0010437X11007445 - D. Constales, H. De Bie, and P. Lian,
*Explicit formulas for the Dunkl dihedral kernel and the $(\kappa , a)$-generalized Fourier kernel*, https://arxiv.org/abs/1610.00098. - H. De Bie,
*The kernel of the radially deformed Fourier transform*, Integral Transforms Spec. Funct.**24**(2013), no. 12, 1000–1008. MR**3172014**, DOI 10.1080/10652469.2013.799467 - Nizar Demni,
*Generalized Bessel function associated with dihedral groups*, J. Lie Theory**22**(2012), no. 1, 81–91. MR**2859027** - Nizar Demni,
*First hitting time of the boundary of the Weyl chamber by radial Dunkl processes*, SIGMA Symmetry Integrability Geom. Methods Appl.**4**(2008), Paper 074, 14. MR**2470522**, DOI 10.3842/SIGMA.2008.074 - Karl Dilcher,
*A generalization of Fibonacci polynomials and a representation of Gegenbauer polynomials of integer order*, Fibonacci Quart.**25**(1987), no. 4, 300–303. MR**911975** - A. Dijksma and T. H. Koornwinder,
*Spherical harmonics and the product of two Jacobi polynomials*, Nederl. Akad. Wetensch. Proc. Ser. A 74=Indag. Math.**33**(1971), 191–196. MR**0291519**, DOI 10.1016/S1385-7258(71)80026-4 - A. Erdélyi, W. Magnus, F. Oberhettinger, and F. G. Tricomi,
*Tables of integral transforms. Vol. I*, McGraw-Hill Book Co., Inc., New York-Toronto-London, 1954. Based, in part, on notes left by Harry Bateman. MR**0061695** - Arthur Erdélyi, Wilhelm Magnus, Fritz Oberhettinger, and Francesco G. Tricomi,
*Higher transcendental functions. Vols. I, II*, McGraw-Hill Book Co., Inc., New York-Toronto-London, 1953. Based, in part, on notes left by Harry Bateman. MR**0058756** - I. S. Gradshteyn and I. M. Ryzhik,
*Table of integrals, series, and products*, 5th ed., Academic Press, Inc., Boston, MA, 1994. Translation edited and with a preface by Alan Jeffrey. MR**1243179** - Toshiyuki Kobayashi and Gen Mano,
*The inversion formula and holomorphic extension of the minimal representation of the conformal group*, Harmonic analysis, group representations, automorphic forms and invariant theory, Lect. Notes Ser. Inst. Math. Sci. Natl. Univ. Singap., vol. 12, World Sci. Publ., Hackensack, NJ, 2007, pp. 151–208. MR**2401813**, DOI 10.1142/9789812770790_{0}006 - J. Koekoek and R. Koekoek,
*The Jacobi inversion formula*, Complex Variables Theory Appl.**39**(1999), no. 1, 1–18. MR**1697414**, DOI 10.1080/17476939908815177 - I. G. Macdonald,
*Symmetric functions and Hall polynomials*, 2nd ed., Oxford Mathematical Monographs, The Clarendon Press, Oxford University Press, New York, 1995. With contributions by A. Zelevinsky; Oxford Science Publications. MR**1354144** - G. N. Watson,
*A treatise on the theory of Bessel functions*, Cambridge Mathematical Library, Cambridge University Press, Cambridge, 1995. Reprint of the second (1944) edition. MR**1349110** - Yuan Xu,
*A product formula for Jacobi polynomials*, Special functions (Hong Kong, 1999) World Sci. Publ., River Edge, NJ, 2000, pp. 423–430. MR**1805999**

## Additional Information

**L. Deleaval**- Affiliation: Laboratoire d’Analyse et de Mathématiques appliquées , Université Paris-Est Marne-la-Vallée , 77454 Marne-la-Vallé Cedex 2, France
- MR Author ID: 882358
- Email: luc.deleaval@u-pem.fr
**N. Demni**- Affiliation: IRMAR, Université de Rennes 1, Campus de Beaulieu, 35042 Rennes cedex, France
- MR Author ID: 816493
- Email: nizar.demni@univ-rennes1.fr
- Received by editor(s): March 31, 2017
- Received by editor(s) in revised form: July 22, 2017, and July 29, 2017
- Published electronically: December 28, 2017
- Communicated by: Yuan Xu
- © Copyright 2017 American Mathematical Society
- Journal: Proc. Amer. Math. Soc.
**146**(2018), 2149-2161 - MSC (2010): Primary 33C45, 33C52, 33C65
- DOI: https://doi.org/10.1090/proc/13914
- MathSciNet review: 3767365