Journal of Lie Theory

no. 1
Generalized Bessel Function Associated with Dihedral Groups
Journal of Lie Theory, Volume 22 (2012) no. 1, pp. 81-91
Motivated by Dunkl operators theory, we consider a generating series involving a modified Bessel function and a Gegenbauer polynomial, that generalizes a known series already considered by L. Gegenbauer. We actually use inversion formulas for Fourier and Radon transforms to derive a closed formula for this series when the parameter of the Gegenbauer polynomial is a positive integer. As a by-product, we get a relatively simple integral representation for the generalized Bessel function associated with dihedral groups Dn, n ≥ 2 when both multiplicities sum to an integer. In particular, we recover a previous result obtained for D4 and we give a special interest to D6. Finally, we derive similar results for odd dihedral groups.
DOI: 10.5802/jolt.661
Classification: 33C52, 33C45, 42C10, 43A85, 43A90
Keywords: Generalized Bessel function, dihedral groups, Jacobi polynomials, Radon Transform 
@article{JOLT_2012_22_1_a2,
     author = {N. Demni},
     title = {Generalized {Bessel} {Function} {Associated} with {Dihedral} {Groups}},
     journal = {Journal of Lie Theory},
     pages = {81--91},
     year = {2012},
     volume = {22},
     number = {1},
     doi = {10.5802/jolt.661},
     zbl = {1253.33016},
     url = {https://jolt.centre-mersenne.org/articles/10.5802/jolt.661/}
}
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N. Demni. Generalized Bessel Function Associated with Dihedral Groups. Journal of Lie Theory, Volume 22 (2012) no. 1, pp. 81-91. doi: 10.5802/jolt.661

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