Geodesic
Modified $q$-Bessel functions and $q$-Macdonald functions
Sbornik. Mathematics, Tome 187 (1996) no. 10, pp. 1525-1544 Cet article a éte moissonné depuis la source Math-Net.Ru

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The $q$-analogues of modified Bessel functions and Macdonald functions are defined in this paper. As in the case of $q$-Bessel functions introduced by Jackson, there are two kinds of these functions. Like their classical prototypes, they arise in harmonic analysis on quantum symmetric spaces. The definition is based on the power series expansion. Recurrence relations, difference equations, and $q$-Wronskians are obtained, as well as analogues of asymptotic expansions, which are convergent series if $q\ne 1$. Some relations for basic hypergeometric series, which result from this, are given. 
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M. A. Olshanetsky; V.-B. K. Rogov. Modified $q$-Bessel functions and $q$-Macdonald functions. Sbornik. Mathematics, Tome 187 (1996) no. 10, pp. 1525-1544. http://geodesic.mathdoc.fr/item/SM_1996_187_10_a5/

[1] Jackson F. H., “The application of basic numbers to Bessel's and Legendre's functions”, Proc. London Math. Soc. (2), 2 (1905), 192–220 | DOI

[2] Floreanini R., Vinet L., Addition formulas for $q$-Bessel functions, Preprint University of Montreal, UdeM-LPN-TH60, 1991

[3] Vaksman L., Korogodskii L., “Algebra ogranichennykh funktsii na kvantovoi gruppe dvizhenii ploskosti i $q$-analogi funktsii Besselya”, DAN SSSR, 304:5 (1989), 1036–1040 | MR | Zbl

[4] Vilenkin N. Ya., Spetsialnye funktsii i teoriya predstavlenii grupp, Nauka, M., 1990 | MR

[5] Dynkin E. B., “Neotritsatelnye sobstvennye funktsii operatora Laplasa–Beltrami na nekotorykh simmetricheskikh prostranstvakh”, DAN SSSR, 141:2 (1961), 288–291 | MR | Zbl

[6] Olshanetsky M., “Martin boundaries for real semisimple Lie groups”, J. Funct. Anal., 126 (1994), 169–216 | DOI | MR | Zbl

[7] Jacquet H. P. B., “Fonctions de Whittaker associees aux groupes de Chevalley”, Bull. Soc. Math. France, 95 (1967), 243–309 | MR | Zbl

[8] Schiffmann G., “Integrales d'entrelacement et fonctions de Whittaker”, Bull. Soc. Math. France, 99 (1971), 3–72 | MR | Zbl

[9] Semenov-Tyan-Shanskii M., “Kvantovanie otkrytoi tsepochki Tody”, Dinamicheskie sistemy – 7, Itogi nauki i tekhniki. Sovremennye problemy matematiki, 16, VINITI, M., 1987, 194–226

[10] Ginsparg P., Moore G., Lectures on 2d Gravity and 2d Strings, Preprint Yale Univ. TCTP-P23-92

[11] Olshanetsky M., Rogov V., “Liouville quantum mechanics on a lattice from geometry of quantum Lorentz group”, J. Phys. A, 27 (1994), 4669–4683 | DOI | MR | Zbl

[12] Gasper Dzh., Rakhman M., Bazisnye gipergeometricheskie ryady, Mir, M., 1993 | MR | Zbl

[13] Beitmen G., Erdeii A., Vysshie transtsendentnye funktsii, T. 2, Nauka, M., 1966 | MR