Remote Access Proceedings of the American Mathematical Society
Green Open Access

Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)

 
 

 

On models of $U_q(sl(2))$ and $q$-Appell functions
using a $q$-integral transformation


Author: Vivek Sahai
Journal: Proc. Amer. Math. Soc. 127 (1999), 3201-3213
MSC (1991): Primary 33D80, 81R50
DOI: https://doi.org/10.1090/S0002-9939-99-04978-3
Published electronically: April 28, 1999
MathSciNet review: 1616625
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: We discuss few models of the quantum universal enveloping algebra of $sl(2)$ from the special function point of view. Two sets of such models are given, one acting on the space of $_1\phi _0$ functions while the other on the space of $q$-Appell functions. These models are closely related through a $q$-integral transformation. Some interesting identities are obtained.


References [Enhancements On Off] (What's this?)

  • 1. L.C.BIEDENHARN, The quantum group $SU_q(2)$ and a $q$-analogue of the Boson operators, J.Phys.A:Math.Gen. 22(1989), L873-L878. MR 90k:17027
  • 2. V.CHARI and A.PRESSLEY, A guide to quantum groups, Cambridge Univ.Press, 1994. MR 95j:17010
  • 3. V.G.DRINFELD, Quantum Groups, Proc. ICM, Vol.1 (1987), 798-820. MR 89f:17017
  • 4. H.EXTON, $q$-Hypergeometric functions and Applications, Ellis-Horwood, 1983. MR 85g:33001
  • 5. L.D.FADDEEV, Quantum groups, Bol.Soc.Brasil Mat. 20(1989) 47-54. MR 92f:17016
  • 6. R.FLOREANINI and L.VINET, $U_q(sl(2))$ and $q$-Special functions, Contemporary Math. 160(1994),85-100. MR 95b:33054
  • 7. - , $q$-orthogonal polynomials and the orthogonal quantum group, Lett.Math.Phys. 22(1991) 45-54. MR 93g:33015
  • 8. - , The metaplectic representation of $su_q(1,1)$ and the $q$-Gegenbauer polynomials, J.Math.Phys. 33(1992), 1358-1363. MR 93f:33020
  • 9. - , Quantum algebras and $q$-special functions, Ann.Phys. 221(1993) 53-70. MR 94d:33014
  • 10. - , On the quantum group and quantum algebras approach to $q$-special functions, Lett.Math.Phys. 27(1993) 179-190. MR 94e:33040
  • 11. - , Using quantum algebras in $q$-special function theory, Phys.Lett.A 170(1992) 21-28. MR 94f:33032
  • 12. - , A model for the continuous $q$-ultraspherical polynomials, J.Math.Phys. 36(1995), 3800-3813. MR 96d:33012
  • 13. G.GASPER and M.RAHMAN, Basic hypergeometric series, Cambridge Univ.Press, 1990. MR 91d:33034
  • 14. M.JIMBO, A $q$-difference analogue of $U(g)$ and the Yang-Baxter equation, Lett. Math. Phys. 10(1985), 63-69. MR 86k:17008
  • 15. J.C.JANTZEN, Lectures on quantum groups, Graduate Series in Maths 6, AMS, 1996. MR 96m:17029
  • 16. E.G.KALNINS, H.L.MANOCHA and W.MILLER, Harmonic analysis and expansion formulas for the two variable hypergeometric functions, Stud.Appl.Math. 66(1982), 69-89. MR 84a:58091
  • 17. - , Models of $q$-algebra representations: Tensor products of special unitary and oscillator algebras, J.Math.Phys. 33(1992), 2365-2383. MR 93d:17021
  • 18. E.G.KALNINS and W.MILLER, Models of $q$-algebra representations: $q$-integral transforms and addition theorems, J.Math.Phys. 35(1994), 1951-1975. MR 95c:33030
  • 19. E.G.KALNINS, W.MILLER and S.MUKHERJEE, Models of $q$-algebra representations: The group of plane motions, SIAM J.Math.Anal. 25(1994), 513-527. MR 95c:33029
  • 20. - , Models of $q$-algebra representations: Matrix elements of the $q$-oscillator algebra, J.Math.Phys. 34(1993), 5333-5356. MR 94j:17012
  • 21. - , Models of $q$-algebra representations: Matrix elements of $U_q(su_2)$, Contemporary Math., AMS 160(1994), 191-208. MR 95b:33055
  • 22. C.KASSEL, Quantum groups, Springer Verlag, 1995. MR 96e:17041
  • 23. P.P.KULISH and N.YuRESHETIKHIN, Quantum linear problem for the Sine-Gordon equation and higher representations, J.Sov.Math. 23(1983) 2435-2441.
  • 24. S.MAJID, Foundations of quantum group theory, Cambridge Univ.Press, 1995. MR 97g:17016
  • 25. H.L.MANOCHA, On models of irreducible $q$-representations of $sl(2,C)$, Appl. Anal. 37 (1990), 19-47. MR 92h:33042
  • 26. A.J.MACFARLANE, On the $q$-analogues of the quantum harmonic oscillator and the quantum group $SU_q(2)$, J.Phys.A:Math.Gen. 22(1989), 4581-4588. MR 91f:81067
  • 27. W.MILLER, Lie theory and Special functions, Academic Press, 1968. MR 41:8736
  • 28. G.RIDEAU and P.WINTERNITZ, Representation of the quantum algebra $su_q(2)$ on a real two dimensional sphere, J.Math.Phys. 34(1993), 6030-6044. MR 94j:33034
  • 29. V.SAHAI, Lie theory of hypergeometric functions $_3F_2$ by means of Euler transformation, J.Indian Math.Soc. 61(1995), 135-152. CMP 97:15
  • 30. - , $q$-Representations of the Lie algebras ${\mathcal G}(a,b)$, J.Indian Math.Soc. 63(1997), 83-98. CMP 98:12
  • 31. E.K.SKLYANIN, Some algebraic structures connected with the Yang-Baxter equation, Funct. Anal. Appl. 16(1982) 27-34, 96. MR 84c:82004
  • 32. - , Some algebraic structures connected with the Yang-Baxter equation. Representations of quantum algebras, Funct.Anal.Appl. 17(1983) 34-48. MR 85k:82011
  • 33. H.M.SRIVASTAVA and H.L.MANOCHA, A treatise on generating functions, Ellis-Horwood, 1984. MR 85m:33016
  • 34. S.L.WORONOWICZ, Compact matrix pseudogroups, Comm.Math.Phys. 111(1987) 613-665. MR 88m:46079
  • 35. - , Twisted $SU(2)$ group. An example of noncommutative differential calculus, Publ.RIMS Kyoto Univ. 23(1987) 117-181. MR 88h:46130

Similar Articles

Retrieve articles in Proceedings of the American Mathematical Society with MSC (1991): 33D80, 81R50

Retrieve articles in all journals with MSC (1991): 33D80, 81R50


Additional Information

Vivek Sahai
Email: vsahai@hotmail.com

DOI: https://doi.org/10.1090/S0002-9939-99-04978-3
Received by editor(s): January 15, 1998
Published electronically: April 28, 1999
Additional Notes: The author is thankful to University Grants Commission, India, for the award of a Visiting Associateship
Communicated by: Hal L. Smith
Article copyright: © Copyright 1999 American Mathematical Society

American Mathematical Society