On integral representations of q -gamma and q -beta functions

Alberto De Sole; Victor G. Kac

Atti della Accademia Nazionale dei Lincei. Classe di Scienze Fisiche, Matematiche e Naturali. Rendiconti Lincei. Matematica e Applicazioni (2005)

  • Volume: 16, Issue: 1, page 11-29
  • ISSN: 1120-6330

Abstract

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We study q -integral representations of the q -gamma and the q -beta functions. As an application of these integral representations, we obtain a simple conceptual proof of a family of identities for Jacobi triple product, including Jacobi's identity, and of Ramanujan's formula for the bilateral hypergeometric series.

How to cite

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De Sole, Alberto, and Kac, Victor G.. "On integral representations of $q$-gamma and $q$-beta functions." Atti della Accademia Nazionale dei Lincei. Classe di Scienze Fisiche, Matematiche e Naturali. Rendiconti Lincei. Matematica e Applicazioni 16.1 (2005): 11-29. <http://eudml.org/doc/252391>.

@article{DeSole2005,
abstract = {We study $q$-integral representations of the $q$-gamma and the $q$-beta functions. As an application of these integral representations, we obtain a simple conceptual proof of a family of identities for Jacobi triple product, including Jacobi's identity, and of Ramanujan's formula for the bilateral hypergeometric series.},
author = {De Sole, Alberto, Kac, Victor G.},
journal = {Atti della Accademia Nazionale dei Lincei. Classe di Scienze Fisiche, Matematiche e Naturali. Rendiconti Lincei. Matematica e Applicazioni},
keywords = {$q$ q -calculus; $q$ q -gamma function; $q$ q -beta function; -calculus; -gamma function; -beta function},
language = {eng},
month = {3},
number = {1},
pages = {11-29},
publisher = {Accademia Nazionale dei Lincei},
title = {On integral representations of $q$-gamma and $q$-beta functions},
url = {http://eudml.org/doc/252391},
volume = {16},
year = {2005},
}

TY - JOUR
AU - De Sole, Alberto
AU - Kac, Victor G.
TI - On integral representations of $q$-gamma and $q$-beta functions
JO - Atti della Accademia Nazionale dei Lincei. Classe di Scienze Fisiche, Matematiche e Naturali. Rendiconti Lincei. Matematica e Applicazioni
DA - 2005/3//
PB - Accademia Nazionale dei Lincei
VL - 16
IS - 1
SP - 11
EP - 29
AB - We study $q$-integral representations of the $q$-gamma and the $q$-beta functions. As an application of these integral representations, we obtain a simple conceptual proof of a family of identities for Jacobi triple product, including Jacobi's identity, and of Ramanujan's formula for the bilateral hypergeometric series.
LA - eng
KW - $q$ q -calculus; $q$ q -gamma function; $q$ q -beta function; -calculus; -gamma function; -beta function
UR - http://eudml.org/doc/252391
ER -

References

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  2. EXTON, H., q -hypergeometric functions and applications. Ellis Horwood Series in Mathematics and its Applications, Ellis Horwood Ltd., Chichester1983, 347 pp. Zbl0514.33001MR708496
  3. GASPER, G. - RAHMAN, M., Basic hypergeometric series. Encyclopedia of Mathematics and its Applications, vol. 35, second edition, Cambridge University Press, Cambridge1990, 456 pp. Zbl1129.33005MR1052153
  4. JACKSON, F.H., A generalization of the functions Γ ⁢ n and x n . Proc. Roy. Soc. London, 74, 1904, 64-72. JFM35.0460.01
  5. JACKSON, F.H., On q -definite integrals. Quart. J. Pure and Applied Math., 41, 1910, 193-203. JFM41.0317.04
  6. KAC, V.G. - CHEUNG, P., Quantum Calculus. Universitext, Springer-Verlag, New York2002, 112 pp. Zbl0986.05001MR1865777DOI10.1007/978-1-4613-0071-7
  7. KAC, V.G., Infinite dimensional Lie algebras. Third edition, Cambridge University Press, Cambridge1990, 400 pp. Zbl0716.17022MR1104219DOI10.1017/CBO9780511626234
  8. KAC, V.G., Vertex algebras for beginners. University Lecture Series, 10, second edition, American Mathematical Society, Providence, RI, 1998, 201 pp. Zbl0861.17017MR1651389
  9. KAC, V.G. - WAKIMOTO, M., Integrable highest weight modules over affine superalgebras and number theory. In: J.-L. BRYLINSKI et al. (eds.), Lie theory and geometry. Progr. Math., 123, Birkhäuser, Boston1994, 415-456. Zbl0854.17028MR1327543
  10. KEMPF, A. - MAJID, S., Algebraic q -integration and Fourier theory on quantum and braided spaces. Journal of Mathematical Physics, 35, 1994, 6802-6837. Zbl0826.17018MR1303081DOI10.1063/1.530644
  11. KOORNWINDER, T.H., q -special functions, a tutorial. Preprint math. CA/9403216, 1994. Zbl0768.33018MR1490602
  12. KOORNWINDER, T.H., Compact quantum groups and q -special functions. In: V. BALDONI - M.A. PICARDELLO (eds.), Representations of Lie groups and quantum groups. Pitman Research Notes in Mathematics Series, 311, Longman Scientific & Technical, 1994, 46-128. Zbl0821.17015MR1431306
  13. KOORNWINDER, T.H., Special functions and q -commuting variables. In: M.E.H. ISMAIL - D.R. MASSON - M. RAHMAN (eds.), Special functions, q -series and related topics (Toronto, ON, 1995). Fields Inst. Commun., vol. 14, Amer. Math. Soc., Providence1997, 131-166. Zbl0882.33014MR1448685
  14. THOMAE, J., Beitrage zur Theorie der durch die Heinesche Reihe. J. reine angew. Math., 70, 1869, 258-281. JFM02.0122.04

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