On the q-Analogues of Srivastava’s Triple Hypergeometric Functions
Abstract
:1. Introduction
2. Definitions
3. Convergence Regions
4. q-Integral Formulas
5. A q-Analogue of a General Reduction Formula by Karlsson
6. Conclusions
References
- Srivastava, H.M. Hypergeometric functions of three variables. Ganita 1964, 15, 97–108. [Google Scholar]
- Srivastava, H.M. Some integrals representing triple hypergeometric functions. Rend. Circolo Mat. Palermo 1967, 16, 99–115. [Google Scholar] [CrossRef]
- Karlsson, P.W. Regions of convergence for hypergeometric series in three variables. Math. Scand. 1974, 34, 241–248. [Google Scholar]
- Ernst, T. Multiple q-Hypergeometric Transformations Involving q-Integrals. In Proceedings of the 9th Annual Conference of the Society for Special Functions and their Applications (SSFA), Gwalior, India, 21–23 June 2010; Volume 9, pp. 91–99.
- Ernst, T. A Comprehensive Treatment of q-Calculus; Birkhäuser: Basel, Switzerland, 2012. [Google Scholar]
- Ernst, T. A method for q-calculus. J. Nonlinear Math. Phys. 2003, 10, 487–525. [Google Scholar] [CrossRef]
- Ernst, T. The different tongues of q-calculus. Proc. Est. Acad. Sci. 2008, 57, 81–99. [Google Scholar] [CrossRef]
- Ernst, T. q-analogues of general reduction formulas by Buschman and Srivastava and an important q-operator reminding of MacRobert. Demonstr. Math. 2011, 44, 285–296. [Google Scholar]
- Gasper, G.; Rahman, M. Basic Hypergeometric Series; Cambridge: Cambridge, UK, 1990. [Google Scholar]
- Srivastava, H.M.; Karlsson, P.W. Multiple Gaussian Hypergeometric Series; Ellis Horwood: New York, NY, USA, 1985. [Google Scholar]
- Ernst, T. Convergence aspects for q–Appell functions I. J. Indian Math. Soc. to be published.
- Ernst, T. Convergence aspects for q-Lauricella functions 1. Adv. Stud. Contemp. Math. 2012, 22, 35–50. [Google Scholar]
- Jackson, F.H. On basic double hypergeometric functions. Quart. J. Math. 1942, 13, 69–82. [Google Scholar] [CrossRef]
- Agarwal, R.P. Some relations between basic hypergeometric functions of two variables. Rend. Circ. Mat. Palermo 1954, 3, 76–82. [Google Scholar] [CrossRef]
- Picard, E. Sur une extension aux fonctions de deux variable du probl‘eme de Riemann relatif aux fonctions hypergeometriques. C. R. Acad. Sci. 1880, XC, 1267. [Google Scholar]
- Horn, J. Über die Convergenz der hypergeometrischen Reihen zweier und dreier Veranderlichen. Math. Ann. 1889, XXXIV, 577–600. [Google Scholar]
- Exton, H. Multiple Hypergeometric Functions and Applications; Mathematics & its Applications. Ellis Horwood Ltd.: Chichester, West Sussex, UK; Halsted Press [John Wiley & Sons, Inc.]: New York, NY, USA; London, UK; Sydney, Australia, 1976. [Google Scholar]
- Choi, J.; Hasanov, A.; Srivastava, H.M.; Turaev, M. Integral representations for Srivastava’s triple hypergeometric functions. Taiwan. J. Math. 2011, 15, 2751–2762. [Google Scholar]
- Karlsson, P.W. Reduction of certain hypergeometric functions of three variables. Glasnik Mat. 1973, 8, 199–204. [Google Scholar]
- Ernst, T. Convergence aspects for q–Horn functions. In preparation.
- Ernst, T. Triple q–hypergeometric functions. In preparation.
- Ernst, T. On certain generalizations of q-hypergeometric functions of two variables. Int. J. Math. Comput. 2012, 16, 1–27. [Google Scholar]
- Hasanov, A.; Srivastava, H.M.; Turaev, M. Decomposition formulas for some triple hypergeometric functions. J. Math. Anal. Appl. 2006, 324, 955–969. [Google Scholar] [CrossRef]
© 2013 by the author; licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution license (http://creativecommons.org/licenses/by/3.0/).
Share and Cite
Ernst, T. On the q-Analogues of Srivastava’s Triple Hypergeometric Functions. Axioms 2013, 2, 85-99. https://doi.org/10.3390/axioms2020085
Ernst T. On the q-Analogues of Srivastava’s Triple Hypergeometric Functions. Axioms. 2013; 2(2):85-99. https://doi.org/10.3390/axioms2020085
Chicago/Turabian StyleErnst, Thomas. 2013. "On the q-Analogues of Srivastava’s Triple Hypergeometric Functions" Axioms 2, no. 2: 85-99. https://doi.org/10.3390/axioms2020085