# RECURSION FORMULAS FOR q-HYPERGEOMETRIC AND q-APPELL SERIES

• Sahai, Vivek (Department of Mathematics and Astronomy Lucknow University) ;
• Verma, Ashish (Department of Mathematical and Statistical Sciences Shri Ramswaroop Memorial University)
• Accepted : 2017.08.04
• Published : 2018.01.31

#### Abstract

We obtain recursion formulas for q-hypergeometric and q-Appell series. We also find recursion formulas for the general double q-hypergeometric series. It is shown that these recursion relations can be expressed in terms of q-derivatives of the respective q-hypergeometric series.

#### References

1. G. Gasper and M. Rahman, Basic Hypergeometric Series, Second edition, Encyclopedia of Mathematics and its Applications, 96. Cambridge University Press, Cambridge, 2004.
2. F. H. Jackson, On basic double hypergeometric functions, Quart. J. Math. (Oxford) 13 (1942), 69-82.
3. S. B. Opps, N. Saad, and H. M. Srivastava, Recursion formulas for Appell's hyperge-ometric function $F_2$ with some applications to radiation field problems, Appl. Math. Comput. 207 (2009), no. 2, 545-558. https://doi.org/10.1016/j.amc.2008.11.006
4. E. D. Rainville, Special Functions, Chelsea Publishing Company, New York, 1960.
5. V. Sahai and A. Verma, Recursion formulas for multivariable hypergeometric functions, Asian-Eur. J. Math. 8 (2015), no. 4, 1550082, 50 pp.
6. V. Sahai and A. Verma, Recursion formulas for Exton's triple hypergeometric functions, Kyungpook Math. J. 56 (2016), no. 2, 473-506. https://doi.org/10.5666/KMJ.2016.56.2.473
7. V. Sahai and A. Verma, Recursion formulas for Srivastava's general triple hypergeometric function, Asian-Eur. J. Math. 9 (2016), no. 3, 1650063, 17 pp. https://doi.org/10.1142/S1793557116500637
8. V. Sahai and A. Verma, Recursion formulas for the Srivastava-Daoust and related multivariable hyper-geometric functions, Asian-Eur. J. Math. 9 (2016), no. 4, 1650081, 35 pp. https://doi.org/10.1142/S1793557116500819
9. H. M. Srivastava and M. C. Daoust, Certain generalized Neumann expansions associated with the Kampe de Feriet function, Indag. Math. 31 (1969), 449-457.
10. H. M. Srivastava and P. W. Karlsson, Multiple Gaussian Hypergeometric Series, Ellis Horwood Series: Mathematics and its Applications. Ellis Horwood Ltd., Chichester; Halsted Press [John Wiley & Sons, Inc.], New York, 1985.
11. R. F. Swarttouw, The contiguous function relations for basic hypergeometric function, J. Math. Anal. Appl. 149 (1990), no. 1, 151-159. https://doi.org/10.1016/0022-247X(90)90292-N
12. X. Wang, Recursion formulas for Appell functions, Integral Transforms Spec. Funct. 23 (2012), no. 6, 421-433. https://doi.org/10.1080/10652469.2011.596483