A NEW LOOK AT q-HYPERGEOMETRIC FUNCTIONS

Abstract

For complex parameters ai, bj , q i = 1, ..., r, j = 1, ..., s, bj ∈ C\{0, −1, −2, ...}, |q| < 1 , define the q-hypergeometric function rΦs a1, ..., ar; b1, ..., bs; q, z by rΦs ai; bj ; q, z = ∑∞ n=0 a1, q n... ar, q n q, q n b1, q n... bs, q n z n r = s + 1; r, s ∈ N0 = N ∪ {0}; z ∈ U where N denote the set of positive integers and a, q n is the q-shifted factorial defined by a, q n = { 1, n = 0; 1 − a 1 − aq 1 − aq2 ... 1 − aqn−1 , n ∈ N. Recently, the authors [7] defined the linear operator M ai, bj ; q f. Using the operator M ai, bj ; q f z f, Aldweby and Darus [13] gave a new integral operator. In this work we highlight a result related to the new integral operator.

Keywords

• q-hypergeometric functions,Integral operators

References

1. Andrews,G.E., (1974), Applications of basic hypergeometric function, SIAM Review, vol. 16,no. 4, pp. 441-484.
2. Dziok, J., Srivastava,H.M., (1999), Classes of analytic functions associated with the generalized hy- pergeometric function, Applied Mathematics and Computation, vol. 103, pp. 1-13.
3. Exton,H., (1983),q-Hypergeometric functions and applications, Ellis Horwood Limited, Chichester, 347 pages.
4. Ghany, H.A., (2009), q-derivative of basic hypergeometric series with respect to parameters, Interna- tional Journal of Mathematical Analysis, vol. 3, no. 33, pp. 1617- 1632.
5. Frasin, B.A, Darus,M.,O (2001), On certain analytic univalent functions, International Journal of Mathematics and Mathematical Sciences, vol. 25, no. 5, pp. 305-310.
6. Ruscheweyh, S., (1975), New criteria for univalent functions, Proceedings of the American Mathemat- ical Society, vol. 49, pp. 109-115.
7. Mohammed, A., Darus,M., (2013), A generalized operator involving the q-hypergeometric function, Matematicki Vesnik. 65, 4, 454465.
8. Gasper, G., Rahman,M., (1990), Basic Hypergeometric Series , Encyclopedia of Mathematics and Its Application, Vol. 35, Cambridge University Press,Cambridge.

9. Carlson, B.C., Shaffer, D.B., (1984), Starlike and prestarlike hypergeometric functions, SIAM Journal on Mathematical Analysis, vol. 15, no. 4, pp. 737-745.
10. Bernardi, S.D., (1969), Convex and starlike univalent functions, Transactions of the American Math- ematical Society, vol. 135, pp. 429-446.
11. Libera, R.J., (1965), Some classes of regular univalent functions, Proceedings of the American Math- ematical Society, vol. 16, pp. 755-758.
12. Livingston,A.E., (1966), On the radius of univalence of certain analytic functions, Proceedings of the American Mathematical Society, vol. 17, pp. 352-357.
13. Aldweby, H., Darus, M., (2013), Univalence of a New General Integral Operator Associated with the q-Hypergeometric Function, International Journal of Mathematics and Mathematical Sciences, 2013, Article ID 769537, 5 pages.