A NEW LOOK AT q-HYPERGEOMETRIC FUNCTIONS

Year 2014, Volume: 4 Issue: 1, 16 - 19, 01.06.2014

M. Darus

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

• Andrews,G.E., (1974), Applications of basic hypergeometric function, SIAM Review, vol. 16,no. 4, pp. 441-484.
• 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.
• Exton,H., (1983),q-Hypergeometric functions and applications, Ellis Horwood Limited, Chichester, 347 pages.
• 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.
• 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.
• Ruscheweyh, S., (1975), New criteria for univalent functions, Proceedings of the American Mathemat- ical Society, vol. 49, pp. 109-115.
• Mohammed, A., Darus,M., (2013), A generalized operator involving the q-hypergeometric function, Matematicki Vesnik. 65, 4, 454465.
• Gasper, G., Rahman,M., (1990), Basic Hypergeometric Series , Encyclopedia of Mathematics and Its Application, Vol. 35, Cambridge University Press,Cambridge.
• Carlson, B.C., Shaffer, D.B., (1984), Starlike and prestarlike hypergeometric functions, SIAM Journal on Mathematical Analysis, vol. 15, no. 4, pp. 737-745.
• Bernardi, S.D., (1969), Convex and starlike univalent functions, Transactions of the American Math- ematical Society, vol. 135, pp. 429-446.
• Libera, R.J., (1965), Some classes of regular univalent functions, Proceedings of the American Math- ematical Society, vol. 16, pp. 755-758.
• Livingston,A.E., (1966), On the radius of univalence of certain analytic functions, Proceedings of the American Mathematical Society, vol. 17, pp. 352-357.
• 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.