Research Article
Certain Properties of -Hypergeometric Functions
Uzoamaka A. Ezeafulukwe
1,2
and Maslina Darus
2
1
Mathematics Department, Faculty of Physical Sciences, University of Nigeria, Nsukka, Nigeria
2
School of Mathematical Sciences, Faculty of Science and Technology, Universiti Kebangsaan Malaysia (UKM), 43600 Bangi,
Selangor Darul Ehsan, Malaysia
Correspondence should be addressed to Maslina Darus; maslina@ukm.edu.my
Received 4 June 2015; Revised 3 August 2015; Accepted 3 August 2015
Academic Editor: Teodor Bulboaca
Copyright © 2015 U. A. Ezeafulukwe and M. Darus. Tis is an open access article distributed under the Creative Commons
Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is
properly cited.
Te quotients of certain -hypergeometric functions are presented as -fractions which converge uniformly in the unit disc. Tese
results lead to the existence of certain -hypergeometric functions in the class of either -convex functions, PC
, or -starlike
functions PS
*
.
1. Introduction and Preliminaries
Let A
be the class of analytic functions in the unit disc
U ={:∈ C,||<1}, normalized by (0)=
(0)−1=0
and of the form
()=+
∞
∑
=2
, (∈ U). (1)
In this paper, we extend some results obtained in the theory
of functions to the -theory and to achieve this we write out
some standard notations and basic defnitions used in this
paper.
Defnition 1. Te -shif factorial, the multiple -shif facto-
rial, and the -binomial coefcients are defned by
(
1
,
2
,...,
;)
=
{
{
{
{
{
{
{
{
{
{
{
{
{
{
{
{
{
1, (=0,=1,
1
=),
−1
∏
=0
(1−
), (=1, ̸ =0,
1
=,∈ N),
∏
=1
(
;)
, (=1,2,...,;∈ N,∈ Z),
[
]
=
{
{
{
{
{
{
{
1, (=0),
(1−
)(1−
−1
)⋅⋅⋅(1−
−+1
)
(;)
, (∈ N),
(2)
where , ∈ C.
In [1, 2] Jackson defned the -derivative operator D
,
as
follows.
Defnition 2. Consider the following:
D
,
()=
()− ()
(1−)
,
(∈ C −{0};0<<1),
D
,
()
=0
=
(0).
(3)
Hindawi Publishing Corporation
International Journal of Mathematics and Mathematical Sciences
Volume 2015, Article ID 489218, 9 pages
http://dx.doi.org/10.1155/2015/489218