International Journal of Mathematics and Mathematical Sciences
Volume 2015 (2015), Article ID 489218, 9 pages
http://dx.doi.org/10.1155/2015/489218

Research Article

Certain Properties of -Hypergeometric Functions

Uzoamaka A. Ezeafulukwe1,2 and Maslina Darus²

¹Mathematics Department, Faculty of Physical Sciences, University of Nigeria, Nsukka, Nigeria
²School of Mathematical Sciences, Faculty of Science and Technology, Universiti Kebangsaan Malaysia (UKM), 43600 Bangi, Selangor Darul Ehsan, Malaysia

Received 4 June 2015; Revised 3 August 2015; Accepted 3 August 2015

Academic Editor: Teodor Bulboaca

Copyright © 2015 Uzoamaka A. Ezeafulukwe and Maslina Darus. This 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.

Abstract.

The quotients of certain -hypergeometric functions are presented as -fractions which converge uniformly in the unit disc. These results lead to the existence of certain -hypergeometric functions in the class of either -convex functions, , or -starlike functions .

1. Introduction and Preliminaries

Let be the class of analytic functions in the unit disc , normalized by and of the form

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 definitions used in this paper.

Definition 1. The -shift factorial, the multiple -shift factorial, and the -binomial coefficients are defined by

where , .

In [1, 2] Jackson defined the -derivative operator as follows.

Definition 2. Consider the following:

From (3), one has

where and as , .

Definition 2 leads to the th order -derivative derived by Sofonea [3] as follows:

where , denotes the identity operator, and denotes the -binomial coefficients defined in Definition 1.

Definition 3. For complex parameters , , and with , , and . denotes the generalized basic (or -) hypergeometric functions

Let , , , with complex parameters , , and . Then (6) becomes the basic (-) hypergeometric functions written as

where is the -shifted factorial defined in Definition 1.

In [4], Srivastava and Owa summarized some properties of functions that belong to the class of -starlike functions in , introduced and investigated by Ismail et al. in [5]. Srivastava and Owa [4] further proposed the study of properties of functions that belong to the class of -starlike functions of order , , and also of the functions that belong to the class of -convex functions in . The authors [4] also defined the class of functions , on the function , as follows:

while [5] defined the class of functions , on the function , as

Meanwhile, Agrawal and Sahoo in [6] defined and studied some properties of functions that belong to the class and also Sahoo and Sharma in [7] (see also [8]) defined and studied the class of -close-to-convex functions. Kanas and Răducanu in [9] also used the Ruscheweyh -differential operator to introduce and study some properties of uniformly starlike functions of order . Other works related to -hypergeometric can also be traced in [10–13].

In addition, in [14] Baricz and Swaminathan used the Alexander duality between starlike and convex function to define the following.

Definition 4. If is the class of -convex functions, then if and only if . Hence, as , , and reduces to , the class of starlike functions.

They used Definition 4 to establish Theorem 5 as follows.

Theorem 5. If satisfies the following two more conditions

(i),(ii),

then

Apart from Theorem 5, to the best of our knowledge, other properties of functions in the class of functions are yet to be studied. For more results and further studies on -calculus see [15–17].

Motivated by the numerous studies, of the abovementioned authors, we aimed, in this paper, at using the parameters , , as real parameters and placing some constraints on quotients of two or more hypergeometric functions to establish the following:

(⧫i),(⧫ii).

We describe the procedures to achieve (⧫i) as follows:

(i)Calculate necessary and sufficient conditions for a function to be in the class .(ii)Calculate suitable, contiguous relations for hypergeometric functions.(iii)Use the contiguous relation to derive equation with a quotient of one or more hypergeometric functions. Let the absolute value of the derived quotient, on the left-hand side of the equation, constrain the function to be in , while the other side of the equation has continued fraction expansion.(iv)Calculate the continued fraction of the hypergeometric quotient on the right-hand side of the equation and convert the continued fractions to a -fraction which converges uniformly.(v)Hence, the -fractions lead to the geometric properties of Pick functions.(vi)Then substituting last result in the third outline with simple calculation on the outcome gives the required results.

The description (i) to (vi) can also be used to establish (⧫ii).

First we write out the known results needed to establish ours. Ismail and Libis noted in [18] that the hypergeometric function

satisfies the -difference equation

They rewrite (13) in the form

iterated the functional relationship (14), and got

They also noted that if these iterations converge, this will give rise to the continued fraction

where

They also modified a result in [19] to Theorem 6. Later on, we will modify Theorem 6 to suit our results.

Theorem 6 (see [18]). Let with exception of the zeros of ; the continued fraction of a meromorphic function of , which is equal to the function , is represented by

throughout the -plane with

It is equal to this ratio in a neighborhood at origin and furnishes the analytic continuation of it throughout the finite -plane.

We review that the sequence of positive numbers, , , with , is called a Hausdorff moment sequence if there exists a positive (Borel) measure on the close interval such that

Equation (20) can also be represented by

with analytic in the slit complex domain , and also belongs to the set of Pick functions.

Lemma 7 bridges the gaps between total monotone sequence, Hausdorff moment sequence, the set of Pick functions, and -fractions.

Lemma 7 (see [14, 20]). Let , be a real sequence; then the following are equivalent:

(i), is totally monotone sequence; that is, for , ,

(ii), is a Hausdorff moment sequence; that is, there exists a positive (Borel) measure on the interval whose sequence of moment is

(iii)The power series is analytic in , has the analytic continuation in the complex plane slit , and also has a corresponding -fraction

 with , , .

For more studies on Hausdorff moment of sequence and hypergeometric mappings see [20–26] and their references. Theorem 8 displays the constraints on the constant that made the -fraction uniformly convergent.

Theorem 8 (see [20]). Letting be constant satisfying one of the following conditions:

(i), (),(ii), (),(iii), (),

then the continued fraction

converges uniformly for , ().

We write Remark 9, which is some comments made by Wall in [20]. This remark enables us to convert our originally derived continued fraction into a -fraction which converges uniformly.

Remark 9. By means of equivalence transformation, the continued fraction of the form

can be transformed into continued fraction of the form (25), by first reducing the partial denominator to unity; hence we obtain

Also (10) converges uniformly if there exist constants , such that

We state necessary and sufficient condition for a function to be in the class established in [5].

Lemma 10 (see [5]). A function is in the class if and only if

We calculate the necessary and sufficient condition for a function to be in the class .

Lemma 11. A function is in the class if and only if

Proof. The proof of Lemma 11 follows from substituting

in (9).

2. Contiguous Relations and Main Results

In this section we write out some contiguous relations of hypergeometric functions, which can be used to derive a continued fraction expansion, for some quotients of two or more hypergeometric functions. We will also state our results. Suppose

Rearranging (13), we have

Substituting the contiguous relation,

calculated in [27] (see also [18]) in (32) gives

Theorem 12. Let , , , be nonnegative real numbers with and for all . Then hypergeometric quotient with exception of the zeros of has the continued fraction

where

Theorem 13. If the hypothesis of Theorem 12 holds and

(i),(ii)   ,(iii)   ,

then the continued fraction (36) can be represented as

where

where is (37) and is (38). Hence the hypergeometric quotient, , converges uniformly.

Theorem 14. If the hypothesis of Theorem 13 holds and there exists an increasing function, , mapping into itself with , then

for all . Further,

for all .

Theorem 15. If the hypothesis of Theorem 14 and inequality (43) holds,

then .

Theorem 16 (see [18]). Let , , with nonzeros of . Then continued fraction of a meromorphic function of , , is represented by

where

Theorem 17. If the hypothesis of Theorem 16 holds and

(i),(ii),

then

with

where are as in (45). Hence converges uniformly.

Theorem 18. If the hypothesis of Theorem 17 holds and there exists an increasing function, , mapping into itself, with , then

for all . Also,

Theorem 19. If the hypothesis of Theorem 18 holds, then

3. Proof of the Main Results

Proof of Theorem 12. To calculate the continued fraction of , we let

We use the following

stated in [18], to calculate the continued fraction of . Solving (52) gives

where

To calculate , we set

Solving (55), we obtain

where

To calculate , we set

Assume ; we rewrite (58) as

where is (37) and is (38). Substituting (56)–(59) in (53), gives the continued fraction (36).

Proof of Theorem 13. To prove Theorem 13, we first reduce the partial denominator of the continued fraction (36) to unity, by setting

where

with

and also

Moreover, (36) can be rewritten as

Assuming , we transform (64) by replacing with , , , and obtain

Since , () and , (65) can be written as

Hence, by Remark 9, the -fraction of converges uniformly in .

Remark 20. The addition of the leading constant does not affect the convergence of the continued fraction . The continued fraction on the right-hand side of (65) converges to the left-hand side provided belongs to the neighborhood of with and is not a pole of the right-hand side. We also note that the continued fraction (65) is called the infinite -fraction.

Proof of Theorem 14. By Lemma 7(iii), the coefficients in the power series expansion about , of the analytic function, on the left-hand side of the continued fraction (65), are Hausdorff moments of an increasing function, in the open interval with infinitely many points of increase (with total increase less than or equal to 1). Hence, there exists a function mapping to itself, satisfying for and its range contains infinitely many points such that

by analytic continuation, with .
Furthermore, since ,

for all .

Proof of Theorem 15. By (35),

By Theorem 14, (69) can be written as

For to be in , we show that

where .
Applying the triangle inequality to the left-hand side of (71), with , we obtain

where

By hypothesis of Theorem 15 and , we need to show that . Hence, Theorem 15 is established.

Proof of Theorem 16. The calculations in Theorem 16 were established in [18].

Proof of Theorem 17. We then need to reduce the partial denominators of (44) to unity, by setting

where

with

with

By Remark 9, (44) can be written as

Hence (78) can be written as

The proofs of Theorems 18 and 19 are the same procedures as those of Theorems 14 and 15 and hence are omitted in this paper.

Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.

Acknowledgment

The work presented here was fully supported by AP-2013-009.

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